Research Article Strong Convergence Algorithm for...

Research ArticleStrong Convergence Algorithm for Split Equilibrium Problemsand Hierarchical Fixed Point Problems

Abdellah Bnouhachem12

1 School of Management Science and Engineering Nanjing University Nanjing 210093 China2 Ibn Zohr University ENSA BP 1136 Agadir Morocco

Correspondence should be addressed to Abdellah Bnouhachem babedallahyahoocom

Received 24 August 2013 Accepted 23 December 2013 Published 20 February 2014

Academic Editors F-S Hsieh K R Kazmi N Petrot and D Xu

Copyright copy 2014 Abdellah Bnouhachem This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The purpose of this paper is to investigate the problem of finding the approximate element of the common set of solutions of asplit equilibrium problem and a hierarchical fixed point problem in a real Hilbert space We establish the strong convergence ofthe proposed method under some mild conditions Several special cases are also discussed Our main result extends and improvessome well-known results in the literature

1 Introduction

Let 119867 be a real Hilbert space whose inner product and normare denoted by ⟨sdot sdot⟩ and sdot Let 119862 be a nonempty closedconvex subset of 119867 We introduce the following definitionswhich are useful in the following analysis

Definition 1 Themapping 119879 119862 rarr 119867 is said to be(a) monotone if

⟨119879119909 minus 119879119910 119909 minus 119910⟩ ge 0 forall119909 119910 isin 119862 (1)

(b) strongly monotone if there exists 120572 gt 0 such that

⟨119879119909 minus 119879119910 119909 minus 119910⟩ ge 1205721003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

forall119909 119910 isin 119862 (2)

(c) 120572-inverse strongly monotone if there exists 120572 gt 0

such that

⟨119879119909 minus 119879119910 119909 minus 119910⟩ ge 1205721003817100381710038171003817119879119909 minus 119879119910

1003817100381710038171003817

2

forall119909 119910 isin 119862 (3)

(d) nonexpansive if1003817100381710038171003817119879119909 minus 119879119910

1003817100381710038171003817 le1003817100381710038171003817119909 minus 119910

1003817100381710038171003817 forall119909 119910 isin 119862 (4)

(e) 119896-Lipschitz continuous if there exists a constant 119896 gt 0

such that1003817100381710038171003817119879119909 minus 119879119910

1003817100381710038171003817 le 1198961003817100381710038171003817119909 minus 119910

1003817100381710038171003817 forall119909 119910 isin 119862 (5)

(f) contraction on 119862 if there exists a constant 0 le 119896 lt 1

such that1003817100381710038171003817119879119909 minus 119879119910

1003817100381710038171003817 le 1198961003817100381710038171003817119909 minus 119910

1003817100381710038171003817 forall119909 119910 isin 119862 (6)

It is easy to observe that every 120572-inverse strongly monotone119879 is monotone and Lipschitz continuous It is well knownthat every nonexpansive operator 119879 119867 rarr 119867 satisfies for all(119909 119910) isin 119867 times 119867 the inequality

⟨(119909 minus 119879 (119909)) minus (119910 minus 119879 (119910)) 119879 (119910) minus 119879 (119909)⟩

le1

2

1003817100381710038171003817(119879(119909) minus 119909) minus (119879(119910) minus 119910)1003817100381710038171003817

2(7)

and therefore we get for all (119909 119910) isin 119867 times Fix(119879)

⟨119909 minus 119879 (119909) 119910 minus 119879 (119909)⟩ le1

2119879 (119909) minus 119909

2 (8)

See for example [1 Theorem 1] and [2 Theorem 3]

Thefixedpoint problem for themapping119879 is to find119909 isin 119862

such that

119879119909 = 119909 (9)

We denote by 119865(119879) the set of solutions of (9) It is well knownthat 119865(119879) is closed and convex and 119875

119865(119879) is well defined

(see [3])

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 390956 12 pageshttpdxdoiorg1011552014390956

2 The Scientific World Journal

The equilibrium problem denoted by EP is to find 119909 isin 119862

such that

119865 (119909 119910) ge 0 forall119910 isin 119862 (10)

The solution set of (10) is denoted by EP(119865) Numerousproblems in physics optimization and economics reduce tofinding a solution of (10) see [4ndash7] In 1997 Combettes andHirstoaga [8] introduced an iterative scheme of finding thebest approximation to the initial data when EP(119865) is non-empty In 2007 Plubtieng and Punpaeng [6] introduced aniterative method for finding the common element of the set119865(119879) cap EP(119865)

Recently Censor et al [9] introduced a new variationalinequality problemwhichwe call the split variational inequal-ity problem (SVIP) Let119867

1and119867

2be two real Hilbert spaces

Given operators 119891 1198671

rarr 1198671and 119892 119867

2rarr 1198672 a bounded

linear operator 119860 1198671

rarr 1198672 and nonempty closed and

convex subsets 119862 sube 1198671and 119876 sube 119867

2 the SVIP is formulated

as follows find a point 119909lowast

isin 119862 such that

⟨119891 (119909lowast) 119909 minus 119909

lowast⟩ ge 0 forall119909 isin 119862 (11)

and such that

119910lowast

= 119860119909lowast

isin 119876 solves ⟨119892 (119910lowast) 119910 minus 119910

lowast⟩ ge 0

forall119910 isin 119876

(12)

In [10]Moudafi introduced an iterativemethodwhich can beregarded as an extension of the method given by Censor et al[9] for the following split monotone variational inclusions

Find 119909lowast

isin 1198671such that 0 isin 119891 (119909

lowast) + 1198611

(119909lowast) (13)

and such that

119910lowast

= 119860119909lowast

isin 1198672solves 0 isin 119892 (119910

lowast) + 1198612

(119910lowast) (14)

where 119861119894119867119894rarr 2119867119894 is a set-valued mapping for 119894 = 1 2 Later

Byrne et al [11] generalized and extended the work of Censoret al [9] and Moudafi [10]

Very recently Kazmi and Rizvi [12] studied the followingpair of equilibrium problems called split equilibrium prob-lem let 119865

1 119862 times 119862 rarr 119877 and 119865

2119876 times 119876 rarr 119877 be nonlinear

bifunctions and let 1198601198671

rarr 1198672be a bounded linear opera-

tor then the split equilibrium problem (SEP) is to find 119909lowast

isin

119862 such that

1198651

(119909lowast 119909) ge 0 forall119909 isin 119862 (15)

and such that

119910lowast

= 119860119909lowast

isin 119876 solves 1198652


The solution set of SEP (15)-(16) is denoted by Λ = 119901 isin

EP(1198651)119860119901 isin EP(119865

2)

Let 119878119862 rarr 119867 be a nonexpansive mapping The followingproblem is called a hierarchical fixed point problem find 119909 isin

119865(119879) such that

⟨119909 minus 119878119909 119910 minus 119909⟩ ge 0 forall119910 isin 119865 (119879) (17)

It is known that the hierarchical fixed point problem (17) linkswith some monotone variational inequalities and convexprogramming problems see [13 14] Various methods [15ndash20] have been proposed to solve the hierarchical fixed pointproblem In 2010 Yao et al [14] introduced the followingstrong convergence iterative algorithm to solve the problem(17)

119910119899

= 120573119899119878119909119899

+ (1 minus 120573119899) 119909119899

119909119899+1

= 119875119862

[120572119899119891 (119909119899) + (1 minus 120572

119899) 119879119910119899] forall119899 ge 0

(18)

where 119891 119862 rarr 119867 is a contraction mapping and 120572119899 and 120573

119899

are two sequences in (0 1) Under some certain restrictionson parameters Yao et al proved that the sequence 119909

119899

generated by (18) converges strongly to 119911 isin 119865(119879) which isthe unique solution of the following variational inequality

⟨(119868 minus 119891) 119911 119910 minus 119911⟩ ge 0 forall119910 isin 119865 (119879) (19)

In 2011 Ceng et al [21] investigated the following iterativemethod

119909119899+1

= 119875119862

[120572119899120588119880 (119909

119899) + (119868 minus 120572

119899120583119865) (119879 (119910

119899))] forall119899 ge 0

(20)

where119880 is a Lipschitzianmapping and119865 is a Lipschitzian andstrongly monotone mapping They proved that under someapproximate assumptions on the operators and parametersthe sequence 119909

119899 generated by (20) converges strongly to the

unique solution of the variational inequality

⟨120588119880 (119911) minus 120583119865 (119911) 119909 minus 119911⟩ ge 0 forall119909 isin Fix (119879) (21)

In the present paper inspired by the above cited works and bythe recent works going in this direction we give an iterativemethod for finding the approximate element of the commonset of solutions of (15)-(16) and (17) in real Hilbert spaceStrong convergence of the iterative algorithm is obtained inthe framework of Hilbert space We would like to mentionthat our proposed method is quite general and flexible andincludes many known results for solving split equilibriumproblems and hierarchical fixed point problems see forexample [13 14 17ndash19 21ndash23] and relevant references citedtherein

2 Preliminaries

In this section we recall some basic definitions and prop-erties which will be frequently used in our later analysisSome useful results proved already in the literature are alsosummarizedThe first lemma provides some basic propertiesof projection onto 119862

Lemma 2 Let119875119862denote the projection of119867 onto119862Then one

has the following inequalities

⟨119911 minus 119875119862 [119911] 119875

119862 [119911] minus V⟩ ge 0 forall119911 isin 119867 V isin 119862

⟨119906 minus V 119875119862 [119906] minus 119875

119862 [V]⟩ ge1003817100381710038171003817119875119862 [119906] minus 119875

119862 [V]10038171003817100381710038172

forall119906 V isin 119867

1003817100381710038171003817119875119862 [119906] minus 119875

119862 [V]1003817100381710038171003817 le 119906 minus V forall119906 V isin 119867

1003817100381710038171003817119906 minus 119875119862 [119911]

1003817100381710038171003817

2

le 119911 minus 1199062

minus1003817100381710038171003817119911 minus 119875

119862 [119911]1003817100381710038171003817

2

forall119911 isin 119867 119906 isin 119862

(22)

The Scientific World Journal 3

Assumption 3 (see [24]) Let 119865 119862 times 119862 rarr R be a bifunctionsatisfying the following assumptions

(i) 119865(119909 119909) = 0 for all 119909 isin 119862(ii) 119865 is monotone that is 119865(119909 119910) + 119865(119910 119909) le 0 for all

119909 119910 isin 119862(iii) for each119909 119910 119911 isin 119862 lim

119905rarr0119865(119905119911+(1minus119905)119909 119910) le 119865(119909 119910)

(iv) for each119909 isin 119862119910 rarr 119865(119909 119910) is convex and lower semi-continuous

(v) for fixed 119903 gt 0 and 119911 isin 119862 there exists a boundedsubset 119870 of 119867

1and 119909 isin 119862 cap 119870 such that

119865 (119910 119909) +1

119903⟨119910 minus 119909 119909 minus 119911⟩ ge 0 forall119910 isin 119862 119870 (23)

Lemma 4 (see [8]) Assume that 1198651119862 times 119862 rarr R satisfies

Assumption 3 For 119903 gt 0 and for all 119909 isin 1198671 define a mapping

1198791198651

1199031198671

rarr 119862 as follows

1198791198651

119903(119909) = 119911 isin 119862119865

1(119911 119910) +

1

119903⟨119910 minus 119911 119911 minus 119909⟩ ge 0


(24)

Then the following hold

(i) 1198791198651

119903is nonempty and single-valued

(ii) 1198791198651

119903is firmly nonexpansive that is

100381710038171003817100381710038171198791198651

119903(119909) minus 119879

1198651

119903(119910)

10038171003817100381710038171003817

2

le ⟨1198791198651

119903(119909) minus 119879

1198651

119903(119910) 119909 minus 119910⟩

forall119909 119910 isin 1198671

(25)

(iii) 119865(1198791198651

119903) = 119864119875(119865

1)

(iv) 119864119875(1198651) is closed and convex

Assume that 1198652 119876 times 119876 rarr R satisfies Assumption 3 For

119904 gt 0 and for all 119906 isin 1198672 define a mapping 119879

1198652

119904 1198672

rarr 119876 asfollows

1198791198652

119904(119906) = V isin 119876 119865

2(V 119908) +

1

119904⟨119908 minus V V minus 119906⟩

ge 0 forall119908 isin 119876

(26)

Then 1198791198652

119904satisfies conditions (i)ndash(iv) of Lemma 4 Consider

119865(1198791198652

119904) = EP(119865

2 119876) where EP(119865

2 119876) is the solution set of the

following equilibrium problem

find 119910lowast

isin 119876 such that 1198652


Lemma 5 (see [25]) Assume that 1198651 119862 times 119862 rarr R satisfies

Assumption 3 and let 1198791198651

119903be defined as in Lemma 4 Let 119909 119910 isin

1198671and 1199031 1199032

gt 0 Then

100381710038171003817100381710038171198791198651

1199032(119910) minus 119879

1198651

1199031(119909)

10038171003817100381710038171003817le

1003817100381710038171003817119910 minus 1199091003817100381710038171003817 +

10038161003816100381610038161003816100381610038161003816

1199032

minus 1199031

1199032

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198651

1199032(119910) minus 119910

10038171003817100381710038171003817

(28)

Lemma6 (see [26]) Let C be a nonempty closed convex subsetof a real Hilbert space H If 119879 119862 rarr 119862 is a nonexpansive map-ping with Fix(119879) = 0 then the mapping 119868 minus 119879 is demiclosed at0 that is if 119909

119899 is a sequence in 119862 weakly converging to 119909 and

if (119868 minus 119879)119909119899 converges strongly to 0 then (119868 minus 119879)119909 = 0

Lemma 7 (see [21]) Let 119880 119862 rarr 119867 be 120591-Lipschitzian map-ping and let 119865 119862 rarr 119867 be a 119896-Lipschitzian and 120578-stronglymonotone mapping then for 0 le 120588120591 lt 120583120578 120583119865 minus 120588119880 is 120583120578 minus 120588120591-strongly monotone that is

⟨(120583119865 minus 120588119880) 119909 minus (120583119865 minus 120588119880) 119910 119909 minus 119910⟩ ge (120583120578 minus 120588120591)1003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

forall119909 119910 isin 119862

(29)

Lemma 8 (see [27]) Suppose that 120582 isin (0 1) and 120583 gt 0 Let119865 119862 rarr 119867 be an 119896-Lipschitzian and 120578-stronglymonotone oper-ator In association with nonexpansive mapping 119879 119862 rarr 119862define the mapping 119879

120582 119862 rarr 119867 by

119879120582119909 = 119879119909 minus 120582120583119865119879 (119909) forall119909 isin 119862 (30)

Then 119879120582 is a contraction provided that 120583 lt (2120578119896

2) that is

10038171003817100381710038171003817119879120582119909 minus 119879120582119910

10038171003817100381710038171003817le (1 minus 120582]) 1003817100381710038171003817119909 minus 119910

1003817100381710038171003817 forall119909 119910 isin 119862 (31)

where ] = 1 minus radic1 minus 120583(2120578 minus 1205831198712)

Lemma 9 (see [28]) Assume that 119886119899 is a sequence of non-

negative real numbers such that

119886119899+1

le (1 minus 120574119899) 119886119899

+ 120575119899 (32)

where 120574119899 is a sequence in (0 1) and 120575

119899is a sequence such that

(1) suminfin

119899=1120574119899

= infin(2) lim sup

119899rarrinfin120575119899120574119899

le 0 or suminfin

119899=1|120575119899| lt infin

Then lim119899rarrinfin

119886119899

= 0

Lemma 10 (see [29]) Let119862 be a closed convex subset of119867 Let119909119899 be a bounded sequence in 119867 Assume that

(i) the weak 119908-limit set 119908119908

(119909119899) sub 119862 where 119908

119908(119909119899) =

119909 119909119899119894

119909(ii) for each 119911 isin 119862 lim

119899rarrinfin119909119899

minus 119911 exists

Then 119909119899 is weakly convergent to a point in 119862

3 The Proposed Method and Some Properties

In this section we suggest and analyze our method and weprove a strong convergence theorem for finding the commonsolutions of the split equilibrium problem (15)-(16) and thehierarchical fixed point problem (17)

Let 1198671and 119867

2be two real Hilbert spaces and let 119862 sube

1198671and 119876 sube 119867

2be nonempty closed convex subsets of

Hilbert spaces 1198671and 119867

2 respectively Let 119860 119867

1rarr 119867

2

be a bounded linear operator Assume that 1198651119862 times 119862 rarr R


and 1198652119876 times 119876 rarr R are the bifunctions satisfying Assump-

tion 3 and 1198652is upper semicontinuous in first argument Let

119878 119879119862 rarr 119862 be a nonexpansive mapping such that Λ cap

119865(119879) = 0 Let 119865 119862 rarr 119862 be an 119896-Lipschitzian mapping and120578-strongly monotone and let 119880 119862 rarr 119862 be 120591-Lipschitzianmapping Nowwe introduce the proposedmethod as follows

Algorithm 11 For a given 1199090

isin 119862 arbitrarily let the iterativesequences 119906

119899 119909119899 and 119910

119899 be generated by

119906119899

= 1198791198651

119903119899(119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)

119910119899

= 120573119899119878119909119899

+ (1 minus 120573119899) 119906119899

119909119899+1

= 119875119862

[120572119899120588119880 (119909

119899) + (119868 minus 120572

119899120583119865) (119879 (119910

119899))] forall119899 ge 0

(33)

where 119903119899 sub (0 2120589) and 120574 isin (0 1119871) 119871 is the spectral radius

of the operator 119860lowast119860 and 119860

lowast is the adjoint of 119860 Suppose thatthe parameters satisfy 0 lt 120583 lt (2120578119896

2) 0 le 120588120591 lt ] where

] = 1 minus radic1 minus 120583(2120578 minus 1205831198962) And 120572119899 and 120573

119899 are sequences

in (0 1) satisfying the following conditions

(a) lim119899rarrinfin

120572119899

= 0 and suminfin

119899=1120572119899

= infin

(b) lim119899rarrinfin

(120573119899120572119899) = 0

(c) suminfin

119899=1|120572119899minus1

minus 120572119899| lt infin and sum

infin

119899=1|120573119899minus1

minus 120573119899| lt infin

(d) lim inf119899rarrinfin

119903119899

lt lim sup119899rarrinfin

119903119899

lt 2120589 and suminfin

119899=1|119903119899minus1

minus

119903119899| lt infin

Remark 12 Our method can be viewed as extension andimprovement for some well-known results as follows

(i) The proposed method is an extension and improve-ment of the method of Wang and Xu [23] for findingthe approximate element of the common set of solu-tions of a split equilibrium problem and a hierarchicalfixed point problem in a real Hilbert space

(ii) If the Lipschitzian mapping 119880 = 119891 119865 = 119868 120588 = 120583 =

1 we obtain an extension and improvement of themethod of Yao et al [14] for finding the approximateelement of the common set of solutions of a splitequilibrium problem and a hierarchical fixed pointproblem in a real Hilbert space

(iii) The contractive mapping 119891 with a coefficient 120572 isin

[0 1) in other papers (see [14 19 22 27]) is extendedto the cases of the Lipschitzian mapping 119880 with acoefficient constant 120574 isin [0 infin)

This shows that Algorithm 11 is quite general and unifying

Lemma 13 Let 119909lowast

isin Λ cap 119865(119879) Then 119909119899 119906119899 and 119910

119899 are

bounded

Proof Let 119909lowast

isin Λ cap 119865(119879) we have 119909lowast

= 1198791198651

119903119899(119909lowast) and 119860119909

lowast=

1198791198652

119903119899(119860119909lowast) Then

1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

=100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899) minus 119909lowast10038171003817100381710038171003817

2

=100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899) minus 1198791198651

119903119899(119909lowast)10038171003817100381710038171003817

2

le10038171003817100381710038171003817119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899minus 119909lowast10038171003817100381710038171003817

2

=1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120574210038171003817100381710038171003817

119860lowast(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

+ 2120574 ⟨119909119899

minus 119909lowast 119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899⟩

=1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 1205742

⟨(1198791198652

119903119899minus 119868) 119860119909

119899 119860119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899⟩

+ 2120574 ⟨119909119899


(1198791198652

119903119899minus 119868) 119860119909

119899⟩

(34)

From the definition of 119871 it follows that

1205742

⟨(1198791198652

119903119899minus 119868) 119860119909

119899 119860119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899⟩

le 1198711205742

⟨(1198791198652

119903119899minus 119868) 119860119909

119899 (1198791198652

119903119899minus 119868) 119860119909

119899⟩

= 119871120574210038171003817100381710038171003817

(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

(35)

It follows from (8) that

2120574 ⟨119909119899


(1198791198652

119903119899minus 119868) 119860119909

119899⟩

= 2120574 ⟨119860 (119909119899

minus 119909lowast) (1198791198652

119903119899minus 119868) 119860119909

119899⟩

= 2120574 ⟨119860 (119909119899

minus 119909lowast) + (119879

1198652

119903119899minus 119868) 119860119909

119899

minus (1198791198652

119903119899minus 119868) 119860119909

119899 (1198791198652

119903119899minus 119868) 119860119909

119899⟩

= 2120574 (⟨1198791198652

119903119899119860119909119899

minus 119860119909lowast (1198791198652

119903119899minus 119868) 119860119909

119899⟩

minus10038171003817100381710038171003817(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

)

le 2120574 (1

2

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

2

minus10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

2

)

= minus12057410038171003817100381710038171003817(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

(36)

Applying (36) and (35) to (34) and from the definition of 120574we get

1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

le1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120574 (119871120574 minus 1)10038171003817100381710038171003817(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

le1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

(37)


Denote 119881119899

= 120572119899120588119880(119909119899) + (119868 minus 120572

119899120583119865)(119879(119910

119899)) Next we prove

that the sequence 119909119899 is bounded without loss of generality

we can assume that 120573119899

le 120572119899for all 119899 ge 1 From (33) we have

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

=1003817100381710038171003817119875119862

[119881119899] minus 119875119862

[119909lowast]1003817100381710038171003817

le1003817100381710038171003817120572119899120588119880 (119909

119899) + (119868 minus 120572

119899120583119865) (119879 (119910

119899)) minus 119909

lowast1003817100381710038171003817

le 120572119899

1003817100381710038171003817120588119880 (119909119899) minus 120583119865 (119909

lowast)1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120572

119899120583119865) (119879 (119910

119899)) minus (119868 minus 120572

119899120583119865) 119879 (119909

lowast)1003817100381710038171003817

= 120572119899

1003817100381710038171003817120588119880 (119909119899) minus 120588119880 (119909

lowast) + (120588119880 minus 120583119865) (119909

lowast)1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120572

119899120583119865) (119879 (119910

119899)) minus (119868 minus 120572

119899120583119865) 119879 (119909

lowast)1003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 + 120572

119899

1003817100381710038171003817(120588119880 minus 120583119865) (119909lowast)1003817100381710038171003817

+ (1 minus 120572119899]) 1003817100381710038171003817119910119899

minus 119909lowast1003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 + 120572

119899

1003817100381710038171003817(120588119880 minus 120583119865) (119909lowast)1003817100381710038171003817

+ (1 minus 120572119899]) 1003817100381710038171003817120573119899119878119909119899

+ (1 minus 120573119899) 119906119899

minus 119909lowast1003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 + 120572

119899

1003817100381710038171003817(120588119880 minus 120583119865) (119909lowast)1003817100381710038171003817

+ (1 minus 120572119899]) (120573119899

1003817100381710038171003817119878119909119899

minus 119878119909lowast1003817100381710038171003817 + 120573

119899

1003817100381710038171003817119878119909lowast

minus 119909lowast1003817100381710038171003817

+ (1 minus 120573119899)

1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817)

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 + 120572

119899

1003817100381710038171003817(120588119880 minus 120583119865) (119909lowast)1003817100381710038171003817

+ (1 minus 120572119899]) (120573119899

1003817100381710038171003817119878119909119899

minus 119878119909lowast1003817100381710038171003817 + 120573

119899

1003817100381710038171003817119878119909lowast

minus 119909lowast1003817100381710038171003817

+ (1 minus 120573119899)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817)

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 + 120572

119899

1003817100381710038171003817(120588119880 minus 120583119865) (119909lowast)1003817100381710038171003817

+ (1 minus 120572119899]) (120573119899

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 + 120573

119899

1003817100381710038171003817119878119909lowast

minus 119909lowast1003817100381710038171003817

+ (1 minus 120573119899)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817)

= (1 minus 120572119899

(] minus 120588120591))1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

+ 120572119899

1003817100381710038171003817(120588119880 minus 120583119865) (119909lowast)1003817100381710038171003817

+ (1 minus 120572119899]) 120573119899

1003817100381710038171003817119878119909lowast

minus 119909lowast1003817100381710038171003817

le (1 minus 120572119899

(] minus 120588120591))1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

+ 120572119899

1003817100381710038171003817(120588119880 minus 120583119865) (119909lowast)1003817100381710038171003817 + 120573119899

1003817100381710038171003817119878119909lowast

minus 119909lowast1003817100381710038171003817

le (1 minus 120572119899

(] minus 120588120591))1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

+ 120572119899

(1003817100381710038171003817(120588119880 minus 120583119865) (119909

lowast)1003817100381710038171003817 +

1003817100381710038171003817119878119909lowast

minus 119909lowast1003817100381710038171003817)

= (1 minus 120572119899

(] minus 120588120591))1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

+120572119899

(] minus 120588120591)

] minus 120588120591(1003817100381710038171003817(120588119880 minus 120583119865) 119909

lowast1003817100381710038171003817 +1003817100381710038171003817119878119909lowast

minus 119909lowast1003817100381710038171003817)

le max1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

1

] minus 120588120591

times (1003817100381710038171003817(120588119880 minus 120583119865) (119909

lowast)1003817100381710038171003817 +

1003817100381710038171003817119878119909lowast

minus 119909lowast1003817100381710038171003817)

(38)

where the third inequality follows from Lemma 8

By induction on 119899 we obtain 119909119899

minus 119909lowast le max119909

0minus

119909lowast (1(1 minus 120588))((120588119880 minus 120583119865)119909

lowast + 119878119909

lowastminus 119909lowast) for 119899 ge 0 and

1199090

isin 119862 Hence 119909119899 is bounded and consequently we deduce

that 119906119899 119910119899 119878(119909

119899) 119879(119909

119899) 119865(119879(119910

119899)) and 119880(119909

119899) are

bounded

Lemma 14 Let119909lowast

isin Λcap119865(119879) and 119909119899 the sequence generated

by the Algorithm 11 Then one has


119909119899+1

minus 119909119899 = 0

(b) the weak 119908-limit set 119908119908

(119909119899) sub 119865(119879) (119908

119908(119909119899) = 119909

119909119899119894

119909)

Proof Since 119906119899

= 1198791198651

119903119899(119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899) and 119906

119899minus1=

1198791198651

119903119899minus1(119909119899minus1

+120574119860lowast(1198791198652

119903119899minus1minus119868)119860119909

119899minus1) it follows fromLemma 5 that

1003817100381710038171003817119906119899

minus 119906119899minus1

1003817100381710038171003817

le10038171003817100381710038171003817119909119899


+120574 (119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899minus 119860lowast

(1198791198652

119903119899minus1minus 119868) 119860119909

119899minus1)10038171003817100381710038171003817

+

10038161003816100381610038161003816100381610038161003816

1 minus119903119899minus1

119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)

minus (119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)10038171003817100381710038171003817

le1003817100381710038171003817119909119899


minus 120574119860lowast119860 (119909119899


)1003817100381710038171003817

+ 120574 119860100381710038171003817100381710038171198791198652

119903119899119860119909119899

minus 1198791198652

119903119899minus1119860119909119899minus1

10038171003817100381710038171003817

+

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)

minus (119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)10038171003817100381710038171003817

le (1003817100381710038171003817119909119899


1003817100381710038171003817

2

minus 21205741003817100381710038171003817119860 (119909119899


)1003817100381710038171003817

2

+120574211986041003817100381710038171003817119909119899


1003817100381710038171003817

2

)12

+ 120574 119860 (1003817100381710038171003817119860 (119909119899


)1003817100381710038171003817

+

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198652

119903119899119860119909119899

minus 119860119909119899

10038171003817100381710038171003817)

+

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)

minus (119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)10038171003817100381710038171003817

le (1 minus 21205741198602

+ 12057421198604)12 1003817100381710038171003817119909119899


1003817100381710038171003817

+ 1205741198602 1003817100381710038171003817119909119899


1003817100381710038171003817

+ 120574 119860

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198652

119903119899119860119909119899

minus 119860119909119899

10038171003817100381710038171003817


+

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)

minus (119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)10038171003817100381710038171003817

= (1 minus 1205741198602)

1003817100381710038171003817119909119899


1003817100381710038171003817

+ 1205741198602 1003817100381710038171003817119909119899


1003817100381710038171003817

+ 120574 119860

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198652

119903119899119860119909119899

minus 119860119909119899

10038171003817100381710038171003817

+

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)

minus (119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)10038171003817100381710038171003817

=1003817100381710038171003817119909119899


1003817100381710038171003817 + 120574 119860

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198652

119903119899119860119909119899

minus 119860119909119899

10038171003817100381710038171003817

+

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)

minus (119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)10038171003817100381710038171003817

=1003817100381710038171003817119909119899


1003817100381710038171003817 +

10038161003816100381610038161003816100381610038161003816

119903119899


119903119899

10038161003816100381610038161003816100381610038161003816

(120574 119860 120590119899

+ 120594119899)

(39)

where 120590119899

= 1198791198652

119903119899119860119909119899

minus 119860119909119899 and 120594

119899= 1198791198651

119903119899(119909119899

+ 120574119860lowast(1198791198652

119903119899minus

119868)119860119909119899) minus (119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899) Without loss of generality

let us assume that there exists a real number 120583 such that 119903119899

gt

120583 gt 0 for all positive integers 119899 Then we get

1003817100381710038171003817119906119899minus1

minus 119906119899

1003817100381710038171003817 le1003817100381710038171003817119909119899minus1

minus 119909119899

1003817100381710038171003817 +1

120583

1003816100381610038161003816119903119899minus1 minus 119903119899

1003816100381610038161003816 (120574 119860 120590119899

+ 120594119899)

(40)

From (33) and the above inequality we get1003817100381710038171003817119910119899


1003817100381710038171003817

=1003817100381710038171003817120573119899119878119909119899

+ (1 minus 120573119899) 119906119899

minus (120573119899minus1

119878119909119899minus1

+ (1 minus 120573119899minus1

) 119906119899minus1

)1003817100381710038171003817

=1003817100381710038171003817120573119899

(119878119909119899

minus 119878119909119899minus1

) + (120573119899


) 119878119909119899minus1

+ (1 minus 120573119899) (119906119899


) + (120573119899minus1

minus 120573119899) 119906119899minus1

1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899


1003817100381710038171003817 + (1 minus 120573119899)

1003817100381710038171003817119906119899


1003817100381710038171003817

+1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 (1003817100381710038171003817119878119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119906119899minus1

1003817100381710038171003817)

le 120573119899

1003817100381710038171003817119909119899


1003817100381710038171003817 + (1 minus 120573119899)

times 1003817100381710038171003817119909119899minus1

minus 119909119899

1003817100381710038171003817 +1

120583

1003816100381610038161003816119903119899minus1 minus 119903119899

1003816100381610038161003816 (120574 119860 120590119899

+ 120594119899)

+1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 (1003817100381710038171003817119878119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119906119899minus1

1003817100381710038171003817)

le1003817100381710038171003817119909119899


1003817100381710038171003817 +1

120583

1003816100381610038161003816119903119899minus1 minus 119903119899

1003816100381610038161003816 (120574 119860 120590119899

+ 120594119899)

+1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 (1003817100381710038171003817119878119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119906119899minus1

1003817100381710038171003817)

(41)

Next we estimate

1003817100381710038171003817119909119899+1

minus 119909119899

1003817100381710038171003817

=1003817100381710038171003817119875119862

[119881119899] minus 119875119862

[119881119899minus1

]1003817100381710038171003817

le1003817100381710038171003817120572119899120588 (119880 (119909

119899) minus 119880 (119909

119899minus1)) + (120572

119899minus 120572119899minus1

) 120588119880 (119909119899minus1

)

+ (119868 minus 120572119899120583119865) (119879 (119910

119899)) minus (119868 minus 120572

119899120583119865) 119879 (119910

119899minus1)

+ (119868 minus 120572119899120583119865) (119879 (119910

119899minus1)) minus (119868 minus 120572

119899minus1120583119865) (119879 (119910

119899minus1))

1003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899


1003817100381710038171003817 + (1 minus 120572119899]) 1003817100381710038171003817119910119899


1003817100381710038171003817

+1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 (1003817100381710038171003817120588119880 (119909

119899minus1)1003817100381710038171003817 +

1003817100381710038171003817120583119865 (119879 (119910119899minus1

))1003817100381710038171003817)

(42)

where the second inequality follows from Lemma 8 From(41) and (42) we have

1003817100381710038171003817119909119899+1

minus 119909119899

1003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899


1003817100381710038171003817 + (1 minus 120572119899])

times 1003817100381710038171003817119909119899


1003817100381710038171003817 +1

120583

1003816100381610038161003816119903119899minus1 minus 119903119899

1003816100381610038161003816 (120574 119860 120590119899

+ 120594119899)

+1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 (1003817100381710038171003817119878119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119906119899minus1

1003817100381710038171003817)

+1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 (1003817100381710038171003817120588119880 (119909

119899minus1)1003817100381710038171003817 +

1003817100381710038171003817120583119865 (119879 (119910119899minus1

))1003817100381710038171003817)

le (1 minus (] minus 120588120591) 120572119899)

1003817100381710038171003817119909119899


1003817100381710038171003817

+1

120583

1003816100381610038161003816119903119899minus1 minus 119903119899

1003816100381610038161003816 (120574 119860 120590119899

+ 120594119899)

+1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 (1003817100381710038171003817119878119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119906119899minus1

1003817100381710038171003817)

+1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 (1003817100381710038171003817120588119880 (119909

119899minus1)1003817100381710038171003817 +

1003817100381710038171003817120583119865 (119879 (119910119899minus1

))1003817100381710038171003817)

le (1 minus (] minus 120588120591) 120572119899)

1003817100381710038171003817119909119899


1003817100381710038171003817

+ 119872 (1

120583

1003816100381610038161003816119903119899minus1 minus 119903119899

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816)

(43)

where

119872 = maxsup119899ge1

(120574 119860 120590119899

+ 120594119899)

sup119899ge1

(1003817100381710038171003817119878119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119906119899minus1

1003817100381710038171003817)

sup119899ge1

(1003817100381710038171003817120588119880 (119909

119899minus1)1003817100381710038171003817 +

1003817100381710038171003817120583119865 (119879 (119910119899minus1

))1003817100381710038171003817)

(44)

It follows from conditions (a)ndash(d) of Algorithm 11 andLemma 9 that

lim119899rarrinfin

1003817100381710038171003817119909119899+1

minus 119909119899

1003817100381710038171003817 = 0 (45)


Next we show that lim119899rarrinfin

119906119899minus119909119899 = 0 Since 119909

lowastisin Λcap119865(119879)

by using (34) and (37) we obtain

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

= ⟨119875119862

[119881119899] minus 119909lowast 119909119899+1

minus 119909lowast⟩

= ⟨119875119862

[119881119899] minus 119881119899 119875119862

[119881119899] minus 119909lowast⟩ + ⟨119881

119899minus 119909lowast 119909119899+1


le ⟨120572119899

(120588119880 (119909119899) minus 120583119865 (119909

lowast) + (119868 minus 120572

119899120583119865) (119879 (119910

119899)))

minus (119868 minus 120572119899120583119865) (119879 (119909

lowast)) 119909119899+1


= ⟨120572119899120588 (119880 (119909

119899) minus 119880 (119909

lowast)) 119909119899+1


+ 120572119899

⟨120588119880 (119909lowast) minus 120583119865 (119909

lowast) 119909119899+1


+ ⟨(119868 minus 120572119899120583119865) (119879 (119910

119899))

minus (119868 minus 120572119899120583119865) (119879 (119909

lowast)) 119909119899+1


le 120572119899120588120591

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

+ 120572119899

⟨120588119880 (119909lowast) minus 120583119865 (119909

lowast) 119909119899+1


+ (1 minus 120572119899]) 1003817100381710038171003817119910119899

minus 119909lowast1003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

le120572119899120588120591

2(1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

)

+ 120572119899

⟨120588119880 (119909lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899])

2(1003817100381710038171003817119910119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

)

le(1 minus 120572

119899(] minus 120588120591))

2

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

+120572119899120588120591

2

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120572119899

⟨120588119880 (119909lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899])

2(120573119899

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+ (1 minus 120573119899)

1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

)

le(1 minus 120572

119899(] minus 120588120591))

2

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

+120572119899120588120591

2

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120572119899

⟨120588119880 (119909lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

2

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+(1 minus 120572

119899]) (1 minus 120573

119899)

2

times 1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120574 (119871120574 minus 1)10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

2

(46)

where the last inequality follows from (37) which implies that

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+(1 minus 120572

119899]) (1 minus 120573

119899)

1 + 120572119899

(] minus 120588120591)

times 1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120574 (119871120574 minus 1)10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

minus(1 minus 120572

119899]) (1 minus 120573

119899) 120574 (1 minus 119871120574)

1 + 120572119899

(] minus 120588120591)

10038171003817100381710038171003817(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

(47)

Then from the above inequality we get

(1 minus 120572119899]) (1 minus 120573

119899) 120574 (1 minus 119871120574)

1 + 120572119899

(] minus 120588120591)

10038171003817100381710038171003817(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)

times ⟨120588119880 (119909lowast) minus 120583119865 (119909

lowast) 119909119899+1


+ 120573119899

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)


lowast) 119909119899+1


+ 120573119899

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+ (1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 +

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817)

times1003817100381710038171003817119909119899+1

minus 119909119899

1003817100381710038171003817

(48)

Since 120574(1 minus 119871120574) gt 0 lim119899rarrinfin

119909119899+1

minus 119909119899 = 0 120572

119899rarr 0 and

120573119899

rarr 0 we obtain

lim119899rarrinfin

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817= 0 (49)


Since 1198791198651

119903119899is firmly nonexpansive we have

1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

=100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899) minus 1198791198651

119903119899(119909lowast)10038171003817100381710038171003817

2

le ⟨119906119899

minus 119909lowast 119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899minus 119909lowast⟩

=1

21003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

+10038171003817100381710038171003817119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899minus 119909lowast10038171003817100381710038171003817

2

minus10038171003817100381710038171003817119906119899

minus 119909lowast

minus [119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899minus 119909lowast]10038171003817100381710038171003817

2

=1

21003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

+10038171003817100381710038171003817119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899minus 119909lowast10038171003817100381710038171003817

2

minus10038171003817100381710038171003817119906119899

minus 119909119899

minus 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

le1

21003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus10038171003817100381710038171003817119906119899

minus 119909119899

minus 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

2

=1

21003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus [1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

+ 120574210038171003817100381710038171003817

119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

2

minus2120574⟨119906119899

minus 119909119899 119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899⟩]

(50)

where the last inequality follows from (34) and (37) Hencewe get

1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

le1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

+ 21205741003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

(51)

From (46) and the above inequality we have1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

le(1 minus 120572

119899(] minus 120588120591))

2

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

+120572119899120588120591

2

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120572119899⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899])

2(120573119899

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+ (1 minus 120573119899)

1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

)

le(1 minus 120572

119899(] minus 120588120591))

2

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

+120572119899120588120591

2

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120572119899⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1

minus 119909lowast⟩ +

(1 minus 120572119899])

2

times 120573119899

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+ (1 minus 120573119899)

times (1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

+21205741003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817)

(52)

which implies that

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)


lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+(1 minus 120572

119899]) (1 minus 120573

119899)

1 + 120572119899

(] minus 120588120591)

times 1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

+21205741003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)


lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+(1 minus 120572

119899]) (1 minus 120573

119899)

1 + 120572119899

(] minus 120588120591)

times minus1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

+ 21205741003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

(53)

Hence

(1 minus 120572119899]) (1 minus 120573

119899)

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+2 (1 minus 120572

119899]) (1 minus 120573

119899) 120574

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+2 (1 minus 120572

119899]) (1 minus 120573

119899) 120574

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

+ (1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 +

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817)

1003817100381710038171003817119909119899+1

minus 119909119899

1003817100381710038171003817

(54)


Since lim119899rarrinfin

119909119899+1

minus 119909119899 = 0 120572

119899rarr 0 120573

119899rarr 0 and

lim119899rarrinfin

(1198791198652

119903119899minus 119868)119860119909

119899 = 0 we obtain

lim119899rarrinfin

1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817 = 0 (55)

Now let 119911 isin Λ cap 119865(119879) since 119879(119909119899) isin 119862 we have

1003817100381710038171003817119909119899

minus 119879 (119909119899)1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119909119899+1

minus 119879 (119909119899)1003817100381710038171003817

=1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119875119862

[119881119899] minus 119875119862

[119879 (119909119899)]

1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817

+1003817100381710038171003817120572119899

(120588119880 (119909119899) minus 120583119865 (119879 (119910

119899)) + 119879 (119910

119899) minus 119879 (119909

119899))

1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817

+ 120572119899

1003817100381710038171003817120588119880 (119909119899) minus 120583119865 (119879 (119910

119899))

1003817100381710038171003817 +1003817100381710038171003817119910119899

minus 119909119899

1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817 + 120572119899

1003817100381710038171003817120588119880 (119909119899) minus 120583119865 (119879 (119910

119899))

1003817100381710038171003817

+1003817100381710038171003817120573119899119878119909119899

+ (1 minus 120573119899) 119906119899

minus 119909119899

1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817 + 120572119899

1003817100381710038171003817120588119880 (119909119899) minus 120583119865 (119879 (119910

119899))

1003817100381710038171003817

+ 120573119899

1003817100381710038171003817119878119909119899

minus 119909119899

1003817100381710038171003817 + (1 minus 120573119899)

1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

(56)


119909119899+1

minus 119909119899 = 0 120572

119899rarr 0 120573

119899rarr 0 120588119880(119909

119899) minus

120583119865(119879(119910119899)) and 119878119909

119899minus 119909119899 are bounded and lim

119899rarrinfin119909119899

minus

119906119899 = 0 we obtain

lim119899rarrinfin

1003817100381710038171003817119909119899

minus 119879 (119909119899)1003817100381710038171003817 = 0 (57)

Since 119909119899 is bounded without loss of generality we can

assume that 119909119899

119909lowast

isin 119862 It follows from Lemma 6 that119909lowast

isin 119865(119879) Therefore 119908119908

(119909119899) sub 119865(119879)

Theorem15 The sequence 119909119899 generated byAlgorithm 11 con-

verges strongly to 119911 which is the unique solution of the varia-tional inequality

⟨120588119880 (119911) minus 120583119865 (119911) 119909 minus 119911⟩ le 0 forall119909 isin Λ cap 119865 (119879) (58)

Proof Since 119909119899 is bounded119909

119899 119908 and fromLemma 14 we

have 119908 isin 119865(119879) Next we show that 119908 isin EP(1198651) Since 119906

119899=

1198791198651

119903119899(119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899) we have

1198651

(119906119899 119910) +

1

119903119899

⟨119910 minus 119906119899 119906119899

minus 119909119899⟩

minus1

119903119899

⟨119910 minus 119906119899 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899⟩ ge 0 forall119910 isin 119862

(59)

It follows from monotonicity of 1198651that

minus1

119903119899

⟨119910 minus 119906119899 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899⟩

+1

119903119899

⟨119910 minus 119906119899 119906119899

minus 119909119899⟩ ge 1198651

(119910 119906119899) forall119910 isin 119862

(60)

minus1

119903119899119896

⟨119910 minus 119906119899119896

120574119860lowast

(1198791198652

119903119899119896

minus 119868) 119860119909119899119896

⟩

+ ⟨119910 minus 119906119899119896

119906119899119896

minus 119909119899119896

119903119899119896

⟩ ge 1198651

(119910 119906119899119896

) forall119910 isin 119862

(61)


119906119899

minus 119909119899 = 0 lim

119899rarrinfin(1198791198652

119903119899minus 119868)119860119909

119899 = 0

and 119909119899

119908 it easy to observe that 119906119899119896

rarr 119908 It follows byAssumption 3(iv) that 119865

1(119910 119908) le 0 for all 119910 isin 119862

For any 0 lt 119905 le 1 and 119910 isin 119862 let 119910119905

= 119905119910 + (1 minus 119905)119908 wehave 119910

119905isin 119862 Then from Assumptions 3((i) and (iv)) we have

0 = 1198651

(119910119905 119910119905)

le 1199051198651

(119910119905 119910) + (1 minus 119905) 119865

1(119910119905 119908)

le 1199051198651

(119910119905 119910)

(62)

Therefore 1198651(119910119905 119910) ge 0 From Assumption 3(iii) we have

1198651(119908 119910) ge 0 which implies that 119908 isin EP(119865

1)

Next we show that 119860119908 isin EP(1198652) Since 119909

119899 is bounded

and 119909119899

119908 there exists a subsequence 119909119899119896

of 119909119899 such

that 119909119899119896

rarr 119908 and since 119860 is a bounded linear operator119860119909119899119896

rarr 119860119908 Now set V119899119896

= 119860119909119899119896

minus 1198791198652

119903119899119896

119860119909119899119896 It follows from

(49) that lim119896rarrinfin

V119899119896

= 0 and119860119909119899119896

minusV119899119896

= 1198791198652

119903119899119896

119860119909119899119896Therefore

from the definition of 1198791198652

119903119899119896

we have

1198652

(119860119909119899119896

minus V119899119896

119910)

+1

119903119899119896

⟨119910 minus (119860119909119899119896

minus V119899119896

)

(119860119909119899119896

minus V119899119896

) minus 119860119909119899119896

⟩ ge 0 forall119910 isin 119862

(63)

Since 1198652is upper semicontinuous in first argument taking

lim sup to above inequality as 119896 rarr infin and using Assump-tion 3(iv) we obtain

1198652

(119860119908 119910) ge 0 forall119910 isin 119862 (64)

which implies that 119860119908 isin EP(1198652) and hence 119908 isin Λ

Thus we have

119908 isin Λ cap 119865 (119879) (65)


Observe that the constants satisfy 0 le 120588120591 lt ] and

119896 ge 120578

lArrrArr 1198962

ge 1205782

lArrrArr 1 minus 2120583120578 + 12058321198962

ge 1 minus 2120583120578 + 12058321205782

lArrrArr radic1 minus 120583 (2120578 minus 1205831198962) ge 1 minus 120583120578

lArrrArr 120583120578 ge 1 minus radic1 minus 120583 (2120578 minus 1205831198962)

lArrrArr 120583120578 ge ]

(66)

Therefore from Lemma 7 the operator 120583119865 minus 120588119880 is 120583120578 minus 120588120591

stronglymonotone and we get the uniqueness of the solutionof the variational inequality (58) and denote it by 119911 isin Λ cap

119865(119879)Next we claim that lim sup

119899rarrinfin⟨120588119880(119911)minus120583119865(119911) 119909

119899minus119911⟩ le

0 Since 119909119899 is bounded there exists a subsequence 119909

119899119896 of

119909119899 such that

lim sup119899rarrinfin

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899

minus 119911⟩

= lim sup119896rarrinfin

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899119896

minus 119911⟩

= ⟨120588119880 (119911) minus 120583119865 (119911) 119908 minus 119911⟩ le 0

(67)

Next we show that 119909119899

rarr 119911 Consider

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

2

= ⟨119875119862

[119881119899] minus 119911 119909

119899+1minus 119911⟩

= ⟨119875119862

[119881119899] minus 119881119899 119875119862

[119881119899] minus 119911⟩ + ⟨119881

119899minus 119911 119909

119899+1minus 119911⟩

le ⟨120572119899

(120588119880 (119909119899) minus 120583119865 (119911)) + (119868 minus 120572

119899120583119865) (119879 (119910

119899))

minus (119868 minus 120572119899120583119865) (119879 (119911)) 119909

119899+1minus 119911⟩

le ⟨120572119899120588 (119880 (119909

119899) minus 119880 (119911)) 119909

119899+1minus 119911⟩

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ ⟨(119868 minus 120572119899120583119865) (119879 (119910

119899)) minus (119868 minus 120572

119899120583119865) (119879 (119911)) 119909

119899+1minus 119911⟩

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ (1 minus 120572119899]) 1003817100381710038171003817119910119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ (1 minus 120572119899]) 120573119899

1003817100381710038171003817119878119909119899

minus 1198781199111003817100381710038171003817 + 120573119899 119878119911 minus 119911

+ (1 minus 120573119899)

1003817100381710038171003817119906119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ (1 minus 120572119899]) 120573119899

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817 + 120573119899 119878119911 minus 119911

+ (1 minus 120573119899)

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

= (1 minus 120572119899

(] minus 120588120591))1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ (1 minus 120572119899]) 120573119899 119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

le1 minus 120572119899

(] minus 120588120591)

2(1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

2

+1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

2

)

+ 120572119899⟨120588119880 (119911) minus 120583119865 (119911) 119909

119899+1minus 119911⟩

+ (1 minus 120572119899]) 120573119899 119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

(68)

which implies that1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

2

le1 minus 120572119899

(] minus 120588120591)

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)⟨120588119880 (119909

119899) minus 120583119865 (119911) 119909

119899+1minus 119911⟩

+2 (1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

le (1 minus 120572119899

(] minus 120588120591))1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

2

+2120572119899

(] minus 120588120591)

1 + 120572119899

(] minus 120588120591)

times 1

] minus 120588120591⟨120588119880 (119911) minus 120583119865 (119911) 119909

119899+1minus 119911⟩

+(1 minus 120572

119899]) 120573119899

120572119899

(] minus 120588120591)119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

(69)

Let 120574119899

= 120572119899(] minus 120588120591) and 120575

119899= (2120572

119899(] minus 120588120591)(1 +

120572119899(] minus 120588120591)))(1(] minus 120588120591))⟨120588119880(119911) minus 120583119865(119911) 119909

119899+1minus 119911⟩ + ((1 minus

120572119899])120573119899120572119899(] minus 120588120591))119878119911 minus 119911119909

119899+1minus 119911

Sinceinfin

sum

119899=1

120572119899

= infin


1

] minus 120588120591⟨120588119880 (119911) minus 120583119865 (119911) 119909

119899+1minus 119911⟩

+(1 minus 120572

119899]) 120573119899

120572119899

(] minus 120588120591)119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817 le 0

(70)

It follows thatinfin

sum

119899=1

120574119899

= infin lim sup119899rarrinfin

120575119899

120574119899

le 0 (71)

Thus all the conditions of Lemma 9 are satisfied Hence wededuce that 119909

119899rarr 119911 This completes the proof


Remark 16 In hierarchical fixed point problem (17) if 119878 = 119868minus

(120588119880 minus 120583119865) then we can get the variational inequality (58) In(58) if119880 = 0 then we get the variational inequality ⟨119865(119911) 119909minus

119911⟩ ge 0 for all 119909 isin Λ cap 119865(119879) which is just the variationalinequality studied by Suzuki [27] extending the common setof solutions of a system of variational inequalities a splitequilibrium problem and a hierarchical fixed point problem

4 Conclusions

In this paper we suggest and analyze an iterative methodfor finding the approximate element of the common set ofsolutions of (15)-(16) and (17) in real Hilbert space whichcan be viewed as a refinement and improvement of someexisting methods for solving a split equilibrium problem anda hierarchical fixed point problem Some existing methods(eg [13 14 17ndash19 21ndash23]) can be viewed as special cases ofAlgorithm 11 Therefore the new algorithm is expected to bewidely applicable

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author would like to thank Professor Omar Halli Rec-tor Ibn Zohr University for providing excellent researchfacilities

References

[1] G Crombez ldquoA geometrical look at iterative methods foroperators with fixed pointsrdquoNumerical Functional Analysis andOptimization vol 26 no 2 pp 157ndash175 2005

[2] G Crombez ldquoA hierarchical presentation of operators withfixed points on Hilbert spacesrdquo Numerical Functional Analysisand Optimization vol 27 pp 259ndash277 2006

[3] H Zhou ldquoConvergence theorems of fixed points for 120581-strictpseudo-contractions in Hilbert spacesrdquo Nonlinear AnalysisTheory Methods and Applications vol 69 no 2 pp 456ndash4622008

[4] S-S Chang H W J Lee and C K Chan ldquoA new method forsolving equilibrium problem fixed point problem and varia-tional inequality problem with application to optimizationrdquoNonlinear Analysis Theory Methods and Applications vol 70no 9 pp 3307ndash3319 2009

[5] P Katchang and P Kumam ldquoA new iterative algorithm of solu-tion for equilibriumproblems variational inequalities and fixedpoint problems in a Hilbert spacerdquo Journal of Applied Mathe-matics and Computing vol 32 no 1 pp 19ndash38 2010

[6] S Plubtieng and R Punpaeng ldquoA general iterative methodfor equilibrium problems and fixed point problems in Hilbertspacesrdquo Journal of Mathematical Analysis and Applications vol336 no 1 pp 455ndash469 2007

[7] X Qin M Shang and Y Su ldquoA general iterative methodfor equilibrium problems and fixed point problems in Hilbertspacesrdquo Nonlinear Analysis Theory Methods and Applicationsvol 69 no 11 pp 3897ndash3909 2008

[8] P L Combettes and S A Hirstoaga ldquoEquilibrium program-ming using proximal like algorithmsrdquo Mathematical Program-ming vol 78 pp 29ndash41 1997

[9] Y Censor A Gibali and S Reich ldquoAlgorithms for the split vari-ational inequality problemrdquo Numerical Algorithms vol 59 no2 pp 301ndash323 2012

[10] A Moudafi ldquoSplit Monotone Variational Inclusionsrdquo Journal ofOptimization Theory and Applications vol 150 no 2 pp 275ndash283 2011

[11] C Byrne Y Censor A Gibali and S Reich ldquoWeak and strongconvergence of algorithms for the split common null pointproblemrdquo httparxivorgabs11085953

[12] K R Kazmi and S H Rizvi ldquoIterative approximation of acommon solution of a split equilibrium problem a variationalinequality problem and a fixed point problemrdquo Journal of theEgyptian Mathematical Society vol 21 pp 44ndash51 2013

[13] G Gu S Wang and Y J Cho ldquoStrong convergence algo-rithms for hierarchical fixed points problems and variationalinequalitiesrdquo Journal of Applied Mathematics vol 2011 ArticleID 164978 17 pages 2011

[14] Y Yao Y J Cho and Y-C Liou ldquoIterative algorithms for hierar-chical fixed points problems and variational inequalitiesrdquoMath-ematical and Computer Modelling vol 52 no 9-10 pp 1697ndash1705 2010

[15] A Bnouhachem and M A Noor ldquoAn iterative method forapproximating the common solutions of a variational inequal-ity a mixed equilibrium problem and a hierarchical fixed pointproblemrdquo Journal of Inequalities and Applications vol 490 pp1ndash25 2013

[16] A Bnouhachem ldquoAlgorithms of common solutions for a varia-tional inequality a split equilibrium problem and a hierarchicalfixed point problemrdquo Fixed Point Theory and Applications vol2013 article 278 pp 1ndash25 2013

[17] F Cianciaruso G Marino L Muglia and Y Yao ldquoOn a two-steps algorithm for hierarchical fixed point problems and varia-tional inequalitiesrdquo Journal of Inequalities and Applications vol2009 Article ID 208692 13 pages 2009

[18] P EMainge andAMoudafi ldquoStrong convergence of an iterativemethod for hierarchical fixed-point problemsrdquoPacific Journal ofOptimization vol 3 no 3 pp 529ndash538 2007

[19] G Marino and H-K Xu ldquoA general iterative method for non-expansive mappings in Hilbert spacesrdquo Journal of MathematicalAnalysis and Applications vol 318 no 1 pp 43ndash52 2006

[20] A Moudafi ldquoKrasnoselski-Mann iteration for hierarchicalfixed-point problemsrdquo Inverse Problems vol 23 no 4 pp 1635ndash1640 2007

[21] L-CCengQHAnsari and J-C Yao ldquoSome iterativemethodsfor finding fixed points and for solving constrained convexminimization problemsrdquo Nonlinear Analysis Theory Methodsand Applications vol 74 no 16 pp 5286ndash5302 2011

[22] M Tian ldquoA general iterative algorithm for nonexpansive map-pings in Hilbert spacesrdquo Nonlinear Analysis Theory Methodsand Applications vol 73 no 3 pp 689ndash694 2010

[23] YWang andW Xu ldquoStrong convergence of a modified iterativealgorithm for hierarchical fixed point problems and variationalinequalitiesrdquo Fixed Point Theory and Applications vol 2013article 121 9 pages 2013

[24] E Blum and W Oettli ldquoFrom optimization and variationalinequalities to equilibriumproblemsrdquoTheMathematics Studentvol 63 pp 123ndash145 1994


[25] F Cianciaruso G Marino L Muglia and Y Yao ldquoA hybridprojection algorithm for finding solutions ofmixed equilibriumproblem and variational inequality problemrdquo Fixed Point The-ory andApplications vol 2010 Article ID 383740 19 pages 2010

[26] Y Yao Y-C Liou and S M Kang ldquoApproach to commonelements of variational inequality problems and fixed pointproblems via a relaxed extragradient methodrdquo Computers andMathematics with Applications vol 59 no 11 pp 3472ndash34802010

[27] T Suzuki ldquoMoudafirsquos viscosity approximations with Meir-Keeler contractionsrdquo Journal of Mathematical Analysis andApplications vol 325 no 1 pp 342ndash352 2007

[28] H-K Xu ldquoIterative algorithms for nonlinear operatorsrdquo Journalof the London Mathematical Society vol 66 no 1 pp 240ndash2562002

[29] G L Acedo and H-K Xu ldquoIterative methods for strict pseudo-contractions in Hilbert spacesrdquo Nonlinear Analysis TheoryMethods and Applications vol 67 no 7 pp 2258ndash2271 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


The equilibrium problem denoted by EP is to find 119909 isin 119862

such that

119865 (119909 119910) ge 0 forall119910 isin 119862 (10)

The solution set of (10) is denoted by EP(119865) Numerousproblems in physics optimization and economics reduce tofinding a solution of (10) see [4ndash7] In 1997 Combettes andHirstoaga [8] introduced an iterative scheme of finding thebest approximation to the initial data when EP(119865) is non-empty In 2007 Plubtieng and Punpaeng [6] introduced aniterative method for finding the common element of the set119865(119879) cap EP(119865)

Recently Censor et al [9] introduced a new variationalinequality problemwhichwe call the split variational inequal-ity problem (SVIP) Let119867

1and119867

2be two real Hilbert spaces

Given operators 119891 1198671

rarr 1198671and 119892 119867

2rarr 1198672 a bounded

linear operator 119860 1198671

rarr 1198672 and nonempty closed and

convex subsets 119862 sube 1198671and 119876 sube 119867

2 the SVIP is formulated

as follows find a point 119909lowast

isin 119862 such that

⟨119891 (119909lowast) 119909 minus 119909

lowast⟩ ge 0 forall119909 isin 119862 (11)

and such that

119910lowast

= 119860119909lowast

isin 119876 solves ⟨119892 (119910lowast) 119910 minus 119910

lowast⟩ ge 0


(12)

In [10]Moudafi introduced an iterativemethodwhich can beregarded as an extension of the method given by Censor et al[9] for the following split monotone variational inclusions

Find 119909lowast

isin 1198671such that 0 isin 119891 (119909

lowast) + 1198611

(119909lowast) (13)

and such that

119910lowast

= 119860119909lowast

isin 1198672solves 0 isin 119892 (119910

lowast) + 1198612

(119910lowast) (14)

where 119861119894119867119894rarr 2119867119894 is a set-valued mapping for 119894 = 1 2 Later

Byrne et al [11] generalized and extended the work of Censoret al [9] and Moudafi [10]

Very recently Kazmi and Rizvi [12] studied the followingpair of equilibrium problems called split equilibrium prob-lem let 119865

1 119862 times 119862 rarr 119877 and 119865

2119876 times 119876 rarr 119877 be nonlinear

bifunctions and let 1198601198671

rarr 1198672be a bounded linear opera-

tor then the split equilibrium problem (SEP) is to find 119909lowast

isin

119862 such that

1198651


and such that

119910lowast

= 119860119909lowast

isin 119876 solves 1198652


The solution set of SEP (15)-(16) is denoted by Λ = 119901 isin

EP(1198651)119860119901 isin EP(119865

2)

Let 119878119862 rarr 119867 be a nonexpansive mapping The followingproblem is called a hierarchical fixed point problem find 119909 isin

119865(119879) such that

⟨119909 minus 119878119909 119910 minus 119909⟩ ge 0 forall119910 isin 119865 (119879) (17)

It is known that the hierarchical fixed point problem (17) linkswith some monotone variational inequalities and convexprogramming problems see [13 14] Various methods [15ndash20] have been proposed to solve the hierarchical fixed pointproblem In 2010 Yao et al [14] introduced the followingstrong convergence iterative algorithm to solve the problem(17)

119910119899

= 120573119899119878119909119899

+ (1 minus 120573119899) 119909119899

119909119899+1

= 119875119862

[120572119899119891 (119909119899) + (1 minus 120572

119899) 119879119910119899] forall119899 ge 0

(18)

where 119891 119862 rarr 119867 is a contraction mapping and 120572119899 and 120573

119899

are two sequences in (0 1) Under some certain restrictionson parameters Yao et al proved that the sequence 119909

119899

generated by (18) converges strongly to 119911 isin 119865(119879) which isthe unique solution of the following variational inequality

⟨(119868 minus 119891) 119911 119910 minus 119911⟩ ge 0 forall119910 isin 119865 (119879) (19)

In 2011 Ceng et al [21] investigated the following iterativemethod

119909119899+1

= 119875119862

[120572119899120588119880 (119909

119899) + (119868 minus 120572

119899120583119865) (119879 (119910

119899))] forall119899 ge 0

(20)

where119880 is a Lipschitzianmapping and119865 is a Lipschitzian andstrongly monotone mapping They proved that under someapproximate assumptions on the operators and parametersthe sequence 119909

119899 generated by (20) converges strongly to the

unique solution of the variational inequality

⟨120588119880 (119911) minus 120583119865 (119911) 119909 minus 119911⟩ ge 0 forall119909 isin Fix (119879) (21)

In the present paper inspired by the above cited works and bythe recent works going in this direction we give an iterativemethod for finding the approximate element of the commonset of solutions of (15)-(16) and (17) in real Hilbert spaceStrong convergence of the iterative algorithm is obtained inthe framework of Hilbert space We would like to mentionthat our proposed method is quite general and flexible andincludes many known results for solving split equilibriumproblems and hierarchical fixed point problems see forexample [13 14 17ndash19 21ndash23] and relevant references citedtherein

2 Preliminaries

In this section we recall some basic definitions and prop-erties which will be frequently used in our later analysisSome useful results proved already in the literature are alsosummarizedThe first lemma provides some basic propertiesof projection onto 119862

Lemma 2 Let119875119862denote the projection of119867 onto119862Then one

has the following inequalities

⟨119911 minus 119875119862 [119911] 119875

119862 [119911] minus V⟩ ge 0 forall119911 isin 119867 V isin 119862

⟨119906 minus V 119875119862 [119906] minus 119875

119862 [V]⟩ ge1003817100381710038171003817119875119862 [119906] minus 119875

119862 [V]10038171003817100381710038172

forall119906 V isin 119867

1003817100381710038171003817119875119862 [119906] minus 119875

119862 [V]1003817100381710038171003817 le 119906 minus V forall119906 V isin 119867

1003817100381710038171003817119906 minus 119875119862 [119911]

1003817100381710038171003817

2

le 119911 minus 1199062

minus1003817100381710038171003817119911 minus 119875

119862 [119911]1003817100381710038171003817

2

forall119911 isin 119867 119906 isin 119862

(22)





119905rarr0119865(119905119911+(1minus119905)119909 119910) le 119865(119909 119910)




119865 (119910 119909) +1




1198791198651

1199031198671


1198791198651

119903(119909) = 119911 isin 119862119865

1(119911 119910) +

1

119903⟨119910 minus 119911 119911 minus 119909⟩ ge 0


(24)


(i) 1198791198651


(ii) 1198791198651


100381710038171003817100381710038171198791198651

119903(119909) minus 119879

1198651

119903(119910)

10038171003817100381710038171003817

2

le ⟨1198791198651

119903(119909) minus 119879

1198651

119903(119910) 119909 minus 119910⟩

forall119909 119910 isin 1198671

(25)

(iii) 119865(1198791198651

119903) = 119864119875(119865

1)




1198652

119904 1198672


1198791198652

119904(119906) = V isin 119876 119865

2(V 119908) +

1

119904⟨119908 minus V V minus 119906⟩


(26)

Then 1198791198652


119865(1198791198652

119904) = EP(119865

2 119876) where EP(119865



find 119910lowast






1198671and 1199031 1199032

gt 0 Then

100381710038171003817100381710038171198791198651

1199032(119910) minus 119879

1198651

1199031(119909)

10038171003817100381710038171003817le

1003817100381710038171003817119910 minus 1199091003817100381710038171003817 +

10038161003816100381610038161003816100381610038161003816

1199032

minus 1199031

1199032

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198651

1199032(119910) minus 119910

10038171003817100381710038171003817

(28)






1003817100381710038171003817

2

forall119909 119910 isin 119862

(29)


120582 119862 rarr 119867 by



2) that is

10038171003817100381710038171003817119879120582119909 minus 119879120582119910

10038171003817100381710038171003817le (1 minus 120582]) 1003817100381710038171003817119909 minus 119910

1003817100381710038171003817 forall119909 119910 isin 119862 (31)




119886119899+1

le (1 minus 120574119899) 119886119899

+ 120575119899 (32)



(1) suminfin

119899=1120574119899

= infin(2) lim sup

119899rarrinfin120575119899120574119899

le 0 or suminfin

119899=1|120575119899| lt infin


119886119899

= 0



(119909119899) sub 119862 where 119908

119908(119909119899) =

119909 119909119899119894


119899rarrinfin119909119899

minus 119911 exists




Let 1198671and 119867


1198671and 119876 sube 119867




1rarr 119867

2









119899 119909119899 and 119910


119906119899

= 1198791198651

119903119899(119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)

119910119899

= 120573119899119878119909119899

+ (1 minus 120573119899) 119906119899

119909119899+1

= 119875119862

[120572119899120588119880 (119909

119899) + (119868 minus 120572

119899120583119865) (119879 (119910

119899))] forall119899 ge 0

(33)




2) 0 le 120588120591 lt ] where





120572119899

= 0 and suminfin

119899=1120572119899

= infin


(120573119899120572119899) = 0

(c) suminfin

119899=1|120572119899minus1


infin

119899=1|120573119899minus1



119903119899


119903119899


119899=1|119903119899minus1

minus

119903119899| lt infin










119899 are

bounded



= 1198791198651

119903119899(119909lowast) and 119860119909

lowast=

1198791198652

119903119899(119860119909lowast) Then

1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

=100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899) minus 119909lowast10038171003817100381710038171003817

2

=100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899) minus 1198791198651

119903119899(119909lowast)10038171003817100381710038171003817

2

le10038171003817100381710038171003817119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899minus 119909lowast10038171003817100381710038171003817

2

=1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120574210038171003817100381710038171003817

119860lowast(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

+ 2120574 ⟨119909119899


(1198791198652

119903119899minus 119868) 119860119909

119899⟩

=1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 1205742

⟨(1198791198652

119903119899minus 119868) 119860119909

119899 119860119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899⟩

+ 2120574 ⟨119909119899


(1198791198652

119903119899minus 119868) 119860119909

119899⟩

(34)


1205742

⟨(1198791198652

119903119899minus 119868) 119860119909

119899 119860119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899⟩

le 1198711205742

⟨(1198791198652

119903119899minus 119868) 119860119909

119899 (1198791198652

119903119899minus 119868) 119860119909

119899⟩

= 119871120574210038171003817100381710038171003817

(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

(35)


2120574 ⟨119909119899


(1198791198652

119903119899minus 119868) 119860119909

119899⟩

= 2120574 ⟨119860 (119909119899

minus 119909lowast) (1198791198652

119903119899minus 119868) 119860119909

119899⟩

= 2120574 ⟨119860 (119909119899

minus 119909lowast) + (119879

1198652

119903119899minus 119868) 119860119909

119899

minus (1198791198652

119903119899minus 119868) 119860119909

119899 (1198791198652

119903119899minus 119868) 119860119909

119899⟩

= 2120574 (⟨1198791198652

119903119899119860119909119899

minus 119860119909lowast (1198791198652

119903119899minus 119868) 119860119909

119899⟩

minus10038171003817100381710038171003817(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

)

le 2120574 (1

2

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

2

minus10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

2

)

= minus12057410038171003817100381710038171003817(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

(36)


1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

le1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120574 (119871120574 minus 1)10038171003817100381710038171003817(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

le1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

(37)


Denote 119881119899

= 120572119899120588119880(119909119899) + (119868 minus 120572

119899120583119865)(119879(119910





1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

=1003817100381710038171003817119875119862

[119881119899] minus 119875119862

[119909lowast]1003817100381710038171003817

le1003817100381710038171003817120572119899120588119880 (119909

119899) + (119868 minus 120572

119899120583119865) (119879 (119910

119899)) minus 119909

lowast1003817100381710038171003817

le 120572119899

1003817100381710038171003817120588119880 (119909119899) minus 120583119865 (119909

lowast)1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120572

119899120583119865) (119879 (119910

119899)) minus (119868 minus 120572

119899120583119865) 119879 (119909

lowast)1003817100381710038171003817

= 120572119899

1003817100381710038171003817120588119880 (119909119899) minus 120588119880 (119909

lowast) + (120588119880 minus 120583119865) (119909

lowast)1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120572

119899120583119865) (119879 (119910

119899)) minus (119868 minus 120572

119899120583119865) 119879 (119909

lowast)1003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 + 120572

119899

1003817100381710038171003817(120588119880 minus 120583119865) (119909lowast)1003817100381710038171003817

+ (1 minus 120572119899]) 1003817100381710038171003817119910119899

minus 119909lowast1003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 + 120572

119899

1003817100381710038171003817(120588119880 minus 120583119865) (119909lowast)1003817100381710038171003817

+ (1 minus 120572119899]) 1003817100381710038171003817120573119899119878119909119899

+ (1 minus 120573119899) 119906119899

minus 119909lowast1003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 + 120572

119899

1003817100381710038171003817(120588119880 minus 120583119865) (119909lowast)1003817100381710038171003817

+ (1 minus 120572119899]) (120573119899

1003817100381710038171003817119878119909119899

minus 119878119909lowast1003817100381710038171003817 + 120573

119899

1003817100381710038171003817119878119909lowast

minus 119909lowast1003817100381710038171003817

+ (1 minus 120573119899)

1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817)

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 + 120572

119899

1003817100381710038171003817(120588119880 minus 120583119865) (119909lowast)1003817100381710038171003817

+ (1 minus 120572119899]) (120573119899

1003817100381710038171003817119878119909119899

minus 119878119909lowast1003817100381710038171003817 + 120573

119899

1003817100381710038171003817119878119909lowast

minus 119909lowast1003817100381710038171003817

+ (1 minus 120573119899)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817)

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 + 120572

119899

1003817100381710038171003817(120588119880 minus 120583119865) (119909lowast)1003817100381710038171003817

+ (1 minus 120572119899]) (120573119899

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 + 120573

119899

1003817100381710038171003817119878119909lowast

minus 119909lowast1003817100381710038171003817

+ (1 minus 120573119899)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817)

= (1 minus 120572119899

(] minus 120588120591))1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

+ 120572119899

1003817100381710038171003817(120588119880 minus 120583119865) (119909lowast)1003817100381710038171003817

+ (1 minus 120572119899]) 120573119899

1003817100381710038171003817119878119909lowast

minus 119909lowast1003817100381710038171003817

le (1 minus 120572119899

(] minus 120588120591))1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

+ 120572119899

1003817100381710038171003817(120588119880 minus 120583119865) (119909lowast)1003817100381710038171003817 + 120573119899

1003817100381710038171003817119878119909lowast

minus 119909lowast1003817100381710038171003817

le (1 minus 120572119899

(] minus 120588120591))1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

+ 120572119899

(1003817100381710038171003817(120588119880 minus 120583119865) (119909

lowast)1003817100381710038171003817 +

1003817100381710038171003817119878119909lowast

minus 119909lowast1003817100381710038171003817)

= (1 minus 120572119899

(] minus 120588120591))1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

+120572119899

(] minus 120588120591)

] minus 120588120591(1003817100381710038171003817(120588119880 minus 120583119865) 119909

lowast1003817100381710038171003817 +1003817100381710038171003817119878119909lowast

minus 119909lowast1003817100381710038171003817)

le max1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

1

] minus 120588120591

times (1003817100381710038171003817(120588119880 minus 120583119865) (119909

lowast)1003817100381710038171003817 +

1003817100381710038171003817119878119909lowast

minus 119909lowast1003817100381710038171003817)

(38)




0minus


lowast + 119878119909


1199090


that 119906119899 119910119899 119878(119909

119899) 119879(119909

119899) 119865(119879(119910

119899)) and 119880(119909

119899) are

bounded





119909119899+1

minus 119909119899 = 0


(119909119899) sub 119865(119879) (119908

119908(119909119899) = 119909

119909119899119894

119909)


= 1198791198651

119903119899(119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899) and 119906

119899minus1=

1198791198651

119903119899minus1(119909119899minus1

+120574119860lowast(1198791198652

119903119899minus1minus119868)119860119909


1003817100381710038171003817119906119899


1003817100381710038171003817

le10038171003817100381710038171003817119909119899


+120574 (119860lowast

(1198791198652

119903119899minus 119868) 119860119909


(1198791198652

119903119899minus1minus 119868) 119860119909

119899minus1)10038171003817100381710038171003817

+

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)

minus (119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)10038171003817100381710038171003817

le1003817100381710038171003817119909119899


minus 120574119860lowast119860 (119909119899


)1003817100381710038171003817

+ 120574 119860100381710038171003817100381710038171198791198652

119903119899119860119909119899

minus 1198791198652

119903119899minus1119860119909119899minus1

10038171003817100381710038171003817

+

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)

minus (119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)10038171003817100381710038171003817

le (1003817100381710038171003817119909119899


1003817100381710038171003817

2

minus 21205741003817100381710038171003817119860 (119909119899


)1003817100381710038171003817

2

+120574211986041003817100381710038171003817119909119899


1003817100381710038171003817

2

)12

+ 120574 119860 (1003817100381710038171003817119860 (119909119899


)1003817100381710038171003817

+

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198652

119903119899119860119909119899

minus 119860119909119899

10038171003817100381710038171003817)

+

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)

minus (119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)10038171003817100381710038171003817

le (1 minus 21205741198602

+ 12057421198604)12 1003817100381710038171003817119909119899


1003817100381710038171003817

+ 1205741198602 1003817100381710038171003817119909119899


1003817100381710038171003817

+ 120574 119860

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198652

119903119899119860119909119899

minus 119860119909119899

10038171003817100381710038171003817


+

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)

minus (119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)10038171003817100381710038171003817

= (1 minus 1205741198602)

1003817100381710038171003817119909119899


1003817100381710038171003817

+ 1205741198602 1003817100381710038171003817119909119899


1003817100381710038171003817

+ 120574 119860

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198652

119903119899119860119909119899

minus 119860119909119899

10038171003817100381710038171003817

+

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)

minus (119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)10038171003817100381710038171003817

=1003817100381710038171003817119909119899


1003817100381710038171003817 + 120574 119860

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198652

119903119899119860119909119899

minus 119860119909119899

10038171003817100381710038171003817

+

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)

minus (119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)10038171003817100381710038171003817

=1003817100381710038171003817119909119899


1003817100381710038171003817 +

10038161003816100381610038161003816100381610038161003816

119903119899


119903119899

10038161003816100381610038161003816100381610038161003816

(120574 119860 120590119899

+ 120594119899)

(39)

where 120590119899

= 1198791198652

119903119899119860119909119899

minus 119860119909119899 and 120594

119899= 1198791198651

119903119899(119909119899

+ 120574119860lowast(1198791198652

119903119899minus

119868)119860119909119899) minus (119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909



gt


1003817100381710038171003817119906119899minus1

minus 119906119899

1003817100381710038171003817 le1003817100381710038171003817119909119899minus1

minus 119909119899

1003817100381710038171003817 +1

120583

1003816100381610038161003816119903119899minus1 minus 119903119899

1003816100381610038161003816 (120574 119860 120590119899

+ 120594119899)

(40)



1003817100381710038171003817

=1003817100381710038171003817120573119899119878119909119899

+ (1 minus 120573119899) 119906119899


119878119909119899minus1

+ (1 minus 120573119899minus1

) 119906119899minus1

)1003817100381710038171003817

=1003817100381710038171003817120573119899

(119878119909119899

minus 119878119909119899minus1

) + (120573119899


) 119878119909119899minus1

+ (1 minus 120573119899) (119906119899


) + (120573119899minus1

minus 120573119899) 119906119899minus1

1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899


1003817100381710038171003817 + (1 minus 120573119899)

1003817100381710038171003817119906119899


1003817100381710038171003817

+1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 (1003817100381710038171003817119878119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119906119899minus1

1003817100381710038171003817)

le 120573119899

1003817100381710038171003817119909119899


1003817100381710038171003817 + (1 minus 120573119899)

times 1003817100381710038171003817119909119899minus1

minus 119909119899

1003817100381710038171003817 +1

120583

1003816100381610038161003816119903119899minus1 minus 119903119899

1003816100381610038161003816 (120574 119860 120590119899

+ 120594119899)

+1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 (1003817100381710038171003817119878119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119906119899minus1

1003817100381710038171003817)

le1003817100381710038171003817119909119899


1003817100381710038171003817 +1

120583

1003816100381610038161003816119903119899minus1 minus 119903119899

1003816100381610038161003816 (120574 119860 120590119899

+ 120594119899)

+1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 (1003817100381710038171003817119878119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119906119899minus1

1003817100381710038171003817)

(41)

Next we estimate

1003817100381710038171003817119909119899+1

minus 119909119899

1003817100381710038171003817

=1003817100381710038171003817119875119862

[119881119899] minus 119875119862

[119881119899minus1

]1003817100381710038171003817

le1003817100381710038171003817120572119899120588 (119880 (119909

119899) minus 119880 (119909

119899minus1)) + (120572

119899minus 120572119899minus1

) 120588119880 (119909119899minus1

)

+ (119868 minus 120572119899120583119865) (119879 (119910

119899)) minus (119868 minus 120572

119899120583119865) 119879 (119910

119899minus1)

+ (119868 minus 120572119899120583119865) (119879 (119910


119899minus1120583119865) (119879 (119910

119899minus1))

1003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899


1003817100381710038171003817 + (1 minus 120572119899]) 1003817100381710038171003817119910119899


1003817100381710038171003817

+1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 (1003817100381710038171003817120588119880 (119909

119899minus1)1003817100381710038171003817 +

1003817100381710038171003817120583119865 (119879 (119910119899minus1

))1003817100381710038171003817)

(42)


1003817100381710038171003817119909119899+1

minus 119909119899

1003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899


1003817100381710038171003817 + (1 minus 120572119899])

times 1003817100381710038171003817119909119899


1003817100381710038171003817 +1

120583

1003816100381610038161003816119903119899minus1 minus 119903119899

1003816100381610038161003816 (120574 119860 120590119899

+ 120594119899)

+1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 (1003817100381710038171003817119878119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119906119899minus1

1003817100381710038171003817)

+1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 (1003817100381710038171003817120588119880 (119909

119899minus1)1003817100381710038171003817 +

1003817100381710038171003817120583119865 (119879 (119910119899minus1

))1003817100381710038171003817)

le (1 minus (] minus 120588120591) 120572119899)

1003817100381710038171003817119909119899


1003817100381710038171003817

+1

120583

1003816100381610038161003816119903119899minus1 minus 119903119899

1003816100381610038161003816 (120574 119860 120590119899

+ 120594119899)

+1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 (1003817100381710038171003817119878119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119906119899minus1

1003817100381710038171003817)

+1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 (1003817100381710038171003817120588119880 (119909

119899minus1)1003817100381710038171003817 +

1003817100381710038171003817120583119865 (119879 (119910119899minus1

))1003817100381710038171003817)

le (1 minus (] minus 120588120591) 120572119899)

1003817100381710038171003817119909119899


1003817100381710038171003817

+ 119872 (1

120583

1003816100381610038161003816119903119899minus1 minus 119903119899

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816)

(43)

where

119872 = maxsup119899ge1

(120574 119860 120590119899

+ 120594119899)

sup119899ge1

(1003817100381710038171003817119878119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119906119899minus1

1003817100381710038171003817)

sup119899ge1

(1003817100381710038171003817120588119880 (119909

119899minus1)1003817100381710038171003817 +

1003817100381710038171003817120583119865 (119879 (119910119899minus1

))1003817100381710038171003817)

(44)


lim119899rarrinfin

1003817100381710038171003817119909119899+1

minus 119909119899

1003817100381710038171003817 = 0 (45)



119906119899minus119909119899 = 0 Since 119909



1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

= ⟨119875119862

[119881119899] minus 119909lowast 119909119899+1


= ⟨119875119862

[119881119899] minus 119881119899 119875119862

[119881119899] minus 119909lowast⟩ + ⟨119881

119899minus 119909lowast 119909119899+1


le ⟨120572119899

(120588119880 (119909119899) minus 120583119865 (119909

lowast) + (119868 minus 120572

119899120583119865) (119879 (119910

119899)))

minus (119868 minus 120572119899120583119865) (119879 (119909

lowast)) 119909119899+1


= ⟨120572119899120588 (119880 (119909

119899) minus 119880 (119909

lowast)) 119909119899+1


+ 120572119899

⟨120588119880 (119909lowast) minus 120583119865 (119909

lowast) 119909119899+1


+ ⟨(119868 minus 120572119899120583119865) (119879 (119910

119899))

minus (119868 minus 120572119899120583119865) (119879 (119909

lowast)) 119909119899+1


le 120572119899120588120591

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

+ 120572119899

⟨120588119880 (119909lowast) minus 120583119865 (119909

lowast) 119909119899+1


+ (1 minus 120572119899]) 1003817100381710038171003817119910119899

minus 119909lowast1003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

le120572119899120588120591

2(1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

)

+ 120572119899

⟨120588119880 (119909lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899])

2(1003817100381710038171003817119910119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

)

le(1 minus 120572

119899(] minus 120588120591))

2

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

+120572119899120588120591

2

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120572119899

⟨120588119880 (119909lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899])

2(120573119899

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+ (1 minus 120573119899)

1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

)

le(1 minus 120572

119899(] minus 120588120591))

2

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

+120572119899120588120591

2

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120572119899

⟨120588119880 (119909lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

2

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+(1 minus 120572

119899]) (1 minus 120573

119899)

2

times 1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120574 (119871120574 minus 1)10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

2

(46)


1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+(1 minus 120572

119899]) (1 minus 120573

119899)

1 + 120572119899

(] minus 120588120591)

times 1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120574 (119871120574 minus 1)10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2


119899]) (1 minus 120573

119899) 120574 (1 minus 119871120574)

1 + 120572119899

(] minus 120588120591)

10038171003817100381710038171003817(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

(47)


(1 minus 120572119899]) (1 minus 120573

119899) 120574 (1 minus 119871120574)

1 + 120572119899

(] minus 120588120591)

10038171003817100381710038171003817(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)


lowast) 119909119899+1


+ 120573119899

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)


lowast) 119909119899+1


+ 120573119899

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+ (1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 +

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817)

times1003817100381710038171003817119909119899+1

minus 119909119899

1003817100381710038171003817

(48)


119909119899+1

minus 119909119899 = 0 120572

119899rarr 0 and

120573119899

rarr 0 we obtain

lim119899rarrinfin

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817= 0 (49)


Since 1198791198651


1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

=100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899) minus 1198791198651

119903119899(119909lowast)10038171003817100381710038171003817

2

le ⟨119906119899

minus 119909lowast 119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909


=1

21003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

+10038171003817100381710038171003817119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899minus 119909lowast10038171003817100381710038171003817

2

minus10038171003817100381710038171003817119906119899

minus 119909lowast

minus [119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899minus 119909lowast]10038171003817100381710038171003817

2

=1

21003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

+10038171003817100381710038171003817119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899minus 119909lowast10038171003817100381710038171003817

2

minus10038171003817100381710038171003817119906119899

minus 119909119899

minus 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

le1

21003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus10038171003817100381710038171003817119906119899

minus 119909119899


(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

2

=1

21003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus [1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

+ 120574210038171003817100381710038171003817

119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

2

minus2120574⟨119906119899

minus 119909119899 119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899⟩]

(50)


1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

le1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

+ 21205741003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

(51)


minus 119909lowast1003817100381710038171003817

2

le(1 minus 120572

119899(] minus 120588120591))

2

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

+120572119899120588120591

2

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120572119899⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899])

2(120573119899

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+ (1 minus 120573119899)

1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

)

le(1 minus 120572

119899(] minus 120588120591))

2

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

+120572119899120588120591

2

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120572119899⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


(1 minus 120572119899])

2

times 120573119899

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+ (1 minus 120573119899)

times (1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

+21205741003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817)

(52)

which implies that

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)


lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+(1 minus 120572

119899]) (1 minus 120573

119899)

1 + 120572119899

(] minus 120588120591)

times 1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

+21205741003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)


lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+(1 minus 120572

119899]) (1 minus 120573

119899)

1 + 120572119899

(] minus 120588120591)

times minus1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

+ 21205741003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

(53)

Hence

(1 minus 120572119899]) (1 minus 120573

119899)

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+2 (1 minus 120572

119899]) (1 minus 120573

119899) 120574

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+2 (1 minus 120572

119899]) (1 minus 120573

119899) 120574

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

+ (1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 +

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817)

1003817100381710038171003817119909119899+1

minus 119909119899

1003817100381710038171003817

(54)



119909119899+1

minus 119909119899 = 0 120572

119899rarr 0 120573

119899rarr 0 and

lim119899rarrinfin

(1198791198652

119903119899minus 119868)119860119909


lim119899rarrinfin

1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817 = 0 (55)


1003817100381710038171003817119909119899

minus 119879 (119909119899)1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119909119899+1

minus 119879 (119909119899)1003817100381710038171003817

=1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119875119862

[119881119899] minus 119875119862

[119879 (119909119899)]

1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817

+1003817100381710038171003817120572119899

(120588119880 (119909119899) minus 120583119865 (119879 (119910

119899)) + 119879 (119910

119899) minus 119879 (119909

119899))

1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817

+ 120572119899

1003817100381710038171003817120588119880 (119909119899) minus 120583119865 (119879 (119910

119899))

1003817100381710038171003817 +1003817100381710038171003817119910119899

minus 119909119899

1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817 + 120572119899

1003817100381710038171003817120588119880 (119909119899) minus 120583119865 (119879 (119910

119899))

1003817100381710038171003817

+1003817100381710038171003817120573119899119878119909119899

+ (1 minus 120573119899) 119906119899

minus 119909119899

1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817 + 120572119899

1003817100381710038171003817120588119880 (119909119899) minus 120583119865 (119879 (119910

119899))

1003817100381710038171003817

+ 120573119899

1003817100381710038171003817119878119909119899

minus 119909119899

1003817100381710038171003817 + (1 minus 120573119899)

1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

(56)


119909119899+1

minus 119909119899 = 0 120572

119899rarr 0 120573

119899rarr 0 120588119880(119909

119899) minus

120583119865(119879(119910119899)) and 119878119909


119899rarrinfin119909119899

minus

119906119899 = 0 we obtain

lim119899rarrinfin

1003817100381710038171003817119909119899

minus 119879 (119909119899)1003817100381710038171003817 = 0 (57)



119909lowast


isin 119865(119879) Therefore 119908119908

(119909119899) sub 119865(119879)







119899=

1198791198651

119903119899(119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899) we have

1198651

(119906119899 119910) +

1

119903119899

⟨119910 minus 119906119899 119906119899

minus 119909119899⟩

minus1

119903119899

⟨119910 minus 119906119899 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899⟩ ge 0 forall119910 isin 119862

(59)


minus1

119903119899

⟨119910 minus 119906119899 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899⟩

+1

119903119899

⟨119910 minus 119906119899 119906119899

minus 119909119899⟩ ge 1198651

(119910 119906119899) forall119910 isin 119862

(60)

minus1

119903119899119896

⟨119910 minus 119906119899119896

120574119860lowast

(1198791198652

119903119899119896

minus 119868) 119860119909119899119896

⟩

+ ⟨119910 minus 119906119899119896

119906119899119896

minus 119909119899119896

119903119899119896

⟩ ge 1198651

(119910 119906119899119896

) forall119910 isin 119862

(61)


119906119899

minus 119909119899 = 0 lim

119899rarrinfin(1198791198652

119903119899minus 119868)119860119909

119899 = 0

and 119909119899



1(119910 119908) le 0 for all 119910 isin 119862


= 119905119910 + (1 minus 119905)119908 wehave 119910


0 = 1198651

(119910119905 119910119905)

le 1199051198651

(119910119905 119910) + (1 minus 119905) 119865

1(119910119905 119908)

le 1199051198651

(119910119905 119910)

(62)



1)


119899 is bounded

and 119909119899


of 119909119899 such

that 119909119899119896


rarr 119860119908 Now set V119899119896

= 119860119909119899119896

minus 1198791198652

119903119899119896

119860119909119899119896 It follows from


V119899119896

= 0 and119860119909119899119896

minusV119899119896

= 1198791198652

119903119899119896

119860119909119899119896Therefore


119903119899119896

we have

1198652

(119860119909119899119896

minus V119899119896

119910)

+1

119903119899119896

⟨119910 minus (119860119909119899119896

minus V119899119896

)

(119860119909119899119896

minus V119899119896

) minus 119860119909119899119896

⟩ ge 0 forall119910 isin 119862

(63)



1198652

(119860119908 119910) ge 0 forall119910 isin 119862 (64)


Thus we have

119908 isin Λ cap 119865 (119879) (65)



119896 ge 120578

lArrrArr 1198962

ge 1205782

lArrrArr 1 minus 2120583120578 + 12058321198962

ge 1 minus 2120583120578 + 12058321205782



lArrrArr 120583120578 ge ]

(66)




119899rarrinfin⟨120588119880(119911)minus120583119865(119911) 119909

119899minus119911⟩ le


119899119896 of

119909119899 such that


⟨120588119880 (119911) minus 120583119865 (119911) 119909119899

minus 119911⟩


⟨120588119880 (119911) minus 120583119865 (119911) 119909119899119896

minus 119911⟩

= ⟨120588119880 (119911) minus 120583119865 (119911) 119908 minus 119911⟩ le 0

(67)



1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

2

= ⟨119875119862

[119881119899] minus 119911 119909

119899+1minus 119911⟩

= ⟨119875119862

[119881119899] minus 119881119899 119875119862

[119881119899] minus 119911⟩ + ⟨119881

119899minus 119911 119909

119899+1minus 119911⟩

le ⟨120572119899

(120588119880 (119909119899) minus 120583119865 (119911)) + (119868 minus 120572

119899120583119865) (119879 (119910

119899))

minus (119868 minus 120572119899120583119865) (119879 (119911)) 119909

119899+1minus 119911⟩

le ⟨120572119899120588 (119880 (119909

119899) minus 119880 (119911)) 119909

119899+1minus 119911⟩

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ ⟨(119868 minus 120572119899120583119865) (119879 (119910

119899)) minus (119868 minus 120572

119899120583119865) (119879 (119911)) 119909

119899+1minus 119911⟩

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ (1 minus 120572119899]) 1003817100381710038171003817119910119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ (1 minus 120572119899]) 120573119899

1003817100381710038171003817119878119909119899

minus 1198781199111003817100381710038171003817 + 120573119899 119878119911 minus 119911

+ (1 minus 120573119899)

1003817100381710038171003817119906119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ (1 minus 120572119899]) 120573119899

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817 + 120573119899 119878119911 minus 119911

+ (1 minus 120573119899)

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

= (1 minus 120572119899

(] minus 120588120591))1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ (1 minus 120572119899]) 120573119899 119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

le1 minus 120572119899

(] minus 120588120591)

2(1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

2

+1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

2

)

+ 120572119899⟨120588119880 (119911) minus 120583119865 (119911) 119909

119899+1minus 119911⟩

+ (1 minus 120572119899]) 120573119899 119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

(68)


minus 1199111003817100381710038171003817

2

le1 minus 120572119899

(] minus 120588120591)

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)⟨120588119880 (119909

119899) minus 120583119865 (119911) 119909

119899+1minus 119911⟩

+2 (1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

le (1 minus 120572119899

(] minus 120588120591))1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

2

+2120572119899

(] minus 120588120591)

1 + 120572119899

(] minus 120588120591)

times 1

] minus 120588120591⟨120588119880 (119911) minus 120583119865 (119911) 119909

119899+1minus 119911⟩

+(1 minus 120572

119899]) 120573119899

120572119899

(] minus 120588120591)119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

(69)

Let 120574119899

= 120572119899(] minus 120588120591) and 120575

119899= (2120572

119899(] minus 120588120591)(1 +


119899+1minus 119911⟩ + ((1 minus

120572119899])120573119899120572119899(] minus 120588120591))119878119911 minus 119911119909

119899+1minus 119911

Sinceinfin

sum

119899=1

120572119899

= infin


1

] minus 120588120591⟨120588119880 (119911) minus 120583119865 (119911) 119909

119899+1minus 119911⟩

+(1 minus 120572

119899]) 120573119899

120572119899

(] minus 120588120591)119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817 le 0

(70)


sum

119899=1

120574119899


120575119899

120574119899

le 0 (71)







4 Conclusions




Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119905rarr0119865(119905119911+(1minus119905)119909 119910) le 119865(119909 119910)




119865 (119910 119909) +1




1198791198651

1199031198671


1198791198651

119903(119909) = 119911 isin 119862119865

1(119911 119910) +

1

119903⟨119910 minus 119911 119911 minus 119909⟩ ge 0


(24)


(i) 1198791198651


(ii) 1198791198651


100381710038171003817100381710038171198791198651

119903(119909) minus 119879

1198651

119903(119910)

10038171003817100381710038171003817

2

le ⟨1198791198651

119903(119909) minus 119879

1198651

119903(119910) 119909 minus 119910⟩

forall119909 119910 isin 1198671

(25)

(iii) 119865(1198791198651

119903) = 119864119875(119865

1)




1198652

119904 1198672


1198791198652

119904(119906) = V isin 119876 119865

2(V 119908) +

1

119904⟨119908 minus V V minus 119906⟩


(26)

Then 1198791198652


119865(1198791198652

119904) = EP(119865

2 119876) where EP(119865



find 119910lowast






1198671and 1199031 1199032

gt 0 Then

100381710038171003817100381710038171198791198651

1199032(119910) minus 119879

1198651

1199031(119909)

10038171003817100381710038171003817le

1003817100381710038171003817119910 minus 1199091003817100381710038171003817 +

10038161003816100381610038161003816100381610038161003816

1199032

minus 1199031

1199032

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198651

1199032(119910) minus 119910

10038171003817100381710038171003817

(28)






1003817100381710038171003817

2

forall119909 119910 isin 119862

(29)


120582 119862 rarr 119867 by



2) that is

10038171003817100381710038171003817119879120582119909 minus 119879120582119910

10038171003817100381710038171003817le (1 minus 120582]) 1003817100381710038171003817119909 minus 119910

1003817100381710038171003817 forall119909 119910 isin 119862 (31)




119886119899+1

le (1 minus 120574119899) 119886119899

+ 120575119899 (32)



(1) suminfin

119899=1120574119899

= infin(2) lim sup

119899rarrinfin120575119899120574119899

le 0 or suminfin

119899=1|120575119899| lt infin


119886119899

= 0



(119909119899) sub 119862 where 119908

119908(119909119899) =

119909 119909119899119894


119899rarrinfin119909119899

minus 119911 exists




Let 1198671and 119867


1198671and 119876 sube 119867




1rarr 119867

2









119899 119909119899 and 119910


119906119899

= 1198791198651

119903119899(119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)

119910119899

= 120573119899119878119909119899

+ (1 minus 120573119899) 119906119899

119909119899+1

= 119875119862

[120572119899120588119880 (119909

119899) + (119868 minus 120572

119899120583119865) (119879 (119910

119899))] forall119899 ge 0

(33)




2) 0 le 120588120591 lt ] where





120572119899

= 0 and suminfin

119899=1120572119899

= infin


(120573119899120572119899) = 0

(c) suminfin

119899=1|120572119899minus1


infin

119899=1|120573119899minus1



119903119899


119903119899


119899=1|119903119899minus1

minus

119903119899| lt infin










119899 are

bounded



= 1198791198651

119903119899(119909lowast) and 119860119909

lowast=

1198791198652

119903119899(119860119909lowast) Then

1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

=100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899) minus 119909lowast10038171003817100381710038171003817

2

=100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899) minus 1198791198651

119903119899(119909lowast)10038171003817100381710038171003817

2

le10038171003817100381710038171003817119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899minus 119909lowast10038171003817100381710038171003817

2

=1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120574210038171003817100381710038171003817

119860lowast(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

+ 2120574 ⟨119909119899


(1198791198652

119903119899minus 119868) 119860119909

119899⟩

=1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 1205742

⟨(1198791198652

119903119899minus 119868) 119860119909

119899 119860119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899⟩

+ 2120574 ⟨119909119899


(1198791198652

119903119899minus 119868) 119860119909

119899⟩

(34)


1205742

⟨(1198791198652

119903119899minus 119868) 119860119909

119899 119860119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899⟩

le 1198711205742

⟨(1198791198652

119903119899minus 119868) 119860119909

119899 (1198791198652

119903119899minus 119868) 119860119909

119899⟩

= 119871120574210038171003817100381710038171003817

(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

(35)


2120574 ⟨119909119899


(1198791198652

119903119899minus 119868) 119860119909

119899⟩

= 2120574 ⟨119860 (119909119899

minus 119909lowast) (1198791198652

119903119899minus 119868) 119860119909

119899⟩

= 2120574 ⟨119860 (119909119899

minus 119909lowast) + (119879

1198652

119903119899minus 119868) 119860119909

119899

minus (1198791198652

119903119899minus 119868) 119860119909

119899 (1198791198652

119903119899minus 119868) 119860119909

119899⟩

= 2120574 (⟨1198791198652

119903119899119860119909119899

minus 119860119909lowast (1198791198652

119903119899minus 119868) 119860119909

119899⟩

minus10038171003817100381710038171003817(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

)

le 2120574 (1

2

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

2

minus10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

2

)

= minus12057410038171003817100381710038171003817(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

(36)


1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

le1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120574 (119871120574 minus 1)10038171003817100381710038171003817(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

le1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

(37)


Denote 119881119899

= 120572119899120588119880(119909119899) + (119868 minus 120572

119899120583119865)(119879(119910





1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

=1003817100381710038171003817119875119862

[119881119899] minus 119875119862

[119909lowast]1003817100381710038171003817

le1003817100381710038171003817120572119899120588119880 (119909

119899) + (119868 minus 120572

119899120583119865) (119879 (119910

119899)) minus 119909

lowast1003817100381710038171003817

le 120572119899

1003817100381710038171003817120588119880 (119909119899) minus 120583119865 (119909

lowast)1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120572

119899120583119865) (119879 (119910

119899)) minus (119868 minus 120572

119899120583119865) 119879 (119909

lowast)1003817100381710038171003817

= 120572119899

1003817100381710038171003817120588119880 (119909119899) minus 120588119880 (119909

lowast) + (120588119880 minus 120583119865) (119909

lowast)1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120572

119899120583119865) (119879 (119910

119899)) minus (119868 minus 120572

119899120583119865) 119879 (119909

lowast)1003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 + 120572

119899

1003817100381710038171003817(120588119880 minus 120583119865) (119909lowast)1003817100381710038171003817

+ (1 minus 120572119899]) 1003817100381710038171003817119910119899

minus 119909lowast1003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 + 120572

119899

1003817100381710038171003817(120588119880 minus 120583119865) (119909lowast)1003817100381710038171003817

+ (1 minus 120572119899]) 1003817100381710038171003817120573119899119878119909119899

+ (1 minus 120573119899) 119906119899

minus 119909lowast1003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 + 120572

119899

1003817100381710038171003817(120588119880 minus 120583119865) (119909lowast)1003817100381710038171003817

+ (1 minus 120572119899]) (120573119899

1003817100381710038171003817119878119909119899

minus 119878119909lowast1003817100381710038171003817 + 120573

119899

1003817100381710038171003817119878119909lowast

minus 119909lowast1003817100381710038171003817

+ (1 minus 120573119899)

1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817)

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 + 120572

119899

1003817100381710038171003817(120588119880 minus 120583119865) (119909lowast)1003817100381710038171003817

+ (1 minus 120572119899]) (120573119899

1003817100381710038171003817119878119909119899

minus 119878119909lowast1003817100381710038171003817 + 120573

119899

1003817100381710038171003817119878119909lowast

minus 119909lowast1003817100381710038171003817

+ (1 minus 120573119899)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817)

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 + 120572

119899

1003817100381710038171003817(120588119880 minus 120583119865) (119909lowast)1003817100381710038171003817

+ (1 minus 120572119899]) (120573119899

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 + 120573

119899

1003817100381710038171003817119878119909lowast

minus 119909lowast1003817100381710038171003817

+ (1 minus 120573119899)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817)

= (1 minus 120572119899

(] minus 120588120591))1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

+ 120572119899

1003817100381710038171003817(120588119880 minus 120583119865) (119909lowast)1003817100381710038171003817

+ (1 minus 120572119899]) 120573119899

1003817100381710038171003817119878119909lowast

minus 119909lowast1003817100381710038171003817

le (1 minus 120572119899

(] minus 120588120591))1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

+ 120572119899

1003817100381710038171003817(120588119880 minus 120583119865) (119909lowast)1003817100381710038171003817 + 120573119899

1003817100381710038171003817119878119909lowast

minus 119909lowast1003817100381710038171003817

le (1 minus 120572119899

(] minus 120588120591))1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

+ 120572119899

(1003817100381710038171003817(120588119880 minus 120583119865) (119909

lowast)1003817100381710038171003817 +

1003817100381710038171003817119878119909lowast

minus 119909lowast1003817100381710038171003817)

= (1 minus 120572119899

(] minus 120588120591))1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

+120572119899

(] minus 120588120591)

] minus 120588120591(1003817100381710038171003817(120588119880 minus 120583119865) 119909

lowast1003817100381710038171003817 +1003817100381710038171003817119878119909lowast

minus 119909lowast1003817100381710038171003817)

le max1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

1

] minus 120588120591

times (1003817100381710038171003817(120588119880 minus 120583119865) (119909

lowast)1003817100381710038171003817 +

1003817100381710038171003817119878119909lowast

minus 119909lowast1003817100381710038171003817)

(38)




0minus


lowast + 119878119909


1199090


that 119906119899 119910119899 119878(119909

119899) 119879(119909

119899) 119865(119879(119910

119899)) and 119880(119909

119899) are

bounded





119909119899+1

minus 119909119899 = 0


(119909119899) sub 119865(119879) (119908

119908(119909119899) = 119909

119909119899119894

119909)


= 1198791198651

119903119899(119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899) and 119906

119899minus1=

1198791198651

119903119899minus1(119909119899minus1

+120574119860lowast(1198791198652

119903119899minus1minus119868)119860119909


1003817100381710038171003817119906119899


1003817100381710038171003817

le10038171003817100381710038171003817119909119899


+120574 (119860lowast

(1198791198652

119903119899minus 119868) 119860119909


(1198791198652

119903119899minus1minus 119868) 119860119909

119899minus1)10038171003817100381710038171003817

+

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)

minus (119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)10038171003817100381710038171003817

le1003817100381710038171003817119909119899


minus 120574119860lowast119860 (119909119899


)1003817100381710038171003817

+ 120574 119860100381710038171003817100381710038171198791198652

119903119899119860119909119899

minus 1198791198652

119903119899minus1119860119909119899minus1

10038171003817100381710038171003817

+

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)

minus (119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)10038171003817100381710038171003817

le (1003817100381710038171003817119909119899


1003817100381710038171003817

2

minus 21205741003817100381710038171003817119860 (119909119899


)1003817100381710038171003817

2

+120574211986041003817100381710038171003817119909119899


1003817100381710038171003817

2

)12

+ 120574 119860 (1003817100381710038171003817119860 (119909119899


)1003817100381710038171003817

+

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198652

119903119899119860119909119899

minus 119860119909119899

10038171003817100381710038171003817)

+

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)

minus (119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)10038171003817100381710038171003817

le (1 minus 21205741198602

+ 12057421198604)12 1003817100381710038171003817119909119899


1003817100381710038171003817

+ 1205741198602 1003817100381710038171003817119909119899


1003817100381710038171003817

+ 120574 119860

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198652

119903119899119860119909119899

minus 119860119909119899

10038171003817100381710038171003817


+

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)

minus (119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)10038171003817100381710038171003817

= (1 minus 1205741198602)

1003817100381710038171003817119909119899


1003817100381710038171003817

+ 1205741198602 1003817100381710038171003817119909119899


1003817100381710038171003817

+ 120574 119860

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198652

119903119899119860119909119899

minus 119860119909119899

10038171003817100381710038171003817

+

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)

minus (119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)10038171003817100381710038171003817

=1003817100381710038171003817119909119899


1003817100381710038171003817 + 120574 119860

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198652

119903119899119860119909119899

minus 119860119909119899

10038171003817100381710038171003817

+

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)

minus (119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)10038171003817100381710038171003817

=1003817100381710038171003817119909119899


1003817100381710038171003817 +

10038161003816100381610038161003816100381610038161003816

119903119899


119903119899

10038161003816100381610038161003816100381610038161003816

(120574 119860 120590119899

+ 120594119899)

(39)

where 120590119899

= 1198791198652

119903119899119860119909119899

minus 119860119909119899 and 120594

119899= 1198791198651

119903119899(119909119899

+ 120574119860lowast(1198791198652

119903119899minus

119868)119860119909119899) minus (119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909



gt


1003817100381710038171003817119906119899minus1

minus 119906119899

1003817100381710038171003817 le1003817100381710038171003817119909119899minus1

minus 119909119899

1003817100381710038171003817 +1

120583

1003816100381610038161003816119903119899minus1 minus 119903119899

1003816100381610038161003816 (120574 119860 120590119899

+ 120594119899)

(40)



1003817100381710038171003817

=1003817100381710038171003817120573119899119878119909119899

+ (1 minus 120573119899) 119906119899


119878119909119899minus1

+ (1 minus 120573119899minus1

) 119906119899minus1

)1003817100381710038171003817

=1003817100381710038171003817120573119899

(119878119909119899

minus 119878119909119899minus1

) + (120573119899


) 119878119909119899minus1

+ (1 minus 120573119899) (119906119899


) + (120573119899minus1

minus 120573119899) 119906119899minus1

1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899


1003817100381710038171003817 + (1 minus 120573119899)

1003817100381710038171003817119906119899


1003817100381710038171003817

+1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 (1003817100381710038171003817119878119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119906119899minus1

1003817100381710038171003817)

le 120573119899

1003817100381710038171003817119909119899


1003817100381710038171003817 + (1 minus 120573119899)

times 1003817100381710038171003817119909119899minus1

minus 119909119899

1003817100381710038171003817 +1

120583

1003816100381610038161003816119903119899minus1 minus 119903119899

1003816100381610038161003816 (120574 119860 120590119899

+ 120594119899)

+1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 (1003817100381710038171003817119878119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119906119899minus1

1003817100381710038171003817)

le1003817100381710038171003817119909119899


1003817100381710038171003817 +1

120583

1003816100381610038161003816119903119899minus1 minus 119903119899

1003816100381610038161003816 (120574 119860 120590119899

+ 120594119899)

+1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 (1003817100381710038171003817119878119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119906119899minus1

1003817100381710038171003817)

(41)

Next we estimate

1003817100381710038171003817119909119899+1

minus 119909119899

1003817100381710038171003817

=1003817100381710038171003817119875119862

[119881119899] minus 119875119862

[119881119899minus1

]1003817100381710038171003817

le1003817100381710038171003817120572119899120588 (119880 (119909

119899) minus 119880 (119909

119899minus1)) + (120572

119899minus 120572119899minus1

) 120588119880 (119909119899minus1

)

+ (119868 minus 120572119899120583119865) (119879 (119910

119899)) minus (119868 minus 120572

119899120583119865) 119879 (119910

119899minus1)

+ (119868 minus 120572119899120583119865) (119879 (119910


119899minus1120583119865) (119879 (119910

119899minus1))

1003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899


1003817100381710038171003817 + (1 minus 120572119899]) 1003817100381710038171003817119910119899


1003817100381710038171003817

+1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 (1003817100381710038171003817120588119880 (119909

119899minus1)1003817100381710038171003817 +

1003817100381710038171003817120583119865 (119879 (119910119899minus1

))1003817100381710038171003817)

(42)


1003817100381710038171003817119909119899+1

minus 119909119899

1003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899


1003817100381710038171003817 + (1 minus 120572119899])

times 1003817100381710038171003817119909119899


1003817100381710038171003817 +1

120583

1003816100381610038161003816119903119899minus1 minus 119903119899

1003816100381610038161003816 (120574 119860 120590119899

+ 120594119899)

+1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 (1003817100381710038171003817119878119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119906119899minus1

1003817100381710038171003817)

+1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 (1003817100381710038171003817120588119880 (119909

119899minus1)1003817100381710038171003817 +

1003817100381710038171003817120583119865 (119879 (119910119899minus1

))1003817100381710038171003817)

le (1 minus (] minus 120588120591) 120572119899)

1003817100381710038171003817119909119899


1003817100381710038171003817

+1

120583

1003816100381610038161003816119903119899minus1 minus 119903119899

1003816100381610038161003816 (120574 119860 120590119899

+ 120594119899)

+1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 (1003817100381710038171003817119878119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119906119899minus1

1003817100381710038171003817)

+1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 (1003817100381710038171003817120588119880 (119909

119899minus1)1003817100381710038171003817 +

1003817100381710038171003817120583119865 (119879 (119910119899minus1

))1003817100381710038171003817)

le (1 minus (] minus 120588120591) 120572119899)

1003817100381710038171003817119909119899


1003817100381710038171003817

+ 119872 (1

120583

1003816100381610038161003816119903119899minus1 minus 119903119899

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816)

(43)

where

119872 = maxsup119899ge1

(120574 119860 120590119899

+ 120594119899)

sup119899ge1

(1003817100381710038171003817119878119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119906119899minus1

1003817100381710038171003817)

sup119899ge1

(1003817100381710038171003817120588119880 (119909

119899minus1)1003817100381710038171003817 +

1003817100381710038171003817120583119865 (119879 (119910119899minus1

))1003817100381710038171003817)

(44)


lim119899rarrinfin

1003817100381710038171003817119909119899+1

minus 119909119899

1003817100381710038171003817 = 0 (45)



119906119899minus119909119899 = 0 Since 119909



1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

= ⟨119875119862

[119881119899] minus 119909lowast 119909119899+1


= ⟨119875119862

[119881119899] minus 119881119899 119875119862

[119881119899] minus 119909lowast⟩ + ⟨119881

119899minus 119909lowast 119909119899+1


le ⟨120572119899

(120588119880 (119909119899) minus 120583119865 (119909

lowast) + (119868 minus 120572

119899120583119865) (119879 (119910

119899)))

minus (119868 minus 120572119899120583119865) (119879 (119909

lowast)) 119909119899+1


= ⟨120572119899120588 (119880 (119909

119899) minus 119880 (119909

lowast)) 119909119899+1


+ 120572119899

⟨120588119880 (119909lowast) minus 120583119865 (119909

lowast) 119909119899+1


+ ⟨(119868 minus 120572119899120583119865) (119879 (119910

119899))

minus (119868 minus 120572119899120583119865) (119879 (119909

lowast)) 119909119899+1


le 120572119899120588120591

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

+ 120572119899

⟨120588119880 (119909lowast) minus 120583119865 (119909

lowast) 119909119899+1


+ (1 minus 120572119899]) 1003817100381710038171003817119910119899

minus 119909lowast1003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

le120572119899120588120591

2(1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

)

+ 120572119899

⟨120588119880 (119909lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899])

2(1003817100381710038171003817119910119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

)

le(1 minus 120572

119899(] minus 120588120591))

2

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

+120572119899120588120591

2

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120572119899

⟨120588119880 (119909lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899])

2(120573119899

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+ (1 minus 120573119899)

1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

)

le(1 minus 120572

119899(] minus 120588120591))

2

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

+120572119899120588120591

2

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120572119899

⟨120588119880 (119909lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

2

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+(1 minus 120572

119899]) (1 minus 120573

119899)

2

times 1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120574 (119871120574 minus 1)10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

2

(46)


1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+(1 minus 120572

119899]) (1 minus 120573

119899)

1 + 120572119899

(] minus 120588120591)

times 1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120574 (119871120574 minus 1)10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2


119899]) (1 minus 120573

119899) 120574 (1 minus 119871120574)

1 + 120572119899

(] minus 120588120591)

10038171003817100381710038171003817(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

(47)


(1 minus 120572119899]) (1 minus 120573

119899) 120574 (1 minus 119871120574)

1 + 120572119899

(] minus 120588120591)

10038171003817100381710038171003817(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)


lowast) 119909119899+1


+ 120573119899

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)


lowast) 119909119899+1


+ 120573119899

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+ (1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 +

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817)

times1003817100381710038171003817119909119899+1

minus 119909119899

1003817100381710038171003817

(48)


119909119899+1

minus 119909119899 = 0 120572

119899rarr 0 and

120573119899

rarr 0 we obtain

lim119899rarrinfin

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817= 0 (49)


Since 1198791198651


1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

=100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899) minus 1198791198651

119903119899(119909lowast)10038171003817100381710038171003817

2

le ⟨119906119899

minus 119909lowast 119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909


=1

21003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

+10038171003817100381710038171003817119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899minus 119909lowast10038171003817100381710038171003817

2

minus10038171003817100381710038171003817119906119899

minus 119909lowast

minus [119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899minus 119909lowast]10038171003817100381710038171003817

2

=1

21003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

+10038171003817100381710038171003817119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899minus 119909lowast10038171003817100381710038171003817

2

minus10038171003817100381710038171003817119906119899

minus 119909119899

minus 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

le1

21003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus10038171003817100381710038171003817119906119899

minus 119909119899


(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

2

=1

21003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus [1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

+ 120574210038171003817100381710038171003817

119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

2

minus2120574⟨119906119899

minus 119909119899 119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899⟩]

(50)


1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

le1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

+ 21205741003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

(51)


minus 119909lowast1003817100381710038171003817

2

le(1 minus 120572

119899(] minus 120588120591))

2

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

+120572119899120588120591

2

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120572119899⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899])

2(120573119899

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+ (1 minus 120573119899)

1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

)

le(1 minus 120572

119899(] minus 120588120591))

2

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

+120572119899120588120591

2

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120572119899⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


(1 minus 120572119899])

2

times 120573119899

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+ (1 minus 120573119899)

times (1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

+21205741003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817)

(52)

which implies that

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)


lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+(1 minus 120572

119899]) (1 minus 120573

119899)

1 + 120572119899

(] minus 120588120591)

times 1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

+21205741003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)


lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+(1 minus 120572

119899]) (1 minus 120573

119899)

1 + 120572119899

(] minus 120588120591)

times minus1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

+ 21205741003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

(53)

Hence

(1 minus 120572119899]) (1 minus 120573

119899)

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+2 (1 minus 120572

119899]) (1 minus 120573

119899) 120574

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+2 (1 minus 120572

119899]) (1 minus 120573

119899) 120574

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

+ (1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 +

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817)

1003817100381710038171003817119909119899+1

minus 119909119899

1003817100381710038171003817

(54)



119909119899+1

minus 119909119899 = 0 120572

119899rarr 0 120573

119899rarr 0 and

lim119899rarrinfin

(1198791198652

119903119899minus 119868)119860119909


lim119899rarrinfin

1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817 = 0 (55)


1003817100381710038171003817119909119899

minus 119879 (119909119899)1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119909119899+1

minus 119879 (119909119899)1003817100381710038171003817

=1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119875119862

[119881119899] minus 119875119862

[119879 (119909119899)]

1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817

+1003817100381710038171003817120572119899

(120588119880 (119909119899) minus 120583119865 (119879 (119910

119899)) + 119879 (119910

119899) minus 119879 (119909

119899))

1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817

+ 120572119899

1003817100381710038171003817120588119880 (119909119899) minus 120583119865 (119879 (119910

119899))

1003817100381710038171003817 +1003817100381710038171003817119910119899

minus 119909119899

1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817 + 120572119899

1003817100381710038171003817120588119880 (119909119899) minus 120583119865 (119879 (119910

119899))

1003817100381710038171003817

+1003817100381710038171003817120573119899119878119909119899

+ (1 minus 120573119899) 119906119899

minus 119909119899

1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817 + 120572119899

1003817100381710038171003817120588119880 (119909119899) minus 120583119865 (119879 (119910

119899))

1003817100381710038171003817

+ 120573119899

1003817100381710038171003817119878119909119899

minus 119909119899

1003817100381710038171003817 + (1 minus 120573119899)

1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

(56)


119909119899+1

minus 119909119899 = 0 120572

119899rarr 0 120573

119899rarr 0 120588119880(119909

119899) minus

120583119865(119879(119910119899)) and 119878119909


119899rarrinfin119909119899

minus

119906119899 = 0 we obtain

lim119899rarrinfin

1003817100381710038171003817119909119899

minus 119879 (119909119899)1003817100381710038171003817 = 0 (57)



119909lowast


isin 119865(119879) Therefore 119908119908

(119909119899) sub 119865(119879)







119899=

1198791198651

119903119899(119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899) we have

1198651

(119906119899 119910) +

1

119903119899

⟨119910 minus 119906119899 119906119899

minus 119909119899⟩

minus1

119903119899

⟨119910 minus 119906119899 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899⟩ ge 0 forall119910 isin 119862

(59)


minus1

119903119899

⟨119910 minus 119906119899 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899⟩

+1

119903119899

⟨119910 minus 119906119899 119906119899

minus 119909119899⟩ ge 1198651

(119910 119906119899) forall119910 isin 119862

(60)

minus1

119903119899119896

⟨119910 minus 119906119899119896

120574119860lowast

(1198791198652

119903119899119896

minus 119868) 119860119909119899119896

⟩

+ ⟨119910 minus 119906119899119896

119906119899119896

minus 119909119899119896

119903119899119896

⟩ ge 1198651

(119910 119906119899119896

) forall119910 isin 119862

(61)


119906119899

minus 119909119899 = 0 lim

119899rarrinfin(1198791198652

119903119899minus 119868)119860119909

119899 = 0

and 119909119899



1(119910 119908) le 0 for all 119910 isin 119862


= 119905119910 + (1 minus 119905)119908 wehave 119910


0 = 1198651

(119910119905 119910119905)

le 1199051198651

(119910119905 119910) + (1 minus 119905) 119865

1(119910119905 119908)

le 1199051198651

(119910119905 119910)

(62)



1)


119899 is bounded

and 119909119899


of 119909119899 such

that 119909119899119896


rarr 119860119908 Now set V119899119896

= 119860119909119899119896

minus 1198791198652

119903119899119896

119860119909119899119896 It follows from


V119899119896

= 0 and119860119909119899119896

minusV119899119896

= 1198791198652

119903119899119896

119860119909119899119896Therefore


119903119899119896

we have

1198652

(119860119909119899119896

minus V119899119896

119910)

+1

119903119899119896

⟨119910 minus (119860119909119899119896

minus V119899119896

)

(119860119909119899119896

minus V119899119896

) minus 119860119909119899119896

⟩ ge 0 forall119910 isin 119862

(63)



1198652

(119860119908 119910) ge 0 forall119910 isin 119862 (64)


Thus we have

119908 isin Λ cap 119865 (119879) (65)



119896 ge 120578

lArrrArr 1198962

ge 1205782

lArrrArr 1 minus 2120583120578 + 12058321198962

ge 1 minus 2120583120578 + 12058321205782



lArrrArr 120583120578 ge ]

(66)




119899rarrinfin⟨120588119880(119911)minus120583119865(119911) 119909

119899minus119911⟩ le


119899119896 of

119909119899 such that


⟨120588119880 (119911) minus 120583119865 (119911) 119909119899

minus 119911⟩


⟨120588119880 (119911) minus 120583119865 (119911) 119909119899119896

minus 119911⟩

= ⟨120588119880 (119911) minus 120583119865 (119911) 119908 minus 119911⟩ le 0

(67)



1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

2

= ⟨119875119862

[119881119899] minus 119911 119909

119899+1minus 119911⟩

= ⟨119875119862

[119881119899] minus 119881119899 119875119862

[119881119899] minus 119911⟩ + ⟨119881

119899minus 119911 119909

119899+1minus 119911⟩

le ⟨120572119899

(120588119880 (119909119899) minus 120583119865 (119911)) + (119868 minus 120572

119899120583119865) (119879 (119910

119899))

minus (119868 minus 120572119899120583119865) (119879 (119911)) 119909

119899+1minus 119911⟩

le ⟨120572119899120588 (119880 (119909

119899) minus 119880 (119911)) 119909

119899+1minus 119911⟩

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ ⟨(119868 minus 120572119899120583119865) (119879 (119910

119899)) minus (119868 minus 120572

119899120583119865) (119879 (119911)) 119909

119899+1minus 119911⟩

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ (1 minus 120572119899]) 1003817100381710038171003817119910119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ (1 minus 120572119899]) 120573119899

1003817100381710038171003817119878119909119899

minus 1198781199111003817100381710038171003817 + 120573119899 119878119911 minus 119911

+ (1 minus 120573119899)

1003817100381710038171003817119906119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ (1 minus 120572119899]) 120573119899

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817 + 120573119899 119878119911 minus 119911

+ (1 minus 120573119899)

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

= (1 minus 120572119899

(] minus 120588120591))1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ (1 minus 120572119899]) 120573119899 119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

le1 minus 120572119899

(] minus 120588120591)

2(1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

2

+1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

2

)

+ 120572119899⟨120588119880 (119911) minus 120583119865 (119911) 119909

119899+1minus 119911⟩

+ (1 minus 120572119899]) 120573119899 119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

(68)


minus 1199111003817100381710038171003817

2

le1 minus 120572119899

(] minus 120588120591)

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)⟨120588119880 (119909

119899) minus 120583119865 (119911) 119909

119899+1minus 119911⟩

+2 (1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

le (1 minus 120572119899

(] minus 120588120591))1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

2

+2120572119899

(] minus 120588120591)

1 + 120572119899

(] minus 120588120591)

times 1

] minus 120588120591⟨120588119880 (119911) minus 120583119865 (119911) 119909

119899+1minus 119911⟩

+(1 minus 120572

119899]) 120573119899

120572119899

(] minus 120588120591)119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

(69)

Let 120574119899

= 120572119899(] minus 120588120591) and 120575

119899= (2120572

119899(] minus 120588120591)(1 +


119899+1minus 119911⟩ + ((1 minus

120572119899])120573119899120572119899(] minus 120588120591))119878119911 minus 119911119909

119899+1minus 119911

Sinceinfin

sum

119899=1

120572119899

= infin


1

] minus 120588120591⟨120588119880 (119911) minus 120583119865 (119911) 119909

119899+1minus 119911⟩

+(1 minus 120572

119899]) 120573119899

120572119899

(] minus 120588120591)119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817 le 0

(70)


sum

119899=1

120574119899


120575119899

120574119899

le 0 (71)







4 Conclusions




Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















119899 119909119899 and 119910


119906119899

= 1198791198651

119903119899(119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)

119910119899

= 120573119899119878119909119899

+ (1 minus 120573119899) 119906119899

119909119899+1

= 119875119862

[120572119899120588119880 (119909

119899) + (119868 minus 120572

119899120583119865) (119879 (119910

119899))] forall119899 ge 0

(33)




2) 0 le 120588120591 lt ] where





120572119899

= 0 and suminfin

119899=1120572119899

= infin


(120573119899120572119899) = 0

(c) suminfin

119899=1|120572119899minus1


infin

119899=1|120573119899minus1



119903119899


119903119899


119899=1|119903119899minus1

minus

119903119899| lt infin










119899 are

bounded



= 1198791198651

119903119899(119909lowast) and 119860119909

lowast=

1198791198652

119903119899(119860119909lowast) Then

1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

=100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899) minus 119909lowast10038171003817100381710038171003817

2

=100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899) minus 1198791198651

119903119899(119909lowast)10038171003817100381710038171003817

2

le10038171003817100381710038171003817119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899minus 119909lowast10038171003817100381710038171003817

2

=1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120574210038171003817100381710038171003817

119860lowast(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

+ 2120574 ⟨119909119899


(1198791198652

119903119899minus 119868) 119860119909

119899⟩

=1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 1205742

⟨(1198791198652

119903119899minus 119868) 119860119909

119899 119860119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899⟩

+ 2120574 ⟨119909119899


(1198791198652

119903119899minus 119868) 119860119909

119899⟩

(34)


1205742

⟨(1198791198652

119903119899minus 119868) 119860119909

119899 119860119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899⟩

le 1198711205742

⟨(1198791198652

119903119899minus 119868) 119860119909

119899 (1198791198652

119903119899minus 119868) 119860119909

119899⟩

= 119871120574210038171003817100381710038171003817

(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

(35)


2120574 ⟨119909119899


(1198791198652

119903119899minus 119868) 119860119909

119899⟩

= 2120574 ⟨119860 (119909119899

minus 119909lowast) (1198791198652

119903119899minus 119868) 119860119909

119899⟩

= 2120574 ⟨119860 (119909119899

minus 119909lowast) + (119879

1198652

119903119899minus 119868) 119860119909

119899

minus (1198791198652

119903119899minus 119868) 119860119909

119899 (1198791198652

119903119899minus 119868) 119860119909

119899⟩

= 2120574 (⟨1198791198652

119903119899119860119909119899

minus 119860119909lowast (1198791198652

119903119899minus 119868) 119860119909

119899⟩

minus10038171003817100381710038171003817(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

)

le 2120574 (1

2

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

2

minus10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

2

)

= minus12057410038171003817100381710038171003817(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

(36)


1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

le1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120574 (119871120574 minus 1)10038171003817100381710038171003817(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

le1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

(37)


Denote 119881119899

= 120572119899120588119880(119909119899) + (119868 minus 120572

119899120583119865)(119879(119910





1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

=1003817100381710038171003817119875119862

[119881119899] minus 119875119862

[119909lowast]1003817100381710038171003817

le1003817100381710038171003817120572119899120588119880 (119909

119899) + (119868 minus 120572

119899120583119865) (119879 (119910

119899)) minus 119909

lowast1003817100381710038171003817

le 120572119899

1003817100381710038171003817120588119880 (119909119899) minus 120583119865 (119909

lowast)1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120572

119899120583119865) (119879 (119910

119899)) minus (119868 minus 120572

119899120583119865) 119879 (119909

lowast)1003817100381710038171003817

= 120572119899

1003817100381710038171003817120588119880 (119909119899) minus 120588119880 (119909

lowast) + (120588119880 minus 120583119865) (119909

lowast)1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120572

119899120583119865) (119879 (119910

119899)) minus (119868 minus 120572

119899120583119865) 119879 (119909

lowast)1003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 + 120572

119899

1003817100381710038171003817(120588119880 minus 120583119865) (119909lowast)1003817100381710038171003817

+ (1 minus 120572119899]) 1003817100381710038171003817119910119899

minus 119909lowast1003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 + 120572

119899

1003817100381710038171003817(120588119880 minus 120583119865) (119909lowast)1003817100381710038171003817

+ (1 minus 120572119899]) 1003817100381710038171003817120573119899119878119909119899

+ (1 minus 120573119899) 119906119899

minus 119909lowast1003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 + 120572

119899

1003817100381710038171003817(120588119880 minus 120583119865) (119909lowast)1003817100381710038171003817

+ (1 minus 120572119899]) (120573119899

1003817100381710038171003817119878119909119899

minus 119878119909lowast1003817100381710038171003817 + 120573

119899

1003817100381710038171003817119878119909lowast

minus 119909lowast1003817100381710038171003817

+ (1 minus 120573119899)

1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817)

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 + 120572

119899

1003817100381710038171003817(120588119880 minus 120583119865) (119909lowast)1003817100381710038171003817

+ (1 minus 120572119899]) (120573119899

1003817100381710038171003817119878119909119899

minus 119878119909lowast1003817100381710038171003817 + 120573

119899

1003817100381710038171003817119878119909lowast

minus 119909lowast1003817100381710038171003817

+ (1 minus 120573119899)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817)

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 + 120572

119899

1003817100381710038171003817(120588119880 minus 120583119865) (119909lowast)1003817100381710038171003817

+ (1 minus 120572119899]) (120573119899

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 + 120573

119899

1003817100381710038171003817119878119909lowast

minus 119909lowast1003817100381710038171003817

+ (1 minus 120573119899)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817)

= (1 minus 120572119899

(] minus 120588120591))1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

+ 120572119899

1003817100381710038171003817(120588119880 minus 120583119865) (119909lowast)1003817100381710038171003817

+ (1 minus 120572119899]) 120573119899

1003817100381710038171003817119878119909lowast

minus 119909lowast1003817100381710038171003817

le (1 minus 120572119899

(] minus 120588120591))1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

+ 120572119899

1003817100381710038171003817(120588119880 minus 120583119865) (119909lowast)1003817100381710038171003817 + 120573119899

1003817100381710038171003817119878119909lowast

minus 119909lowast1003817100381710038171003817

le (1 minus 120572119899

(] minus 120588120591))1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

+ 120572119899

(1003817100381710038171003817(120588119880 minus 120583119865) (119909

lowast)1003817100381710038171003817 +

1003817100381710038171003817119878119909lowast

minus 119909lowast1003817100381710038171003817)

= (1 minus 120572119899

(] minus 120588120591))1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

+120572119899

(] minus 120588120591)

] minus 120588120591(1003817100381710038171003817(120588119880 minus 120583119865) 119909

lowast1003817100381710038171003817 +1003817100381710038171003817119878119909lowast

minus 119909lowast1003817100381710038171003817)

le max1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

1

] minus 120588120591

times (1003817100381710038171003817(120588119880 minus 120583119865) (119909

lowast)1003817100381710038171003817 +

1003817100381710038171003817119878119909lowast

minus 119909lowast1003817100381710038171003817)

(38)




0minus


lowast + 119878119909


1199090


that 119906119899 119910119899 119878(119909

119899) 119879(119909

119899) 119865(119879(119910

119899)) and 119880(119909

119899) are

bounded





119909119899+1

minus 119909119899 = 0


(119909119899) sub 119865(119879) (119908

119908(119909119899) = 119909

119909119899119894

119909)


= 1198791198651

119903119899(119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899) and 119906

119899minus1=

1198791198651

119903119899minus1(119909119899minus1

+120574119860lowast(1198791198652

119903119899minus1minus119868)119860119909


1003817100381710038171003817119906119899


1003817100381710038171003817

le10038171003817100381710038171003817119909119899


+120574 (119860lowast

(1198791198652

119903119899minus 119868) 119860119909


(1198791198652

119903119899minus1minus 119868) 119860119909

119899minus1)10038171003817100381710038171003817

+

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)

minus (119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)10038171003817100381710038171003817

le1003817100381710038171003817119909119899


minus 120574119860lowast119860 (119909119899


)1003817100381710038171003817

+ 120574 119860100381710038171003817100381710038171198791198652

119903119899119860119909119899

minus 1198791198652

119903119899minus1119860119909119899minus1

10038171003817100381710038171003817

+

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)

minus (119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)10038171003817100381710038171003817

le (1003817100381710038171003817119909119899


1003817100381710038171003817

2

minus 21205741003817100381710038171003817119860 (119909119899


)1003817100381710038171003817

2

+120574211986041003817100381710038171003817119909119899


1003817100381710038171003817

2

)12

+ 120574 119860 (1003817100381710038171003817119860 (119909119899


)1003817100381710038171003817

+

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198652

119903119899119860119909119899

minus 119860119909119899

10038171003817100381710038171003817)

+

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)

minus (119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)10038171003817100381710038171003817

le (1 minus 21205741198602

+ 12057421198604)12 1003817100381710038171003817119909119899


1003817100381710038171003817

+ 1205741198602 1003817100381710038171003817119909119899


1003817100381710038171003817

+ 120574 119860

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198652

119903119899119860119909119899

minus 119860119909119899

10038171003817100381710038171003817


+

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)

minus (119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)10038171003817100381710038171003817

= (1 minus 1205741198602)

1003817100381710038171003817119909119899


1003817100381710038171003817

+ 1205741198602 1003817100381710038171003817119909119899


1003817100381710038171003817

+ 120574 119860

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198652

119903119899119860119909119899

minus 119860119909119899

10038171003817100381710038171003817

+

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)

minus (119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)10038171003817100381710038171003817

=1003817100381710038171003817119909119899


1003817100381710038171003817 + 120574 119860

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198652

119903119899119860119909119899

minus 119860119909119899

10038171003817100381710038171003817

+

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)

minus (119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)10038171003817100381710038171003817

=1003817100381710038171003817119909119899


1003817100381710038171003817 +

10038161003816100381610038161003816100381610038161003816

119903119899


119903119899

10038161003816100381610038161003816100381610038161003816

(120574 119860 120590119899

+ 120594119899)

(39)

where 120590119899

= 1198791198652

119903119899119860119909119899

minus 119860119909119899 and 120594

119899= 1198791198651

119903119899(119909119899

+ 120574119860lowast(1198791198652

119903119899minus

119868)119860119909119899) minus (119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909



gt


1003817100381710038171003817119906119899minus1

minus 119906119899

1003817100381710038171003817 le1003817100381710038171003817119909119899minus1

minus 119909119899

1003817100381710038171003817 +1

120583

1003816100381610038161003816119903119899minus1 minus 119903119899

1003816100381610038161003816 (120574 119860 120590119899

+ 120594119899)

(40)



1003817100381710038171003817

=1003817100381710038171003817120573119899119878119909119899

+ (1 minus 120573119899) 119906119899


119878119909119899minus1

+ (1 minus 120573119899minus1

) 119906119899minus1

)1003817100381710038171003817

=1003817100381710038171003817120573119899

(119878119909119899

minus 119878119909119899minus1

) + (120573119899


) 119878119909119899minus1

+ (1 minus 120573119899) (119906119899


) + (120573119899minus1

minus 120573119899) 119906119899minus1

1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899


1003817100381710038171003817 + (1 minus 120573119899)

1003817100381710038171003817119906119899


1003817100381710038171003817

+1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 (1003817100381710038171003817119878119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119906119899minus1

1003817100381710038171003817)

le 120573119899

1003817100381710038171003817119909119899


1003817100381710038171003817 + (1 minus 120573119899)

times 1003817100381710038171003817119909119899minus1

minus 119909119899

1003817100381710038171003817 +1

120583

1003816100381610038161003816119903119899minus1 minus 119903119899

1003816100381610038161003816 (120574 119860 120590119899

+ 120594119899)

+1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 (1003817100381710038171003817119878119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119906119899minus1

1003817100381710038171003817)

le1003817100381710038171003817119909119899


1003817100381710038171003817 +1

120583

1003816100381610038161003816119903119899minus1 minus 119903119899

1003816100381610038161003816 (120574 119860 120590119899

+ 120594119899)

+1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 (1003817100381710038171003817119878119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119906119899minus1

1003817100381710038171003817)

(41)

Next we estimate

1003817100381710038171003817119909119899+1

minus 119909119899

1003817100381710038171003817

=1003817100381710038171003817119875119862

[119881119899] minus 119875119862

[119881119899minus1

]1003817100381710038171003817

le1003817100381710038171003817120572119899120588 (119880 (119909

119899) minus 119880 (119909

119899minus1)) + (120572

119899minus 120572119899minus1

) 120588119880 (119909119899minus1

)

+ (119868 minus 120572119899120583119865) (119879 (119910

119899)) minus (119868 minus 120572

119899120583119865) 119879 (119910

119899minus1)

+ (119868 minus 120572119899120583119865) (119879 (119910


119899minus1120583119865) (119879 (119910

119899minus1))

1003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899


1003817100381710038171003817 + (1 minus 120572119899]) 1003817100381710038171003817119910119899


1003817100381710038171003817

+1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 (1003817100381710038171003817120588119880 (119909

119899minus1)1003817100381710038171003817 +

1003817100381710038171003817120583119865 (119879 (119910119899minus1

))1003817100381710038171003817)

(42)


1003817100381710038171003817119909119899+1

minus 119909119899

1003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899


1003817100381710038171003817 + (1 minus 120572119899])

times 1003817100381710038171003817119909119899


1003817100381710038171003817 +1

120583

1003816100381610038161003816119903119899minus1 minus 119903119899

1003816100381610038161003816 (120574 119860 120590119899

+ 120594119899)

+1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 (1003817100381710038171003817119878119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119906119899minus1

1003817100381710038171003817)

+1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 (1003817100381710038171003817120588119880 (119909

119899minus1)1003817100381710038171003817 +

1003817100381710038171003817120583119865 (119879 (119910119899minus1

))1003817100381710038171003817)

le (1 minus (] minus 120588120591) 120572119899)

1003817100381710038171003817119909119899


1003817100381710038171003817

+1

120583

1003816100381610038161003816119903119899minus1 minus 119903119899

1003816100381610038161003816 (120574 119860 120590119899

+ 120594119899)

+1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 (1003817100381710038171003817119878119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119906119899minus1

1003817100381710038171003817)

+1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 (1003817100381710038171003817120588119880 (119909

119899minus1)1003817100381710038171003817 +

1003817100381710038171003817120583119865 (119879 (119910119899minus1

))1003817100381710038171003817)

le (1 minus (] minus 120588120591) 120572119899)

1003817100381710038171003817119909119899


1003817100381710038171003817

+ 119872 (1

120583

1003816100381610038161003816119903119899minus1 minus 119903119899

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816)

(43)

where

119872 = maxsup119899ge1

(120574 119860 120590119899

+ 120594119899)

sup119899ge1

(1003817100381710038171003817119878119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119906119899minus1

1003817100381710038171003817)

sup119899ge1

(1003817100381710038171003817120588119880 (119909

119899minus1)1003817100381710038171003817 +

1003817100381710038171003817120583119865 (119879 (119910119899minus1

))1003817100381710038171003817)

(44)


lim119899rarrinfin

1003817100381710038171003817119909119899+1

minus 119909119899

1003817100381710038171003817 = 0 (45)



119906119899minus119909119899 = 0 Since 119909



1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

= ⟨119875119862

[119881119899] minus 119909lowast 119909119899+1


= ⟨119875119862

[119881119899] minus 119881119899 119875119862

[119881119899] minus 119909lowast⟩ + ⟨119881

119899minus 119909lowast 119909119899+1


le ⟨120572119899

(120588119880 (119909119899) minus 120583119865 (119909

lowast) + (119868 minus 120572

119899120583119865) (119879 (119910

119899)))

minus (119868 minus 120572119899120583119865) (119879 (119909

lowast)) 119909119899+1


= ⟨120572119899120588 (119880 (119909

119899) minus 119880 (119909

lowast)) 119909119899+1


+ 120572119899

⟨120588119880 (119909lowast) minus 120583119865 (119909

lowast) 119909119899+1


+ ⟨(119868 minus 120572119899120583119865) (119879 (119910

119899))

minus (119868 minus 120572119899120583119865) (119879 (119909

lowast)) 119909119899+1


le 120572119899120588120591

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

+ 120572119899

⟨120588119880 (119909lowast) minus 120583119865 (119909

lowast) 119909119899+1


+ (1 minus 120572119899]) 1003817100381710038171003817119910119899

minus 119909lowast1003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

le120572119899120588120591

2(1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

)

+ 120572119899

⟨120588119880 (119909lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899])

2(1003817100381710038171003817119910119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

)

le(1 minus 120572

119899(] minus 120588120591))

2

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

+120572119899120588120591

2

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120572119899

⟨120588119880 (119909lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899])

2(120573119899

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+ (1 minus 120573119899)

1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

)

le(1 minus 120572

119899(] minus 120588120591))

2

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

+120572119899120588120591

2

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120572119899

⟨120588119880 (119909lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

2

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+(1 minus 120572

119899]) (1 minus 120573

119899)

2

times 1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120574 (119871120574 minus 1)10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

2

(46)


1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+(1 minus 120572

119899]) (1 minus 120573

119899)

1 + 120572119899

(] minus 120588120591)

times 1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120574 (119871120574 minus 1)10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2


119899]) (1 minus 120573

119899) 120574 (1 minus 119871120574)

1 + 120572119899

(] minus 120588120591)

10038171003817100381710038171003817(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

(47)


(1 minus 120572119899]) (1 minus 120573

119899) 120574 (1 minus 119871120574)

1 + 120572119899

(] minus 120588120591)

10038171003817100381710038171003817(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)


lowast) 119909119899+1


+ 120573119899

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)


lowast) 119909119899+1


+ 120573119899

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+ (1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 +

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817)

times1003817100381710038171003817119909119899+1

minus 119909119899

1003817100381710038171003817

(48)


119909119899+1

minus 119909119899 = 0 120572

119899rarr 0 and

120573119899

rarr 0 we obtain

lim119899rarrinfin

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817= 0 (49)


Since 1198791198651


1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

=100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899) minus 1198791198651

119903119899(119909lowast)10038171003817100381710038171003817

2

le ⟨119906119899

minus 119909lowast 119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909


=1

21003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

+10038171003817100381710038171003817119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899minus 119909lowast10038171003817100381710038171003817

2

minus10038171003817100381710038171003817119906119899

minus 119909lowast

minus [119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899minus 119909lowast]10038171003817100381710038171003817

2

=1

21003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

+10038171003817100381710038171003817119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899minus 119909lowast10038171003817100381710038171003817

2

minus10038171003817100381710038171003817119906119899

minus 119909119899

minus 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

le1

21003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus10038171003817100381710038171003817119906119899

minus 119909119899


(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

2

=1

21003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus [1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

+ 120574210038171003817100381710038171003817

119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

2

minus2120574⟨119906119899

minus 119909119899 119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899⟩]

(50)


1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

le1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

+ 21205741003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

(51)


minus 119909lowast1003817100381710038171003817

2

le(1 minus 120572

119899(] minus 120588120591))

2

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

+120572119899120588120591

2

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120572119899⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899])

2(120573119899

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+ (1 minus 120573119899)

1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

)

le(1 minus 120572

119899(] minus 120588120591))

2

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

+120572119899120588120591

2

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120572119899⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


(1 minus 120572119899])

2

times 120573119899

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+ (1 minus 120573119899)

times (1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

+21205741003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817)

(52)

which implies that

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)


lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+(1 minus 120572

119899]) (1 minus 120573

119899)

1 + 120572119899

(] minus 120588120591)

times 1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

+21205741003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)


lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+(1 minus 120572

119899]) (1 minus 120573

119899)

1 + 120572119899

(] minus 120588120591)

times minus1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

+ 21205741003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

(53)

Hence

(1 minus 120572119899]) (1 minus 120573

119899)

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+2 (1 minus 120572

119899]) (1 minus 120573

119899) 120574

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+2 (1 minus 120572

119899]) (1 minus 120573

119899) 120574

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

+ (1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 +

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817)

1003817100381710038171003817119909119899+1

minus 119909119899

1003817100381710038171003817

(54)



119909119899+1

minus 119909119899 = 0 120572

119899rarr 0 120573

119899rarr 0 and

lim119899rarrinfin

(1198791198652

119903119899minus 119868)119860119909


lim119899rarrinfin

1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817 = 0 (55)


1003817100381710038171003817119909119899

minus 119879 (119909119899)1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119909119899+1

minus 119879 (119909119899)1003817100381710038171003817

=1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119875119862

[119881119899] minus 119875119862

[119879 (119909119899)]

1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817

+1003817100381710038171003817120572119899

(120588119880 (119909119899) minus 120583119865 (119879 (119910

119899)) + 119879 (119910

119899) minus 119879 (119909

119899))

1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817

+ 120572119899

1003817100381710038171003817120588119880 (119909119899) minus 120583119865 (119879 (119910

119899))

1003817100381710038171003817 +1003817100381710038171003817119910119899

minus 119909119899

1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817 + 120572119899

1003817100381710038171003817120588119880 (119909119899) minus 120583119865 (119879 (119910

119899))

1003817100381710038171003817

+1003817100381710038171003817120573119899119878119909119899

+ (1 minus 120573119899) 119906119899

minus 119909119899

1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817 + 120572119899

1003817100381710038171003817120588119880 (119909119899) minus 120583119865 (119879 (119910

119899))

1003817100381710038171003817

+ 120573119899

1003817100381710038171003817119878119909119899

minus 119909119899

1003817100381710038171003817 + (1 minus 120573119899)

1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

(56)


119909119899+1

minus 119909119899 = 0 120572

119899rarr 0 120573

119899rarr 0 120588119880(119909

119899) minus

120583119865(119879(119910119899)) and 119878119909


119899rarrinfin119909119899

minus

119906119899 = 0 we obtain

lim119899rarrinfin

1003817100381710038171003817119909119899

minus 119879 (119909119899)1003817100381710038171003817 = 0 (57)



119909lowast


isin 119865(119879) Therefore 119908119908

(119909119899) sub 119865(119879)







119899=

1198791198651

119903119899(119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899) we have

1198651

(119906119899 119910) +

1

119903119899

⟨119910 minus 119906119899 119906119899

minus 119909119899⟩

minus1

119903119899

⟨119910 minus 119906119899 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899⟩ ge 0 forall119910 isin 119862

(59)


minus1

119903119899

⟨119910 minus 119906119899 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899⟩

+1

119903119899

⟨119910 minus 119906119899 119906119899

minus 119909119899⟩ ge 1198651

(119910 119906119899) forall119910 isin 119862

(60)

minus1

119903119899119896

⟨119910 minus 119906119899119896

120574119860lowast

(1198791198652

119903119899119896

minus 119868) 119860119909119899119896

⟩

+ ⟨119910 minus 119906119899119896

119906119899119896

minus 119909119899119896

119903119899119896

⟩ ge 1198651

(119910 119906119899119896

) forall119910 isin 119862

(61)


119906119899

minus 119909119899 = 0 lim

119899rarrinfin(1198791198652

119903119899minus 119868)119860119909

119899 = 0

and 119909119899



1(119910 119908) le 0 for all 119910 isin 119862


= 119905119910 + (1 minus 119905)119908 wehave 119910


0 = 1198651

(119910119905 119910119905)

le 1199051198651

(119910119905 119910) + (1 minus 119905) 119865

1(119910119905 119908)

le 1199051198651

(119910119905 119910)

(62)



1)


119899 is bounded

and 119909119899


of 119909119899 such

that 119909119899119896


rarr 119860119908 Now set V119899119896

= 119860119909119899119896

minus 1198791198652

119903119899119896

119860119909119899119896 It follows from


V119899119896

= 0 and119860119909119899119896

minusV119899119896

= 1198791198652

119903119899119896

119860119909119899119896Therefore


119903119899119896

we have

1198652

(119860119909119899119896

minus V119899119896

119910)

+1

119903119899119896

⟨119910 minus (119860119909119899119896

minus V119899119896

)

(119860119909119899119896

minus V119899119896

) minus 119860119909119899119896

⟩ ge 0 forall119910 isin 119862

(63)



1198652

(119860119908 119910) ge 0 forall119910 isin 119862 (64)


Thus we have

119908 isin Λ cap 119865 (119879) (65)



119896 ge 120578

lArrrArr 1198962

ge 1205782

lArrrArr 1 minus 2120583120578 + 12058321198962

ge 1 minus 2120583120578 + 12058321205782



lArrrArr 120583120578 ge ]

(66)




119899rarrinfin⟨120588119880(119911)minus120583119865(119911) 119909

119899minus119911⟩ le


119899119896 of

119909119899 such that


⟨120588119880 (119911) minus 120583119865 (119911) 119909119899

minus 119911⟩


⟨120588119880 (119911) minus 120583119865 (119911) 119909119899119896

minus 119911⟩

= ⟨120588119880 (119911) minus 120583119865 (119911) 119908 minus 119911⟩ le 0

(67)



1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

2

= ⟨119875119862

[119881119899] minus 119911 119909

119899+1minus 119911⟩

= ⟨119875119862

[119881119899] minus 119881119899 119875119862

[119881119899] minus 119911⟩ + ⟨119881

119899minus 119911 119909

119899+1minus 119911⟩

le ⟨120572119899

(120588119880 (119909119899) minus 120583119865 (119911)) + (119868 minus 120572

119899120583119865) (119879 (119910

119899))

minus (119868 minus 120572119899120583119865) (119879 (119911)) 119909

119899+1minus 119911⟩

le ⟨120572119899120588 (119880 (119909

119899) minus 119880 (119911)) 119909

119899+1minus 119911⟩

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ ⟨(119868 minus 120572119899120583119865) (119879 (119910

119899)) minus (119868 minus 120572

119899120583119865) (119879 (119911)) 119909

119899+1minus 119911⟩

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ (1 minus 120572119899]) 1003817100381710038171003817119910119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ (1 minus 120572119899]) 120573119899

1003817100381710038171003817119878119909119899

minus 1198781199111003817100381710038171003817 + 120573119899 119878119911 minus 119911

+ (1 minus 120573119899)

1003817100381710038171003817119906119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ (1 minus 120572119899]) 120573119899

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817 + 120573119899 119878119911 minus 119911

+ (1 minus 120573119899)

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

= (1 minus 120572119899

(] minus 120588120591))1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ (1 minus 120572119899]) 120573119899 119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

le1 minus 120572119899

(] minus 120588120591)

2(1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

2

+1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

2

)

+ 120572119899⟨120588119880 (119911) minus 120583119865 (119911) 119909

119899+1minus 119911⟩

+ (1 minus 120572119899]) 120573119899 119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

(68)


minus 1199111003817100381710038171003817

2

le1 minus 120572119899

(] minus 120588120591)

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)⟨120588119880 (119909

119899) minus 120583119865 (119911) 119909

119899+1minus 119911⟩

+2 (1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

le (1 minus 120572119899

(] minus 120588120591))1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

2

+2120572119899

(] minus 120588120591)

1 + 120572119899

(] minus 120588120591)

times 1

] minus 120588120591⟨120588119880 (119911) minus 120583119865 (119911) 119909

119899+1minus 119911⟩

+(1 minus 120572

119899]) 120573119899

120572119899

(] minus 120588120591)119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

(69)

Let 120574119899

= 120572119899(] minus 120588120591) and 120575

119899= (2120572

119899(] minus 120588120591)(1 +


119899+1minus 119911⟩ + ((1 minus

120572119899])120573119899120572119899(] minus 120588120591))119878119911 minus 119911119909

119899+1minus 119911

Sinceinfin

sum

119899=1

120572119899

= infin


1

] minus 120588120591⟨120588119880 (119911) minus 120583119865 (119911) 119909

119899+1minus 119911⟩

+(1 minus 120572

119899]) 120573119899

120572119899

(] minus 120588120591)119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817 le 0

(70)


sum

119899=1

120574119899


120575119899

120574119899

le 0 (71)







4 Conclusions




Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Denote 119881119899

= 120572119899120588119880(119909119899) + (119868 minus 120572

119899120583119865)(119879(119910





1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

=1003817100381710038171003817119875119862

[119881119899] minus 119875119862

[119909lowast]1003817100381710038171003817

le1003817100381710038171003817120572119899120588119880 (119909

119899) + (119868 minus 120572

119899120583119865) (119879 (119910

119899)) minus 119909

lowast1003817100381710038171003817

le 120572119899

1003817100381710038171003817120588119880 (119909119899) minus 120583119865 (119909

lowast)1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120572

119899120583119865) (119879 (119910

119899)) minus (119868 minus 120572

119899120583119865) 119879 (119909

lowast)1003817100381710038171003817

= 120572119899

1003817100381710038171003817120588119880 (119909119899) minus 120588119880 (119909

lowast) + (120588119880 minus 120583119865) (119909

lowast)1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120572

119899120583119865) (119879 (119910

119899)) minus (119868 minus 120572

119899120583119865) 119879 (119909

lowast)1003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 + 120572

119899

1003817100381710038171003817(120588119880 minus 120583119865) (119909lowast)1003817100381710038171003817

+ (1 minus 120572119899]) 1003817100381710038171003817119910119899

minus 119909lowast1003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 + 120572

119899

1003817100381710038171003817(120588119880 minus 120583119865) (119909lowast)1003817100381710038171003817

+ (1 minus 120572119899]) 1003817100381710038171003817120573119899119878119909119899

+ (1 minus 120573119899) 119906119899

minus 119909lowast1003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 + 120572

119899

1003817100381710038171003817(120588119880 minus 120583119865) (119909lowast)1003817100381710038171003817

+ (1 minus 120572119899]) (120573119899

1003817100381710038171003817119878119909119899

minus 119878119909lowast1003817100381710038171003817 + 120573

119899

1003817100381710038171003817119878119909lowast

minus 119909lowast1003817100381710038171003817

+ (1 minus 120573119899)

1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817)

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 + 120572

119899

1003817100381710038171003817(120588119880 minus 120583119865) (119909lowast)1003817100381710038171003817

+ (1 minus 120572119899]) (120573119899

1003817100381710038171003817119878119909119899

minus 119878119909lowast1003817100381710038171003817 + 120573

119899

1003817100381710038171003817119878119909lowast

minus 119909lowast1003817100381710038171003817

+ (1 minus 120573119899)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817)

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 + 120572

119899

1003817100381710038171003817(120588119880 minus 120583119865) (119909lowast)1003817100381710038171003817

+ (1 minus 120572119899]) (120573119899

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 + 120573

119899

1003817100381710038171003817119878119909lowast

minus 119909lowast1003817100381710038171003817

+ (1 minus 120573119899)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817)

= (1 minus 120572119899

(] minus 120588120591))1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

+ 120572119899

1003817100381710038171003817(120588119880 minus 120583119865) (119909lowast)1003817100381710038171003817

+ (1 minus 120572119899]) 120573119899

1003817100381710038171003817119878119909lowast

minus 119909lowast1003817100381710038171003817

le (1 minus 120572119899

(] minus 120588120591))1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

+ 120572119899

1003817100381710038171003817(120588119880 minus 120583119865) (119909lowast)1003817100381710038171003817 + 120573119899

1003817100381710038171003817119878119909lowast

minus 119909lowast1003817100381710038171003817

le (1 minus 120572119899

(] minus 120588120591))1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

+ 120572119899

(1003817100381710038171003817(120588119880 minus 120583119865) (119909

lowast)1003817100381710038171003817 +

1003817100381710038171003817119878119909lowast

minus 119909lowast1003817100381710038171003817)

= (1 minus 120572119899

(] minus 120588120591))1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

+120572119899

(] minus 120588120591)

] minus 120588120591(1003817100381710038171003817(120588119880 minus 120583119865) 119909

lowast1003817100381710038171003817 +1003817100381710038171003817119878119909lowast

minus 119909lowast1003817100381710038171003817)

le max1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

1

] minus 120588120591

times (1003817100381710038171003817(120588119880 minus 120583119865) (119909

lowast)1003817100381710038171003817 +

1003817100381710038171003817119878119909lowast

minus 119909lowast1003817100381710038171003817)

(38)




0minus


lowast + 119878119909


1199090


that 119906119899 119910119899 119878(119909

119899) 119879(119909

119899) 119865(119879(119910

119899)) and 119880(119909

119899) are

bounded





119909119899+1

minus 119909119899 = 0


(119909119899) sub 119865(119879) (119908

119908(119909119899) = 119909

119909119899119894

119909)


= 1198791198651

119903119899(119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899) and 119906

119899minus1=

1198791198651

119903119899minus1(119909119899minus1

+120574119860lowast(1198791198652

119903119899minus1minus119868)119860119909


1003817100381710038171003817119906119899


1003817100381710038171003817

le10038171003817100381710038171003817119909119899


+120574 (119860lowast

(1198791198652

119903119899minus 119868) 119860119909


(1198791198652

119903119899minus1minus 119868) 119860119909

119899minus1)10038171003817100381710038171003817

+

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)

minus (119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)10038171003817100381710038171003817

le1003817100381710038171003817119909119899


minus 120574119860lowast119860 (119909119899


)1003817100381710038171003817

+ 120574 119860100381710038171003817100381710038171198791198652

119903119899119860119909119899

minus 1198791198652

119903119899minus1119860119909119899minus1

10038171003817100381710038171003817

+

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)

minus (119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)10038171003817100381710038171003817

le (1003817100381710038171003817119909119899


1003817100381710038171003817

2

minus 21205741003817100381710038171003817119860 (119909119899


)1003817100381710038171003817

2

+120574211986041003817100381710038171003817119909119899


1003817100381710038171003817

2

)12

+ 120574 119860 (1003817100381710038171003817119860 (119909119899


)1003817100381710038171003817

+

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198652

119903119899119860119909119899

minus 119860119909119899

10038171003817100381710038171003817)

+

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)

minus (119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)10038171003817100381710038171003817

le (1 minus 21205741198602

+ 12057421198604)12 1003817100381710038171003817119909119899


1003817100381710038171003817

+ 1205741198602 1003817100381710038171003817119909119899


1003817100381710038171003817

+ 120574 119860

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198652

119903119899119860119909119899

minus 119860119909119899

10038171003817100381710038171003817


+

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)

minus (119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)10038171003817100381710038171003817

= (1 minus 1205741198602)

1003817100381710038171003817119909119899


1003817100381710038171003817

+ 1205741198602 1003817100381710038171003817119909119899


1003817100381710038171003817

+ 120574 119860

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198652

119903119899119860119909119899

minus 119860119909119899

10038171003817100381710038171003817

+

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)

minus (119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)10038171003817100381710038171003817

=1003817100381710038171003817119909119899


1003817100381710038171003817 + 120574 119860

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198652

119903119899119860119909119899

minus 119860119909119899

10038171003817100381710038171003817

+

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)

minus (119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)10038171003817100381710038171003817

=1003817100381710038171003817119909119899


1003817100381710038171003817 +

10038161003816100381610038161003816100381610038161003816

119903119899


119903119899

10038161003816100381610038161003816100381610038161003816

(120574 119860 120590119899

+ 120594119899)

(39)

where 120590119899

= 1198791198652

119903119899119860119909119899

minus 119860119909119899 and 120594

119899= 1198791198651

119903119899(119909119899

+ 120574119860lowast(1198791198652

119903119899minus

119868)119860119909119899) minus (119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909



gt


1003817100381710038171003817119906119899minus1

minus 119906119899

1003817100381710038171003817 le1003817100381710038171003817119909119899minus1

minus 119909119899

1003817100381710038171003817 +1

120583

1003816100381610038161003816119903119899minus1 minus 119903119899

1003816100381610038161003816 (120574 119860 120590119899

+ 120594119899)

(40)



1003817100381710038171003817

=1003817100381710038171003817120573119899119878119909119899

+ (1 minus 120573119899) 119906119899


119878119909119899minus1

+ (1 minus 120573119899minus1

) 119906119899minus1

)1003817100381710038171003817

=1003817100381710038171003817120573119899

(119878119909119899

minus 119878119909119899minus1

) + (120573119899


) 119878119909119899minus1

+ (1 minus 120573119899) (119906119899


) + (120573119899minus1

minus 120573119899) 119906119899minus1

1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899


1003817100381710038171003817 + (1 minus 120573119899)

1003817100381710038171003817119906119899


1003817100381710038171003817

+1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 (1003817100381710038171003817119878119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119906119899minus1

1003817100381710038171003817)

le 120573119899

1003817100381710038171003817119909119899


1003817100381710038171003817 + (1 minus 120573119899)

times 1003817100381710038171003817119909119899minus1

minus 119909119899

1003817100381710038171003817 +1

120583

1003816100381610038161003816119903119899minus1 minus 119903119899

1003816100381610038161003816 (120574 119860 120590119899

+ 120594119899)

+1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 (1003817100381710038171003817119878119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119906119899minus1

1003817100381710038171003817)

le1003817100381710038171003817119909119899


1003817100381710038171003817 +1

120583

1003816100381610038161003816119903119899minus1 minus 119903119899

1003816100381610038161003816 (120574 119860 120590119899

+ 120594119899)

+1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 (1003817100381710038171003817119878119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119906119899minus1

1003817100381710038171003817)

(41)

Next we estimate

1003817100381710038171003817119909119899+1

minus 119909119899

1003817100381710038171003817

=1003817100381710038171003817119875119862

[119881119899] minus 119875119862

[119881119899minus1

]1003817100381710038171003817

le1003817100381710038171003817120572119899120588 (119880 (119909

119899) minus 119880 (119909

119899minus1)) + (120572

119899minus 120572119899minus1

) 120588119880 (119909119899minus1

)

+ (119868 minus 120572119899120583119865) (119879 (119910

119899)) minus (119868 minus 120572

119899120583119865) 119879 (119910

119899minus1)

+ (119868 minus 120572119899120583119865) (119879 (119910


119899minus1120583119865) (119879 (119910

119899minus1))

1003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899


1003817100381710038171003817 + (1 minus 120572119899]) 1003817100381710038171003817119910119899


1003817100381710038171003817

+1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 (1003817100381710038171003817120588119880 (119909

119899minus1)1003817100381710038171003817 +

1003817100381710038171003817120583119865 (119879 (119910119899minus1

))1003817100381710038171003817)

(42)


1003817100381710038171003817119909119899+1

minus 119909119899

1003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899


1003817100381710038171003817 + (1 minus 120572119899])

times 1003817100381710038171003817119909119899


1003817100381710038171003817 +1

120583

1003816100381610038161003816119903119899minus1 minus 119903119899

1003816100381610038161003816 (120574 119860 120590119899

+ 120594119899)

+1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 (1003817100381710038171003817119878119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119906119899minus1

1003817100381710038171003817)

+1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 (1003817100381710038171003817120588119880 (119909

119899minus1)1003817100381710038171003817 +

1003817100381710038171003817120583119865 (119879 (119910119899minus1

))1003817100381710038171003817)

le (1 minus (] minus 120588120591) 120572119899)

1003817100381710038171003817119909119899


1003817100381710038171003817

+1

120583

1003816100381610038161003816119903119899minus1 minus 119903119899

1003816100381610038161003816 (120574 119860 120590119899

+ 120594119899)

+1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 (1003817100381710038171003817119878119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119906119899minus1

1003817100381710038171003817)

+1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 (1003817100381710038171003817120588119880 (119909

119899minus1)1003817100381710038171003817 +

1003817100381710038171003817120583119865 (119879 (119910119899minus1

))1003817100381710038171003817)

le (1 minus (] minus 120588120591) 120572119899)

1003817100381710038171003817119909119899


1003817100381710038171003817

+ 119872 (1

120583

1003816100381610038161003816119903119899minus1 minus 119903119899

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816)

(43)

where

119872 = maxsup119899ge1

(120574 119860 120590119899

+ 120594119899)

sup119899ge1

(1003817100381710038171003817119878119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119906119899minus1

1003817100381710038171003817)

sup119899ge1

(1003817100381710038171003817120588119880 (119909

119899minus1)1003817100381710038171003817 +

1003817100381710038171003817120583119865 (119879 (119910119899minus1

))1003817100381710038171003817)

(44)


lim119899rarrinfin

1003817100381710038171003817119909119899+1

minus 119909119899

1003817100381710038171003817 = 0 (45)



119906119899minus119909119899 = 0 Since 119909



1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

= ⟨119875119862

[119881119899] minus 119909lowast 119909119899+1


= ⟨119875119862

[119881119899] minus 119881119899 119875119862

[119881119899] minus 119909lowast⟩ + ⟨119881

119899minus 119909lowast 119909119899+1


le ⟨120572119899

(120588119880 (119909119899) minus 120583119865 (119909

lowast) + (119868 minus 120572

119899120583119865) (119879 (119910

119899)))

minus (119868 minus 120572119899120583119865) (119879 (119909

lowast)) 119909119899+1


= ⟨120572119899120588 (119880 (119909

119899) minus 119880 (119909

lowast)) 119909119899+1


+ 120572119899

⟨120588119880 (119909lowast) minus 120583119865 (119909

lowast) 119909119899+1


+ ⟨(119868 minus 120572119899120583119865) (119879 (119910

119899))

minus (119868 minus 120572119899120583119865) (119879 (119909

lowast)) 119909119899+1


le 120572119899120588120591

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

+ 120572119899

⟨120588119880 (119909lowast) minus 120583119865 (119909

lowast) 119909119899+1


+ (1 minus 120572119899]) 1003817100381710038171003817119910119899

minus 119909lowast1003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

le120572119899120588120591

2(1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

)

+ 120572119899

⟨120588119880 (119909lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899])

2(1003817100381710038171003817119910119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

)

le(1 minus 120572

119899(] minus 120588120591))

2

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

+120572119899120588120591

2

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120572119899

⟨120588119880 (119909lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899])

2(120573119899

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+ (1 minus 120573119899)

1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

)

le(1 minus 120572

119899(] minus 120588120591))

2

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

+120572119899120588120591

2

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120572119899

⟨120588119880 (119909lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

2

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+(1 minus 120572

119899]) (1 minus 120573

119899)

2

times 1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120574 (119871120574 minus 1)10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

2

(46)


1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+(1 minus 120572

119899]) (1 minus 120573

119899)

1 + 120572119899

(] minus 120588120591)

times 1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120574 (119871120574 minus 1)10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2


119899]) (1 minus 120573

119899) 120574 (1 minus 119871120574)

1 + 120572119899

(] minus 120588120591)

10038171003817100381710038171003817(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

(47)


(1 minus 120572119899]) (1 minus 120573

119899) 120574 (1 minus 119871120574)

1 + 120572119899

(] minus 120588120591)

10038171003817100381710038171003817(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)


lowast) 119909119899+1


+ 120573119899

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)


lowast) 119909119899+1


+ 120573119899

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+ (1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 +

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817)

times1003817100381710038171003817119909119899+1

minus 119909119899

1003817100381710038171003817

(48)


119909119899+1

minus 119909119899 = 0 120572

119899rarr 0 and

120573119899

rarr 0 we obtain

lim119899rarrinfin

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817= 0 (49)


Since 1198791198651


1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

=100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899) minus 1198791198651

119903119899(119909lowast)10038171003817100381710038171003817

2

le ⟨119906119899

minus 119909lowast 119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909


=1

21003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

+10038171003817100381710038171003817119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899minus 119909lowast10038171003817100381710038171003817

2

minus10038171003817100381710038171003817119906119899

minus 119909lowast

minus [119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899minus 119909lowast]10038171003817100381710038171003817

2

=1

21003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

+10038171003817100381710038171003817119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899minus 119909lowast10038171003817100381710038171003817

2

minus10038171003817100381710038171003817119906119899

minus 119909119899

minus 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

le1

21003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus10038171003817100381710038171003817119906119899

minus 119909119899


(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

2

=1

21003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus [1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

+ 120574210038171003817100381710038171003817

119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

2

minus2120574⟨119906119899

minus 119909119899 119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899⟩]

(50)


1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

le1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

+ 21205741003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

(51)


minus 119909lowast1003817100381710038171003817

2

le(1 minus 120572

119899(] minus 120588120591))

2

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

+120572119899120588120591

2

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120572119899⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899])

2(120573119899

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+ (1 minus 120573119899)

1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

)

le(1 minus 120572

119899(] minus 120588120591))

2

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

+120572119899120588120591

2

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120572119899⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


(1 minus 120572119899])

2

times 120573119899

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+ (1 minus 120573119899)

times (1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

+21205741003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817)

(52)

which implies that

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)


lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+(1 minus 120572

119899]) (1 minus 120573

119899)

1 + 120572119899

(] minus 120588120591)

times 1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

+21205741003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)


lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+(1 minus 120572

119899]) (1 minus 120573

119899)

1 + 120572119899

(] minus 120588120591)

times minus1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

+ 21205741003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

(53)

Hence

(1 minus 120572119899]) (1 minus 120573

119899)

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+2 (1 minus 120572

119899]) (1 minus 120573

119899) 120574

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+2 (1 minus 120572

119899]) (1 minus 120573

119899) 120574

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

+ (1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 +

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817)

1003817100381710038171003817119909119899+1

minus 119909119899

1003817100381710038171003817

(54)



119909119899+1

minus 119909119899 = 0 120572

119899rarr 0 120573

119899rarr 0 and

lim119899rarrinfin

(1198791198652

119903119899minus 119868)119860119909


lim119899rarrinfin

1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817 = 0 (55)


1003817100381710038171003817119909119899

minus 119879 (119909119899)1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119909119899+1

minus 119879 (119909119899)1003817100381710038171003817

=1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119875119862

[119881119899] minus 119875119862

[119879 (119909119899)]

1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817

+1003817100381710038171003817120572119899

(120588119880 (119909119899) minus 120583119865 (119879 (119910

119899)) + 119879 (119910

119899) minus 119879 (119909

119899))

1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817

+ 120572119899

1003817100381710038171003817120588119880 (119909119899) minus 120583119865 (119879 (119910

119899))

1003817100381710038171003817 +1003817100381710038171003817119910119899

minus 119909119899

1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817 + 120572119899

1003817100381710038171003817120588119880 (119909119899) minus 120583119865 (119879 (119910

119899))

1003817100381710038171003817

+1003817100381710038171003817120573119899119878119909119899

+ (1 minus 120573119899) 119906119899

minus 119909119899

1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817 + 120572119899

1003817100381710038171003817120588119880 (119909119899) minus 120583119865 (119879 (119910

119899))

1003817100381710038171003817

+ 120573119899

1003817100381710038171003817119878119909119899

minus 119909119899

1003817100381710038171003817 + (1 minus 120573119899)

1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

(56)


119909119899+1

minus 119909119899 = 0 120572

119899rarr 0 120573

119899rarr 0 120588119880(119909

119899) minus

120583119865(119879(119910119899)) and 119878119909


119899rarrinfin119909119899

minus

119906119899 = 0 we obtain

lim119899rarrinfin

1003817100381710038171003817119909119899

minus 119879 (119909119899)1003817100381710038171003817 = 0 (57)



119909lowast


isin 119865(119879) Therefore 119908119908

(119909119899) sub 119865(119879)







119899=

1198791198651

119903119899(119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899) we have

1198651

(119906119899 119910) +

1

119903119899

⟨119910 minus 119906119899 119906119899

minus 119909119899⟩

minus1

119903119899

⟨119910 minus 119906119899 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899⟩ ge 0 forall119910 isin 119862

(59)


minus1

119903119899

⟨119910 minus 119906119899 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899⟩

+1

119903119899

⟨119910 minus 119906119899 119906119899

minus 119909119899⟩ ge 1198651

(119910 119906119899) forall119910 isin 119862

(60)

minus1

119903119899119896

⟨119910 minus 119906119899119896

120574119860lowast

(1198791198652

119903119899119896

minus 119868) 119860119909119899119896

⟩

+ ⟨119910 minus 119906119899119896

119906119899119896

minus 119909119899119896

119903119899119896

⟩ ge 1198651

(119910 119906119899119896

) forall119910 isin 119862

(61)


119906119899

minus 119909119899 = 0 lim

119899rarrinfin(1198791198652

119903119899minus 119868)119860119909

119899 = 0

and 119909119899



1(119910 119908) le 0 for all 119910 isin 119862


= 119905119910 + (1 minus 119905)119908 wehave 119910


0 = 1198651

(119910119905 119910119905)

le 1199051198651

(119910119905 119910) + (1 minus 119905) 119865

1(119910119905 119908)

le 1199051198651

(119910119905 119910)

(62)



1)


119899 is bounded

and 119909119899


of 119909119899 such

that 119909119899119896


rarr 119860119908 Now set V119899119896

= 119860119909119899119896

minus 1198791198652

119903119899119896

119860119909119899119896 It follows from


V119899119896

= 0 and119860119909119899119896

minusV119899119896

= 1198791198652

119903119899119896

119860119909119899119896Therefore


119903119899119896

we have

1198652

(119860119909119899119896

minus V119899119896

119910)

+1

119903119899119896

⟨119910 minus (119860119909119899119896

minus V119899119896

)

(119860119909119899119896

minus V119899119896

) minus 119860119909119899119896

⟩ ge 0 forall119910 isin 119862

(63)



1198652

(119860119908 119910) ge 0 forall119910 isin 119862 (64)


Thus we have

119908 isin Λ cap 119865 (119879) (65)



119896 ge 120578

lArrrArr 1198962

ge 1205782

lArrrArr 1 minus 2120583120578 + 12058321198962

ge 1 minus 2120583120578 + 12058321205782



lArrrArr 120583120578 ge ]

(66)




119899rarrinfin⟨120588119880(119911)minus120583119865(119911) 119909

119899minus119911⟩ le


119899119896 of

119909119899 such that


⟨120588119880 (119911) minus 120583119865 (119911) 119909119899

minus 119911⟩


⟨120588119880 (119911) minus 120583119865 (119911) 119909119899119896

minus 119911⟩

= ⟨120588119880 (119911) minus 120583119865 (119911) 119908 minus 119911⟩ le 0

(67)



1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

2

= ⟨119875119862

[119881119899] minus 119911 119909

119899+1minus 119911⟩

= ⟨119875119862

[119881119899] minus 119881119899 119875119862

[119881119899] minus 119911⟩ + ⟨119881

119899minus 119911 119909

119899+1minus 119911⟩

le ⟨120572119899

(120588119880 (119909119899) minus 120583119865 (119911)) + (119868 minus 120572

119899120583119865) (119879 (119910

119899))

minus (119868 minus 120572119899120583119865) (119879 (119911)) 119909

119899+1minus 119911⟩

le ⟨120572119899120588 (119880 (119909

119899) minus 119880 (119911)) 119909

119899+1minus 119911⟩

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ ⟨(119868 minus 120572119899120583119865) (119879 (119910

119899)) minus (119868 minus 120572

119899120583119865) (119879 (119911)) 119909

119899+1minus 119911⟩

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ (1 minus 120572119899]) 1003817100381710038171003817119910119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ (1 minus 120572119899]) 120573119899

1003817100381710038171003817119878119909119899

minus 1198781199111003817100381710038171003817 + 120573119899 119878119911 minus 119911

+ (1 minus 120573119899)

1003817100381710038171003817119906119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ (1 minus 120572119899]) 120573119899

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817 + 120573119899 119878119911 minus 119911

+ (1 minus 120573119899)

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

= (1 minus 120572119899

(] minus 120588120591))1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ (1 minus 120572119899]) 120573119899 119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

le1 minus 120572119899

(] minus 120588120591)

2(1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

2

+1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

2

)

+ 120572119899⟨120588119880 (119911) minus 120583119865 (119911) 119909

119899+1minus 119911⟩

+ (1 minus 120572119899]) 120573119899 119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

(68)


minus 1199111003817100381710038171003817

2

le1 minus 120572119899

(] minus 120588120591)

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)⟨120588119880 (119909

119899) minus 120583119865 (119911) 119909

119899+1minus 119911⟩

+2 (1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

le (1 minus 120572119899

(] minus 120588120591))1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

2

+2120572119899

(] minus 120588120591)

1 + 120572119899

(] minus 120588120591)

times 1

] minus 120588120591⟨120588119880 (119911) minus 120583119865 (119911) 119909

119899+1minus 119911⟩

+(1 minus 120572

119899]) 120573119899

120572119899

(] minus 120588120591)119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

(69)

Let 120574119899

= 120572119899(] minus 120588120591) and 120575

119899= (2120572

119899(] minus 120588120591)(1 +


119899+1minus 119911⟩ + ((1 minus

120572119899])120573119899120572119899(] minus 120588120591))119878119911 minus 119911119909

119899+1minus 119911

Sinceinfin

sum

119899=1

120572119899

= infin


1

] minus 120588120591⟨120588119880 (119911) minus 120583119865 (119911) 119909

119899+1minus 119911⟩

+(1 minus 120572

119899]) 120573119899

120572119899

(] minus 120588120591)119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817 le 0

(70)


sum

119899=1

120574119899


120575119899

120574119899

le 0 (71)







4 Conclusions




Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)

minus (119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)10038171003817100381710038171003817

= (1 minus 1205741198602)

1003817100381710038171003817119909119899


1003817100381710038171003817

+ 1205741198602 1003817100381710038171003817119909119899


1003817100381710038171003817

+ 120574 119860

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198652

119903119899119860119909119899

minus 119860119909119899

10038171003817100381710038171003817

+

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)

minus (119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)10038171003817100381710038171003817

=1003817100381710038171003817119909119899


1003817100381710038171003817 + 120574 119860

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198652

119903119899119860119909119899

minus 119860119909119899

10038171003817100381710038171003817

+

10038161003816100381610038161003816100381610038161003816


119903119899

10038161003816100381610038161003816100381610038161003816

100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)

minus (119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899)10038171003817100381710038171003817

=1003817100381710038171003817119909119899


1003817100381710038171003817 +

10038161003816100381610038161003816100381610038161003816

119903119899


119903119899

10038161003816100381610038161003816100381610038161003816

(120574 119860 120590119899

+ 120594119899)

(39)

where 120590119899

= 1198791198652

119903119899119860119909119899

minus 119860119909119899 and 120594

119899= 1198791198651

119903119899(119909119899

+ 120574119860lowast(1198791198652

119903119899minus

119868)119860119909119899) minus (119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909



gt


1003817100381710038171003817119906119899minus1

minus 119906119899

1003817100381710038171003817 le1003817100381710038171003817119909119899minus1

minus 119909119899

1003817100381710038171003817 +1

120583

1003816100381610038161003816119903119899minus1 minus 119903119899

1003816100381610038161003816 (120574 119860 120590119899

+ 120594119899)

(40)



1003817100381710038171003817

=1003817100381710038171003817120573119899119878119909119899

+ (1 minus 120573119899) 119906119899


119878119909119899minus1

+ (1 minus 120573119899minus1

) 119906119899minus1

)1003817100381710038171003817

=1003817100381710038171003817120573119899

(119878119909119899

minus 119878119909119899minus1

) + (120573119899


) 119878119909119899minus1

+ (1 minus 120573119899) (119906119899


) + (120573119899minus1

minus 120573119899) 119906119899minus1

1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899


1003817100381710038171003817 + (1 minus 120573119899)

1003817100381710038171003817119906119899


1003817100381710038171003817

+1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 (1003817100381710038171003817119878119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119906119899minus1

1003817100381710038171003817)

le 120573119899

1003817100381710038171003817119909119899


1003817100381710038171003817 + (1 minus 120573119899)

times 1003817100381710038171003817119909119899minus1

minus 119909119899

1003817100381710038171003817 +1

120583

1003816100381610038161003816119903119899minus1 minus 119903119899

1003816100381610038161003816 (120574 119860 120590119899

+ 120594119899)

+1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 (1003817100381710038171003817119878119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119906119899minus1

1003817100381710038171003817)

le1003817100381710038171003817119909119899


1003817100381710038171003817 +1

120583

1003816100381610038161003816119903119899minus1 minus 119903119899

1003816100381610038161003816 (120574 119860 120590119899

+ 120594119899)

+1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 (1003817100381710038171003817119878119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119906119899minus1

1003817100381710038171003817)

(41)

Next we estimate

1003817100381710038171003817119909119899+1

minus 119909119899

1003817100381710038171003817

=1003817100381710038171003817119875119862

[119881119899] minus 119875119862

[119881119899minus1

]1003817100381710038171003817

le1003817100381710038171003817120572119899120588 (119880 (119909

119899) minus 119880 (119909

119899minus1)) + (120572

119899minus 120572119899minus1

) 120588119880 (119909119899minus1

)

+ (119868 minus 120572119899120583119865) (119879 (119910

119899)) minus (119868 minus 120572

119899120583119865) 119879 (119910

119899minus1)

+ (119868 minus 120572119899120583119865) (119879 (119910


119899minus1120583119865) (119879 (119910

119899minus1))

1003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899


1003817100381710038171003817 + (1 minus 120572119899]) 1003817100381710038171003817119910119899


1003817100381710038171003817

+1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 (1003817100381710038171003817120588119880 (119909

119899minus1)1003817100381710038171003817 +

1003817100381710038171003817120583119865 (119879 (119910119899minus1

))1003817100381710038171003817)

(42)


1003817100381710038171003817119909119899+1

minus 119909119899

1003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899


1003817100381710038171003817 + (1 minus 120572119899])

times 1003817100381710038171003817119909119899


1003817100381710038171003817 +1

120583

1003816100381610038161003816119903119899minus1 minus 119903119899

1003816100381610038161003816 (120574 119860 120590119899

+ 120594119899)

+1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 (1003817100381710038171003817119878119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119906119899minus1

1003817100381710038171003817)

+1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 (1003817100381710038171003817120588119880 (119909

119899minus1)1003817100381710038171003817 +

1003817100381710038171003817120583119865 (119879 (119910119899minus1

))1003817100381710038171003817)

le (1 minus (] minus 120588120591) 120572119899)

1003817100381710038171003817119909119899


1003817100381710038171003817

+1

120583

1003816100381610038161003816119903119899minus1 minus 119903119899

1003816100381610038161003816 (120574 119860 120590119899

+ 120594119899)

+1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 (1003817100381710038171003817119878119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119906119899minus1

1003817100381710038171003817)

+1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 (1003817100381710038171003817120588119880 (119909

119899minus1)1003817100381710038171003817 +

1003817100381710038171003817120583119865 (119879 (119910119899minus1

))1003817100381710038171003817)

le (1 minus (] minus 120588120591) 120572119899)

1003817100381710038171003817119909119899


1003817100381710038171003817

+ 119872 (1

120583

1003816100381610038161003816119903119899minus1 minus 119903119899

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816)

(43)

where

119872 = maxsup119899ge1

(120574 119860 120590119899

+ 120594119899)

sup119899ge1

(1003817100381710038171003817119878119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119906119899minus1

1003817100381710038171003817)

sup119899ge1

(1003817100381710038171003817120588119880 (119909

119899minus1)1003817100381710038171003817 +

1003817100381710038171003817120583119865 (119879 (119910119899minus1

))1003817100381710038171003817)

(44)


lim119899rarrinfin

1003817100381710038171003817119909119899+1

minus 119909119899

1003817100381710038171003817 = 0 (45)



119906119899minus119909119899 = 0 Since 119909



1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

= ⟨119875119862

[119881119899] minus 119909lowast 119909119899+1


= ⟨119875119862

[119881119899] minus 119881119899 119875119862

[119881119899] minus 119909lowast⟩ + ⟨119881

119899minus 119909lowast 119909119899+1


le ⟨120572119899

(120588119880 (119909119899) minus 120583119865 (119909

lowast) + (119868 minus 120572

119899120583119865) (119879 (119910

119899)))

minus (119868 minus 120572119899120583119865) (119879 (119909

lowast)) 119909119899+1


= ⟨120572119899120588 (119880 (119909

119899) minus 119880 (119909

lowast)) 119909119899+1


+ 120572119899

⟨120588119880 (119909lowast) minus 120583119865 (119909

lowast) 119909119899+1


+ ⟨(119868 minus 120572119899120583119865) (119879 (119910

119899))

minus (119868 minus 120572119899120583119865) (119879 (119909

lowast)) 119909119899+1


le 120572119899120588120591

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

+ 120572119899

⟨120588119880 (119909lowast) minus 120583119865 (119909

lowast) 119909119899+1


+ (1 minus 120572119899]) 1003817100381710038171003817119910119899

minus 119909lowast1003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

le120572119899120588120591

2(1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

)

+ 120572119899

⟨120588119880 (119909lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899])

2(1003817100381710038171003817119910119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

)

le(1 minus 120572

119899(] minus 120588120591))

2

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

+120572119899120588120591

2

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120572119899

⟨120588119880 (119909lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899])

2(120573119899

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+ (1 minus 120573119899)

1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

)

le(1 minus 120572

119899(] minus 120588120591))

2

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

+120572119899120588120591

2

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120572119899

⟨120588119880 (119909lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

2

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+(1 minus 120572

119899]) (1 minus 120573

119899)

2

times 1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120574 (119871120574 minus 1)10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

2

(46)


1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+(1 minus 120572

119899]) (1 minus 120573

119899)

1 + 120572119899

(] minus 120588120591)

times 1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120574 (119871120574 minus 1)10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2


119899]) (1 minus 120573

119899) 120574 (1 minus 119871120574)

1 + 120572119899

(] minus 120588120591)

10038171003817100381710038171003817(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

(47)


(1 minus 120572119899]) (1 minus 120573

119899) 120574 (1 minus 119871120574)

1 + 120572119899

(] minus 120588120591)

10038171003817100381710038171003817(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)


lowast) 119909119899+1


+ 120573119899

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)


lowast) 119909119899+1


+ 120573119899

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+ (1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 +

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817)

times1003817100381710038171003817119909119899+1

minus 119909119899

1003817100381710038171003817

(48)


119909119899+1

minus 119909119899 = 0 120572

119899rarr 0 and

120573119899

rarr 0 we obtain

lim119899rarrinfin

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817= 0 (49)


Since 1198791198651


1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

=100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899) minus 1198791198651

119903119899(119909lowast)10038171003817100381710038171003817

2

le ⟨119906119899

minus 119909lowast 119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909


=1

21003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

+10038171003817100381710038171003817119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899minus 119909lowast10038171003817100381710038171003817

2

minus10038171003817100381710038171003817119906119899

minus 119909lowast

minus [119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899minus 119909lowast]10038171003817100381710038171003817

2

=1

21003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

+10038171003817100381710038171003817119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899minus 119909lowast10038171003817100381710038171003817

2

minus10038171003817100381710038171003817119906119899

minus 119909119899

minus 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

le1

21003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus10038171003817100381710038171003817119906119899

minus 119909119899


(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

2

=1

21003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus [1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

+ 120574210038171003817100381710038171003817

119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

2

minus2120574⟨119906119899

minus 119909119899 119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899⟩]

(50)


1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

le1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

+ 21205741003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

(51)


minus 119909lowast1003817100381710038171003817

2

le(1 minus 120572

119899(] minus 120588120591))

2

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

+120572119899120588120591

2

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120572119899⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899])

2(120573119899

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+ (1 minus 120573119899)

1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

)

le(1 minus 120572

119899(] minus 120588120591))

2

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

+120572119899120588120591

2

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120572119899⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


(1 minus 120572119899])

2

times 120573119899

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+ (1 minus 120573119899)

times (1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

+21205741003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817)

(52)

which implies that

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)


lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+(1 minus 120572

119899]) (1 minus 120573

119899)

1 + 120572119899

(] minus 120588120591)

times 1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

+21205741003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)


lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+(1 minus 120572

119899]) (1 minus 120573

119899)

1 + 120572119899

(] minus 120588120591)

times minus1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

+ 21205741003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

(53)

Hence

(1 minus 120572119899]) (1 minus 120573

119899)

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+2 (1 minus 120572

119899]) (1 minus 120573

119899) 120574

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+2 (1 minus 120572

119899]) (1 minus 120573

119899) 120574

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

+ (1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 +

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817)

1003817100381710038171003817119909119899+1

minus 119909119899

1003817100381710038171003817

(54)



119909119899+1

minus 119909119899 = 0 120572

119899rarr 0 120573

119899rarr 0 and

lim119899rarrinfin

(1198791198652

119903119899minus 119868)119860119909


lim119899rarrinfin

1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817 = 0 (55)


1003817100381710038171003817119909119899

minus 119879 (119909119899)1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119909119899+1

minus 119879 (119909119899)1003817100381710038171003817

=1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119875119862

[119881119899] minus 119875119862

[119879 (119909119899)]

1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817

+1003817100381710038171003817120572119899

(120588119880 (119909119899) minus 120583119865 (119879 (119910

119899)) + 119879 (119910

119899) minus 119879 (119909

119899))

1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817

+ 120572119899

1003817100381710038171003817120588119880 (119909119899) minus 120583119865 (119879 (119910

119899))

1003817100381710038171003817 +1003817100381710038171003817119910119899

minus 119909119899

1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817 + 120572119899

1003817100381710038171003817120588119880 (119909119899) minus 120583119865 (119879 (119910

119899))

1003817100381710038171003817

+1003817100381710038171003817120573119899119878119909119899

+ (1 minus 120573119899) 119906119899

minus 119909119899

1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817 + 120572119899

1003817100381710038171003817120588119880 (119909119899) minus 120583119865 (119879 (119910

119899))

1003817100381710038171003817

+ 120573119899

1003817100381710038171003817119878119909119899

minus 119909119899

1003817100381710038171003817 + (1 minus 120573119899)

1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

(56)


119909119899+1

minus 119909119899 = 0 120572

119899rarr 0 120573

119899rarr 0 120588119880(119909

119899) minus

120583119865(119879(119910119899)) and 119878119909


119899rarrinfin119909119899

minus

119906119899 = 0 we obtain

lim119899rarrinfin

1003817100381710038171003817119909119899

minus 119879 (119909119899)1003817100381710038171003817 = 0 (57)



119909lowast


isin 119865(119879) Therefore 119908119908

(119909119899) sub 119865(119879)







119899=

1198791198651

119903119899(119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899) we have

1198651

(119906119899 119910) +

1

119903119899

⟨119910 minus 119906119899 119906119899

minus 119909119899⟩

minus1

119903119899

⟨119910 minus 119906119899 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899⟩ ge 0 forall119910 isin 119862

(59)


minus1

119903119899

⟨119910 minus 119906119899 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899⟩

+1

119903119899

⟨119910 minus 119906119899 119906119899

minus 119909119899⟩ ge 1198651

(119910 119906119899) forall119910 isin 119862

(60)

minus1

119903119899119896

⟨119910 minus 119906119899119896

120574119860lowast

(1198791198652

119903119899119896

minus 119868) 119860119909119899119896

⟩

+ ⟨119910 minus 119906119899119896

119906119899119896

minus 119909119899119896

119903119899119896

⟩ ge 1198651

(119910 119906119899119896

) forall119910 isin 119862

(61)


119906119899

minus 119909119899 = 0 lim

119899rarrinfin(1198791198652

119903119899minus 119868)119860119909

119899 = 0

and 119909119899



1(119910 119908) le 0 for all 119910 isin 119862


= 119905119910 + (1 minus 119905)119908 wehave 119910


0 = 1198651

(119910119905 119910119905)

le 1199051198651

(119910119905 119910) + (1 minus 119905) 119865

1(119910119905 119908)

le 1199051198651

(119910119905 119910)

(62)



1)


119899 is bounded

and 119909119899


of 119909119899 such

that 119909119899119896


rarr 119860119908 Now set V119899119896

= 119860119909119899119896

minus 1198791198652

119903119899119896

119860119909119899119896 It follows from


V119899119896

= 0 and119860119909119899119896

minusV119899119896

= 1198791198652

119903119899119896

119860119909119899119896Therefore


119903119899119896

we have

1198652

(119860119909119899119896

minus V119899119896

119910)

+1

119903119899119896

⟨119910 minus (119860119909119899119896

minus V119899119896

)

(119860119909119899119896

minus V119899119896

) minus 119860119909119899119896

⟩ ge 0 forall119910 isin 119862

(63)



1198652

(119860119908 119910) ge 0 forall119910 isin 119862 (64)


Thus we have

119908 isin Λ cap 119865 (119879) (65)



119896 ge 120578

lArrrArr 1198962

ge 1205782

lArrrArr 1 minus 2120583120578 + 12058321198962

ge 1 minus 2120583120578 + 12058321205782



lArrrArr 120583120578 ge ]

(66)




119899rarrinfin⟨120588119880(119911)minus120583119865(119911) 119909

119899minus119911⟩ le


119899119896 of

119909119899 such that


⟨120588119880 (119911) minus 120583119865 (119911) 119909119899

minus 119911⟩


⟨120588119880 (119911) minus 120583119865 (119911) 119909119899119896

minus 119911⟩

= ⟨120588119880 (119911) minus 120583119865 (119911) 119908 minus 119911⟩ le 0

(67)



1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

2

= ⟨119875119862

[119881119899] minus 119911 119909

119899+1minus 119911⟩

= ⟨119875119862

[119881119899] minus 119881119899 119875119862

[119881119899] minus 119911⟩ + ⟨119881

119899minus 119911 119909

119899+1minus 119911⟩

le ⟨120572119899

(120588119880 (119909119899) minus 120583119865 (119911)) + (119868 minus 120572

119899120583119865) (119879 (119910

119899))

minus (119868 minus 120572119899120583119865) (119879 (119911)) 119909

119899+1minus 119911⟩

le ⟨120572119899120588 (119880 (119909

119899) minus 119880 (119911)) 119909

119899+1minus 119911⟩

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ ⟨(119868 minus 120572119899120583119865) (119879 (119910

119899)) minus (119868 minus 120572

119899120583119865) (119879 (119911)) 119909

119899+1minus 119911⟩

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ (1 minus 120572119899]) 1003817100381710038171003817119910119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ (1 minus 120572119899]) 120573119899

1003817100381710038171003817119878119909119899

minus 1198781199111003817100381710038171003817 + 120573119899 119878119911 minus 119911

+ (1 minus 120573119899)

1003817100381710038171003817119906119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ (1 minus 120572119899]) 120573119899

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817 + 120573119899 119878119911 minus 119911

+ (1 minus 120573119899)

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

= (1 minus 120572119899

(] minus 120588120591))1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ (1 minus 120572119899]) 120573119899 119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

le1 minus 120572119899

(] minus 120588120591)

2(1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

2

+1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

2

)

+ 120572119899⟨120588119880 (119911) minus 120583119865 (119911) 119909

119899+1minus 119911⟩

+ (1 minus 120572119899]) 120573119899 119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

(68)


minus 1199111003817100381710038171003817

2

le1 minus 120572119899

(] minus 120588120591)

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)⟨120588119880 (119909

119899) minus 120583119865 (119911) 119909

119899+1minus 119911⟩

+2 (1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

le (1 minus 120572119899

(] minus 120588120591))1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

2

+2120572119899

(] minus 120588120591)

1 + 120572119899

(] minus 120588120591)

times 1

] minus 120588120591⟨120588119880 (119911) minus 120583119865 (119911) 119909

119899+1minus 119911⟩

+(1 minus 120572

119899]) 120573119899

120572119899

(] minus 120588120591)119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

(69)

Let 120574119899

= 120572119899(] minus 120588120591) and 120575

119899= (2120572

119899(] minus 120588120591)(1 +


119899+1minus 119911⟩ + ((1 minus

120572119899])120573119899120572119899(] minus 120588120591))119878119911 minus 119911119909

119899+1minus 119911

Sinceinfin

sum

119899=1

120572119899

= infin


1

] minus 120588120591⟨120588119880 (119911) minus 120583119865 (119911) 119909

119899+1minus 119911⟩

+(1 minus 120572

119899]) 120573119899

120572119899

(] minus 120588120591)119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817 le 0

(70)


sum

119899=1

120574119899


120575119899

120574119899

le 0 (71)







4 Conclusions




Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119906119899minus119909119899 = 0 Since 119909



1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

= ⟨119875119862

[119881119899] minus 119909lowast 119909119899+1


= ⟨119875119862

[119881119899] minus 119881119899 119875119862

[119881119899] minus 119909lowast⟩ + ⟨119881

119899minus 119909lowast 119909119899+1


le ⟨120572119899

(120588119880 (119909119899) minus 120583119865 (119909

lowast) + (119868 minus 120572

119899120583119865) (119879 (119910

119899)))

minus (119868 minus 120572119899120583119865) (119879 (119909

lowast)) 119909119899+1


= ⟨120572119899120588 (119880 (119909

119899) minus 119880 (119909

lowast)) 119909119899+1


+ 120572119899

⟨120588119880 (119909lowast) minus 120583119865 (119909

lowast) 119909119899+1


+ ⟨(119868 minus 120572119899120583119865) (119879 (119910

119899))

minus (119868 minus 120572119899120583119865) (119879 (119909

lowast)) 119909119899+1


le 120572119899120588120591

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

+ 120572119899

⟨120588119880 (119909lowast) minus 120583119865 (119909

lowast) 119909119899+1


+ (1 minus 120572119899]) 1003817100381710038171003817119910119899

minus 119909lowast1003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

le120572119899120588120591

2(1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

)

+ 120572119899

⟨120588119880 (119909lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899])

2(1003817100381710038171003817119910119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

)

le(1 minus 120572

119899(] minus 120588120591))

2

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

+120572119899120588120591

2

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120572119899

⟨120588119880 (119909lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899])

2(120573119899

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+ (1 minus 120573119899)

1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

)

le(1 minus 120572

119899(] minus 120588120591))

2

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

+120572119899120588120591

2

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120572119899

⟨120588119880 (119909lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

2

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+(1 minus 120572

119899]) (1 minus 120573

119899)

2

times 1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120574 (119871120574 minus 1)10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

2

(46)


1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+(1 minus 120572

119899]) (1 minus 120573

119899)

1 + 120572119899

(] minus 120588120591)

times 1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120574 (119871120574 minus 1)10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2


119899]) (1 minus 120573

119899) 120574 (1 minus 119871120574)

1 + 120572119899

(] minus 120588120591)

10038171003817100381710038171003817(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

(47)


(1 minus 120572119899]) (1 minus 120573

119899) 120574 (1 minus 119871120574)

1 + 120572119899

(] minus 120588120591)

10038171003817100381710038171003817(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)


lowast) 119909119899+1


+ 120573119899

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)


lowast) 119909119899+1


+ 120573119899

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+ (1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 +

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817)

times1003817100381710038171003817119909119899+1

minus 119909119899

1003817100381710038171003817

(48)


119909119899+1

minus 119909119899 = 0 120572

119899rarr 0 and

120573119899

rarr 0 we obtain

lim119899rarrinfin

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817= 0 (49)


Since 1198791198651


1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

=100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899) minus 1198791198651

119903119899(119909lowast)10038171003817100381710038171003817

2

le ⟨119906119899

minus 119909lowast 119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909


=1

21003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

+10038171003817100381710038171003817119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899minus 119909lowast10038171003817100381710038171003817

2

minus10038171003817100381710038171003817119906119899

minus 119909lowast

minus [119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899minus 119909lowast]10038171003817100381710038171003817

2

=1

21003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

+10038171003817100381710038171003817119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899minus 119909lowast10038171003817100381710038171003817

2

minus10038171003817100381710038171003817119906119899

minus 119909119899

minus 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

le1

21003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus10038171003817100381710038171003817119906119899

minus 119909119899


(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

2

=1

21003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus [1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

+ 120574210038171003817100381710038171003817

119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

2

minus2120574⟨119906119899

minus 119909119899 119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899⟩]

(50)


1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

le1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

+ 21205741003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

(51)


minus 119909lowast1003817100381710038171003817

2

le(1 minus 120572

119899(] minus 120588120591))

2

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

+120572119899120588120591

2

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120572119899⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899])

2(120573119899

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+ (1 minus 120573119899)

1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

)

le(1 minus 120572

119899(] minus 120588120591))

2

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

+120572119899120588120591

2

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120572119899⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


(1 minus 120572119899])

2

times 120573119899

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+ (1 minus 120573119899)

times (1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

+21205741003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817)

(52)

which implies that

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)


lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+(1 minus 120572

119899]) (1 minus 120573

119899)

1 + 120572119899

(] minus 120588120591)

times 1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

+21205741003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)


lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+(1 minus 120572

119899]) (1 minus 120573

119899)

1 + 120572119899

(] minus 120588120591)

times minus1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

+ 21205741003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

(53)

Hence

(1 minus 120572119899]) (1 minus 120573

119899)

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+2 (1 minus 120572

119899]) (1 minus 120573

119899) 120574

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+2 (1 minus 120572

119899]) (1 minus 120573

119899) 120574

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

+ (1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 +

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817)

1003817100381710038171003817119909119899+1

minus 119909119899

1003817100381710038171003817

(54)



119909119899+1

minus 119909119899 = 0 120572

119899rarr 0 120573

119899rarr 0 and

lim119899rarrinfin

(1198791198652

119903119899minus 119868)119860119909


lim119899rarrinfin

1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817 = 0 (55)


1003817100381710038171003817119909119899

minus 119879 (119909119899)1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119909119899+1

minus 119879 (119909119899)1003817100381710038171003817

=1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119875119862

[119881119899] minus 119875119862

[119879 (119909119899)]

1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817

+1003817100381710038171003817120572119899

(120588119880 (119909119899) minus 120583119865 (119879 (119910

119899)) + 119879 (119910

119899) minus 119879 (119909

119899))

1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817

+ 120572119899

1003817100381710038171003817120588119880 (119909119899) minus 120583119865 (119879 (119910

119899))

1003817100381710038171003817 +1003817100381710038171003817119910119899

minus 119909119899

1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817 + 120572119899

1003817100381710038171003817120588119880 (119909119899) minus 120583119865 (119879 (119910

119899))

1003817100381710038171003817

+1003817100381710038171003817120573119899119878119909119899

+ (1 minus 120573119899) 119906119899

minus 119909119899

1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817 + 120572119899

1003817100381710038171003817120588119880 (119909119899) minus 120583119865 (119879 (119910

119899))

1003817100381710038171003817

+ 120573119899

1003817100381710038171003817119878119909119899

minus 119909119899

1003817100381710038171003817 + (1 minus 120573119899)

1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

(56)


119909119899+1

minus 119909119899 = 0 120572

119899rarr 0 120573

119899rarr 0 120588119880(119909

119899) minus

120583119865(119879(119910119899)) and 119878119909


119899rarrinfin119909119899

minus

119906119899 = 0 we obtain

lim119899rarrinfin

1003817100381710038171003817119909119899

minus 119879 (119909119899)1003817100381710038171003817 = 0 (57)



119909lowast


isin 119865(119879) Therefore 119908119908

(119909119899) sub 119865(119879)







119899=

1198791198651

119903119899(119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899) we have

1198651

(119906119899 119910) +

1

119903119899

⟨119910 minus 119906119899 119906119899

minus 119909119899⟩

minus1

119903119899

⟨119910 minus 119906119899 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899⟩ ge 0 forall119910 isin 119862

(59)


minus1

119903119899

⟨119910 minus 119906119899 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899⟩

+1

119903119899

⟨119910 minus 119906119899 119906119899

minus 119909119899⟩ ge 1198651

(119910 119906119899) forall119910 isin 119862

(60)

minus1

119903119899119896

⟨119910 minus 119906119899119896

120574119860lowast

(1198791198652

119903119899119896

minus 119868) 119860119909119899119896

⟩

+ ⟨119910 minus 119906119899119896

119906119899119896

minus 119909119899119896

119903119899119896

⟩ ge 1198651

(119910 119906119899119896

) forall119910 isin 119862

(61)


119906119899

minus 119909119899 = 0 lim

119899rarrinfin(1198791198652

119903119899minus 119868)119860119909

119899 = 0

and 119909119899



1(119910 119908) le 0 for all 119910 isin 119862


= 119905119910 + (1 minus 119905)119908 wehave 119910


0 = 1198651

(119910119905 119910119905)

le 1199051198651

(119910119905 119910) + (1 minus 119905) 119865

1(119910119905 119908)

le 1199051198651

(119910119905 119910)

(62)



1)


119899 is bounded

and 119909119899


of 119909119899 such

that 119909119899119896


rarr 119860119908 Now set V119899119896

= 119860119909119899119896

minus 1198791198652

119903119899119896

119860119909119899119896 It follows from


V119899119896

= 0 and119860119909119899119896

minusV119899119896

= 1198791198652

119903119899119896

119860119909119899119896Therefore


119903119899119896

we have

1198652

(119860119909119899119896

minus V119899119896

119910)

+1

119903119899119896

⟨119910 minus (119860119909119899119896

minus V119899119896

)

(119860119909119899119896

minus V119899119896

) minus 119860119909119899119896

⟩ ge 0 forall119910 isin 119862

(63)



1198652

(119860119908 119910) ge 0 forall119910 isin 119862 (64)


Thus we have

119908 isin Λ cap 119865 (119879) (65)



119896 ge 120578

lArrrArr 1198962

ge 1205782

lArrrArr 1 minus 2120583120578 + 12058321198962

ge 1 minus 2120583120578 + 12058321205782



lArrrArr 120583120578 ge ]

(66)




119899rarrinfin⟨120588119880(119911)minus120583119865(119911) 119909

119899minus119911⟩ le


119899119896 of

119909119899 such that


⟨120588119880 (119911) minus 120583119865 (119911) 119909119899

minus 119911⟩


⟨120588119880 (119911) minus 120583119865 (119911) 119909119899119896

minus 119911⟩

= ⟨120588119880 (119911) minus 120583119865 (119911) 119908 minus 119911⟩ le 0

(67)



1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

2

= ⟨119875119862

[119881119899] minus 119911 119909

119899+1minus 119911⟩

= ⟨119875119862

[119881119899] minus 119881119899 119875119862

[119881119899] minus 119911⟩ + ⟨119881

119899minus 119911 119909

119899+1minus 119911⟩

le ⟨120572119899

(120588119880 (119909119899) minus 120583119865 (119911)) + (119868 minus 120572

119899120583119865) (119879 (119910

119899))

minus (119868 minus 120572119899120583119865) (119879 (119911)) 119909

119899+1minus 119911⟩

le ⟨120572119899120588 (119880 (119909

119899) minus 119880 (119911)) 119909

119899+1minus 119911⟩

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ ⟨(119868 minus 120572119899120583119865) (119879 (119910

119899)) minus (119868 minus 120572

119899120583119865) (119879 (119911)) 119909

119899+1minus 119911⟩

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ (1 minus 120572119899]) 1003817100381710038171003817119910119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ (1 minus 120572119899]) 120573119899

1003817100381710038171003817119878119909119899

minus 1198781199111003817100381710038171003817 + 120573119899 119878119911 minus 119911

+ (1 minus 120573119899)

1003817100381710038171003817119906119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ (1 minus 120572119899]) 120573119899

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817 + 120573119899 119878119911 minus 119911

+ (1 minus 120573119899)

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

= (1 minus 120572119899

(] minus 120588120591))1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ (1 minus 120572119899]) 120573119899 119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

le1 minus 120572119899

(] minus 120588120591)

2(1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

2

+1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

2

)

+ 120572119899⟨120588119880 (119911) minus 120583119865 (119911) 119909

119899+1minus 119911⟩

+ (1 minus 120572119899]) 120573119899 119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

(68)


minus 1199111003817100381710038171003817

2

le1 minus 120572119899

(] minus 120588120591)

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)⟨120588119880 (119909

119899) minus 120583119865 (119911) 119909

119899+1minus 119911⟩

+2 (1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

le (1 minus 120572119899

(] minus 120588120591))1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

2

+2120572119899

(] minus 120588120591)

1 + 120572119899

(] minus 120588120591)

times 1

] minus 120588120591⟨120588119880 (119911) minus 120583119865 (119911) 119909

119899+1minus 119911⟩

+(1 minus 120572

119899]) 120573119899

120572119899

(] minus 120588120591)119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

(69)

Let 120574119899

= 120572119899(] minus 120588120591) and 120575

119899= (2120572

119899(] minus 120588120591)(1 +


119899+1minus 119911⟩ + ((1 minus

120572119899])120573119899120572119899(] minus 120588120591))119878119911 minus 119911119909

119899+1minus 119911

Sinceinfin

sum

119899=1

120572119899

= infin


1

] minus 120588120591⟨120588119880 (119911) minus 120583119865 (119911) 119909

119899+1minus 119911⟩

+(1 minus 120572

119899]) 120573119899

120572119899

(] minus 120588120591)119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817 le 0

(70)


sum

119899=1

120574119899


120575119899

120574119899

le 0 (71)







4 Conclusions




Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Since 1198791198651


1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

=100381710038171003817100381710038171198791198651

119903119899(119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899) minus 1198791198651

119903119899(119909lowast)10038171003817100381710038171003817

2

le ⟨119906119899

minus 119909lowast 119909119899

+ 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909


=1

21003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

+10038171003817100381710038171003817119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899minus 119909lowast10038171003817100381710038171003817

2

minus10038171003817100381710038171003817119906119899

minus 119909lowast

minus [119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899minus 119909lowast]10038171003817100381710038171003817

2

=1

21003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

+10038171003817100381710038171003817119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899minus 119909lowast10038171003817100381710038171003817

2

minus10038171003817100381710038171003817119906119899

minus 119909119899

minus 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899

10038171003817100381710038171003817

2

le1

21003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus10038171003817100381710038171003817119906119899

minus 119909119899


(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

2

=1

21003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus [1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

+ 120574210038171003817100381710038171003817

119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

2

minus2120574⟨119906119899

minus 119909119899 119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899⟩]

(50)


1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

le1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

+ 21205741003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

(51)


minus 119909lowast1003817100381710038171003817

2

le(1 minus 120572

119899(] minus 120588120591))

2

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

+120572119899120588120591

2

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120572119899⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899])

2(120573119899

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+ (1 minus 120573119899)

1003817100381710038171003817119906119899

minus 119909lowast1003817100381710038171003817

2

)

le(1 minus 120572

119899(] minus 120588120591))

2

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

+120572119899120588120591

2

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+ 120572119899⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


(1 minus 120572119899])

2

times 120573119899

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+ (1 minus 120573119899)

times (1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

+21205741003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817)

(52)

which implies that

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)


lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+(1 minus 120572

119899]) (1 minus 120573

119899)

1 + 120572119899

(] minus 120588120591)

times 1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

+21205741003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)


lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+(1 minus 120572

119899]) (1 minus 120573

119899)

1 + 120572119899

(] minus 120588120591)

times minus1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

+ 21205741003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

(53)

Hence

(1 minus 120572119899]) (1 minus 120573

119899)

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+2 (1 minus 120572

119899]) (1 minus 120573

119899) 120574

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

+1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

minus1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817

2

le120572119899120588120591

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)⟨120588119880 (119909

lowast) minus 120583119865 (119909

lowast) 119909119899+1


+(1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119878119909119899

minus 119909lowast1003817100381710038171003817

2

+2 (1 minus 120572

119899]) (1 minus 120573

119899) 120574

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119860119906119899

minus 119860119909119899

1003817100381710038171003817

10038171003817100381710038171003817(1198791198652

119903119899minus 119868) 119860119909

119899

10038171003817100381710038171003817

+ (1003817100381710038171003817119909119899

minus 119909lowast1003817100381710038171003817 +

1003817100381710038171003817119909119899+1

minus 119909lowast1003817100381710038171003817)

1003817100381710038171003817119909119899+1

minus 119909119899

1003817100381710038171003817

(54)



119909119899+1

minus 119909119899 = 0 120572

119899rarr 0 120573

119899rarr 0 and

lim119899rarrinfin

(1198791198652

119903119899minus 119868)119860119909


lim119899rarrinfin

1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817 = 0 (55)


1003817100381710038171003817119909119899

minus 119879 (119909119899)1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119909119899+1

minus 119879 (119909119899)1003817100381710038171003817

=1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119875119862

[119881119899] minus 119875119862

[119879 (119909119899)]

1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817

+1003817100381710038171003817120572119899

(120588119880 (119909119899) minus 120583119865 (119879 (119910

119899)) + 119879 (119910

119899) minus 119879 (119909

119899))

1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817

+ 120572119899

1003817100381710038171003817120588119880 (119909119899) minus 120583119865 (119879 (119910

119899))

1003817100381710038171003817 +1003817100381710038171003817119910119899

minus 119909119899

1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817 + 120572119899

1003817100381710038171003817120588119880 (119909119899) minus 120583119865 (119879 (119910

119899))

1003817100381710038171003817

+1003817100381710038171003817120573119899119878119909119899

+ (1 minus 120573119899) 119906119899

minus 119909119899

1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817 + 120572119899

1003817100381710038171003817120588119880 (119909119899) minus 120583119865 (119879 (119910

119899))

1003817100381710038171003817

+ 120573119899

1003817100381710038171003817119878119909119899

minus 119909119899

1003817100381710038171003817 + (1 minus 120573119899)

1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

(56)


119909119899+1

minus 119909119899 = 0 120572

119899rarr 0 120573

119899rarr 0 120588119880(119909

119899) minus

120583119865(119879(119910119899)) and 119878119909


119899rarrinfin119909119899

minus

119906119899 = 0 we obtain

lim119899rarrinfin

1003817100381710038171003817119909119899

minus 119879 (119909119899)1003817100381710038171003817 = 0 (57)



119909lowast


isin 119865(119879) Therefore 119908119908

(119909119899) sub 119865(119879)







119899=

1198791198651

119903119899(119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899) we have

1198651

(119906119899 119910) +

1

119903119899

⟨119910 minus 119906119899 119906119899

minus 119909119899⟩

minus1

119903119899

⟨119910 minus 119906119899 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899⟩ ge 0 forall119910 isin 119862

(59)


minus1

119903119899

⟨119910 minus 119906119899 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899⟩

+1

119903119899

⟨119910 minus 119906119899 119906119899

minus 119909119899⟩ ge 1198651

(119910 119906119899) forall119910 isin 119862

(60)

minus1

119903119899119896

⟨119910 minus 119906119899119896

120574119860lowast

(1198791198652

119903119899119896

minus 119868) 119860119909119899119896

⟩

+ ⟨119910 minus 119906119899119896

119906119899119896

minus 119909119899119896

119903119899119896

⟩ ge 1198651

(119910 119906119899119896

) forall119910 isin 119862

(61)


119906119899

minus 119909119899 = 0 lim

119899rarrinfin(1198791198652

119903119899minus 119868)119860119909

119899 = 0

and 119909119899



1(119910 119908) le 0 for all 119910 isin 119862


= 119905119910 + (1 minus 119905)119908 wehave 119910


0 = 1198651

(119910119905 119910119905)

le 1199051198651

(119910119905 119910) + (1 minus 119905) 119865

1(119910119905 119908)

le 1199051198651

(119910119905 119910)

(62)



1)


119899 is bounded

and 119909119899


of 119909119899 such

that 119909119899119896


rarr 119860119908 Now set V119899119896

= 119860119909119899119896

minus 1198791198652

119903119899119896

119860119909119899119896 It follows from


V119899119896

= 0 and119860119909119899119896

minusV119899119896

= 1198791198652

119903119899119896

119860119909119899119896Therefore


119903119899119896

we have

1198652

(119860119909119899119896

minus V119899119896

119910)

+1

119903119899119896

⟨119910 minus (119860119909119899119896

minus V119899119896

)

(119860119909119899119896

minus V119899119896

) minus 119860119909119899119896

⟩ ge 0 forall119910 isin 119862

(63)



1198652

(119860119908 119910) ge 0 forall119910 isin 119862 (64)


Thus we have

119908 isin Λ cap 119865 (119879) (65)



119896 ge 120578

lArrrArr 1198962

ge 1205782

lArrrArr 1 minus 2120583120578 + 12058321198962

ge 1 minus 2120583120578 + 12058321205782



lArrrArr 120583120578 ge ]

(66)




119899rarrinfin⟨120588119880(119911)minus120583119865(119911) 119909

119899minus119911⟩ le


119899119896 of

119909119899 such that


⟨120588119880 (119911) minus 120583119865 (119911) 119909119899

minus 119911⟩


⟨120588119880 (119911) minus 120583119865 (119911) 119909119899119896

minus 119911⟩

= ⟨120588119880 (119911) minus 120583119865 (119911) 119908 minus 119911⟩ le 0

(67)



1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

2

= ⟨119875119862

[119881119899] minus 119911 119909

119899+1minus 119911⟩

= ⟨119875119862

[119881119899] minus 119881119899 119875119862

[119881119899] minus 119911⟩ + ⟨119881

119899minus 119911 119909

119899+1minus 119911⟩

le ⟨120572119899

(120588119880 (119909119899) minus 120583119865 (119911)) + (119868 minus 120572

119899120583119865) (119879 (119910

119899))

minus (119868 minus 120572119899120583119865) (119879 (119911)) 119909

119899+1minus 119911⟩

le ⟨120572119899120588 (119880 (119909

119899) minus 119880 (119911)) 119909

119899+1minus 119911⟩

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ ⟨(119868 minus 120572119899120583119865) (119879 (119910

119899)) minus (119868 minus 120572

119899120583119865) (119879 (119911)) 119909

119899+1minus 119911⟩

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ (1 minus 120572119899]) 1003817100381710038171003817119910119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ (1 minus 120572119899]) 120573119899

1003817100381710038171003817119878119909119899

minus 1198781199111003817100381710038171003817 + 120573119899 119878119911 minus 119911

+ (1 minus 120573119899)

1003817100381710038171003817119906119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ (1 minus 120572119899]) 120573119899

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817 + 120573119899 119878119911 minus 119911

+ (1 minus 120573119899)

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

= (1 minus 120572119899

(] minus 120588120591))1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ (1 minus 120572119899]) 120573119899 119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

le1 minus 120572119899

(] minus 120588120591)

2(1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

2

+1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

2

)

+ 120572119899⟨120588119880 (119911) minus 120583119865 (119911) 119909

119899+1minus 119911⟩

+ (1 minus 120572119899]) 120573119899 119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

(68)


minus 1199111003817100381710038171003817

2

le1 minus 120572119899

(] minus 120588120591)

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)⟨120588119880 (119909

119899) minus 120583119865 (119911) 119909

119899+1minus 119911⟩

+2 (1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

le (1 minus 120572119899

(] minus 120588120591))1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

2

+2120572119899

(] minus 120588120591)

1 + 120572119899

(] minus 120588120591)

times 1

] minus 120588120591⟨120588119880 (119911) minus 120583119865 (119911) 119909

119899+1minus 119911⟩

+(1 minus 120572

119899]) 120573119899

120572119899

(] minus 120588120591)119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

(69)

Let 120574119899

= 120572119899(] minus 120588120591) and 120575

119899= (2120572

119899(] minus 120588120591)(1 +


119899+1minus 119911⟩ + ((1 minus

120572119899])120573119899120572119899(] minus 120588120591))119878119911 minus 119911119909

119899+1minus 119911

Sinceinfin

sum

119899=1

120572119899

= infin


1

] minus 120588120591⟨120588119880 (119911) minus 120583119865 (119911) 119909

119899+1minus 119911⟩

+(1 minus 120572

119899]) 120573119899

120572119899

(] minus 120588120591)119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817 le 0

(70)


sum

119899=1

120574119899


120575119899

120574119899

le 0 (71)







4 Conclusions




Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119909119899+1

minus 119909119899 = 0 120572

119899rarr 0 120573

119899rarr 0 and

lim119899rarrinfin

(1198791198652

119903119899minus 119868)119860119909


lim119899rarrinfin

1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817 = 0 (55)


1003817100381710038171003817119909119899

minus 119879 (119909119899)1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119909119899+1

minus 119879 (119909119899)1003817100381710038171003817

=1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119875119862

[119881119899] minus 119875119862

[119879 (119909119899)]

1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817

+1003817100381710038171003817120572119899

(120588119880 (119909119899) minus 120583119865 (119879 (119910

119899)) + 119879 (119910

119899) minus 119879 (119909

119899))

1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817

+ 120572119899

1003817100381710038171003817120588119880 (119909119899) minus 120583119865 (119879 (119910

119899))

1003817100381710038171003817 +1003817100381710038171003817119910119899

minus 119909119899

1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817 + 120572119899

1003817100381710038171003817120588119880 (119909119899) minus 120583119865 (119879 (119910

119899))

1003817100381710038171003817

+1003817100381710038171003817120573119899119878119909119899

+ (1 minus 120573119899) 119906119899

minus 119909119899

1003817100381710038171003817

le1003817100381710038171003817119909119899

minus 119909119899+1

1003817100381710038171003817 + 120572119899

1003817100381710038171003817120588119880 (119909119899) minus 120583119865 (119879 (119910

119899))

1003817100381710038171003817

+ 120573119899

1003817100381710038171003817119878119909119899

minus 119909119899

1003817100381710038171003817 + (1 minus 120573119899)

1003817100381710038171003817119906119899

minus 119909119899

1003817100381710038171003817

(56)


119909119899+1

minus 119909119899 = 0 120572

119899rarr 0 120573

119899rarr 0 120588119880(119909

119899) minus

120583119865(119879(119910119899)) and 119878119909


119899rarrinfin119909119899

minus

119906119899 = 0 we obtain

lim119899rarrinfin

1003817100381710038171003817119909119899

minus 119879 (119909119899)1003817100381710038171003817 = 0 (57)



119909lowast


isin 119865(119879) Therefore 119908119908

(119909119899) sub 119865(119879)







119899=

1198791198651

119903119899(119909119899

+ 120574119860lowast(1198791198652

119903119899minus 119868)119860119909

119899) we have

1198651

(119906119899 119910) +

1

119903119899

⟨119910 minus 119906119899 119906119899

minus 119909119899⟩

minus1

119903119899

⟨119910 minus 119906119899 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899⟩ ge 0 forall119910 isin 119862

(59)


minus1

119903119899

⟨119910 minus 119906119899 120574119860lowast

(1198791198652

119903119899minus 119868) 119860119909

119899⟩

+1

119903119899

⟨119910 minus 119906119899 119906119899

minus 119909119899⟩ ge 1198651

(119910 119906119899) forall119910 isin 119862

(60)

minus1

119903119899119896

⟨119910 minus 119906119899119896

120574119860lowast

(1198791198652

119903119899119896

minus 119868) 119860119909119899119896

⟩

+ ⟨119910 minus 119906119899119896

119906119899119896

minus 119909119899119896

119903119899119896

⟩ ge 1198651

(119910 119906119899119896

) forall119910 isin 119862

(61)


119906119899

minus 119909119899 = 0 lim

119899rarrinfin(1198791198652

119903119899minus 119868)119860119909

119899 = 0

and 119909119899



1(119910 119908) le 0 for all 119910 isin 119862


= 119905119910 + (1 minus 119905)119908 wehave 119910


0 = 1198651

(119910119905 119910119905)

le 1199051198651

(119910119905 119910) + (1 minus 119905) 119865

1(119910119905 119908)

le 1199051198651

(119910119905 119910)

(62)



1)


119899 is bounded

and 119909119899


of 119909119899 such

that 119909119899119896


rarr 119860119908 Now set V119899119896

= 119860119909119899119896

minus 1198791198652

119903119899119896

119860119909119899119896 It follows from


V119899119896

= 0 and119860119909119899119896

minusV119899119896

= 1198791198652

119903119899119896

119860119909119899119896Therefore


119903119899119896

we have

1198652

(119860119909119899119896

minus V119899119896

119910)

+1

119903119899119896

⟨119910 minus (119860119909119899119896

minus V119899119896

)

(119860119909119899119896

minus V119899119896

) minus 119860119909119899119896

⟩ ge 0 forall119910 isin 119862

(63)



1198652

(119860119908 119910) ge 0 forall119910 isin 119862 (64)


Thus we have

119908 isin Λ cap 119865 (119879) (65)



119896 ge 120578

lArrrArr 1198962

ge 1205782

lArrrArr 1 minus 2120583120578 + 12058321198962

ge 1 minus 2120583120578 + 12058321205782



lArrrArr 120583120578 ge ]

(66)




119899rarrinfin⟨120588119880(119911)minus120583119865(119911) 119909

119899minus119911⟩ le


119899119896 of

119909119899 such that


⟨120588119880 (119911) minus 120583119865 (119911) 119909119899

minus 119911⟩


⟨120588119880 (119911) minus 120583119865 (119911) 119909119899119896

minus 119911⟩

= ⟨120588119880 (119911) minus 120583119865 (119911) 119908 minus 119911⟩ le 0

(67)



1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

2

= ⟨119875119862

[119881119899] minus 119911 119909

119899+1minus 119911⟩

= ⟨119875119862

[119881119899] minus 119881119899 119875119862

[119881119899] minus 119911⟩ + ⟨119881

119899minus 119911 119909

119899+1minus 119911⟩

le ⟨120572119899

(120588119880 (119909119899) minus 120583119865 (119911)) + (119868 minus 120572

119899120583119865) (119879 (119910

119899))

minus (119868 minus 120572119899120583119865) (119879 (119911)) 119909

119899+1minus 119911⟩

le ⟨120572119899120588 (119880 (119909

119899) minus 119880 (119911)) 119909

119899+1minus 119911⟩

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ ⟨(119868 minus 120572119899120583119865) (119879 (119910

119899)) minus (119868 minus 120572

119899120583119865) (119879 (119911)) 119909

119899+1minus 119911⟩

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ (1 minus 120572119899]) 1003817100381710038171003817119910119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ (1 minus 120572119899]) 120573119899

1003817100381710038171003817119878119909119899

minus 1198781199111003817100381710038171003817 + 120573119899 119878119911 minus 119911

+ (1 minus 120573119899)

1003817100381710038171003817119906119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ (1 minus 120572119899]) 120573119899

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817 + 120573119899 119878119911 minus 119911

+ (1 minus 120573119899)

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

= (1 minus 120572119899

(] minus 120588120591))1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ (1 minus 120572119899]) 120573119899 119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

le1 minus 120572119899

(] minus 120588120591)

2(1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

2

+1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

2

)

+ 120572119899⟨120588119880 (119911) minus 120583119865 (119911) 119909

119899+1minus 119911⟩

+ (1 minus 120572119899]) 120573119899 119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

(68)


minus 1199111003817100381710038171003817

2

le1 minus 120572119899

(] minus 120588120591)

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)⟨120588119880 (119909

119899) minus 120583119865 (119911) 119909

119899+1minus 119911⟩

+2 (1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

le (1 minus 120572119899

(] minus 120588120591))1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

2

+2120572119899

(] minus 120588120591)

1 + 120572119899

(] minus 120588120591)

times 1

] minus 120588120591⟨120588119880 (119911) minus 120583119865 (119911) 119909

119899+1minus 119911⟩

+(1 minus 120572

119899]) 120573119899

120572119899

(] minus 120588120591)119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

(69)

Let 120574119899

= 120572119899(] minus 120588120591) and 120575

119899= (2120572

119899(] minus 120588120591)(1 +


119899+1minus 119911⟩ + ((1 minus

120572119899])120573119899120572119899(] minus 120588120591))119878119911 minus 119911119909

119899+1minus 119911

Sinceinfin

sum

119899=1

120572119899

= infin


1

] minus 120588120591⟨120588119880 (119911) minus 120583119865 (119911) 119909

119899+1minus 119911⟩

+(1 minus 120572

119899]) 120573119899

120572119899

(] minus 120588120591)119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817 le 0

(70)


sum

119899=1

120574119899


120575119899

120574119899

le 0 (71)







4 Conclusions




Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119896 ge 120578

lArrrArr 1198962

ge 1205782

lArrrArr 1 minus 2120583120578 + 12058321198962

ge 1 minus 2120583120578 + 12058321205782



lArrrArr 120583120578 ge ]

(66)




119899rarrinfin⟨120588119880(119911)minus120583119865(119911) 119909

119899minus119911⟩ le


119899119896 of

119909119899 such that


⟨120588119880 (119911) minus 120583119865 (119911) 119909119899

minus 119911⟩


⟨120588119880 (119911) minus 120583119865 (119911) 119909119899119896

minus 119911⟩

= ⟨120588119880 (119911) minus 120583119865 (119911) 119908 minus 119911⟩ le 0

(67)



1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

2

= ⟨119875119862

[119881119899] minus 119911 119909

119899+1minus 119911⟩

= ⟨119875119862

[119881119899] minus 119881119899 119875119862

[119881119899] minus 119911⟩ + ⟨119881

119899minus 119911 119909

119899+1minus 119911⟩

le ⟨120572119899

(120588119880 (119909119899) minus 120583119865 (119911)) + (119868 minus 120572

119899120583119865) (119879 (119910

119899))

minus (119868 minus 120572119899120583119865) (119879 (119911)) 119909

119899+1minus 119911⟩

le ⟨120572119899120588 (119880 (119909

119899) minus 119880 (119911)) 119909

119899+1minus 119911⟩

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ ⟨(119868 minus 120572119899120583119865) (119879 (119910

119899)) minus (119868 minus 120572

119899120583119865) (119879 (119911)) 119909

119899+1minus 119911⟩

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ (1 minus 120572119899]) 1003817100381710038171003817119910119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ (1 minus 120572119899]) 120573119899

1003817100381710038171003817119878119909119899

minus 1198781199111003817100381710038171003817 + 120573119899 119878119911 minus 119911

+ (1 minus 120573119899)

1003817100381710038171003817119906119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

le 120572119899120588120591

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ (1 minus 120572119899]) 120573119899

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817 + 120573119899 119878119911 minus 119911

+ (1 minus 120573119899)

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

= (1 minus 120572119899

(] minus 120588120591))1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

+ 120572119899

⟨120588119880 (119911) minus 120583119865 (119911) 119909119899+1

minus 119911⟩

+ (1 minus 120572119899]) 120573119899 119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

le1 minus 120572119899

(] minus 120588120591)

2(1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

2

+1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

2

)

+ 120572119899⟨120588119880 (119911) minus 120583119865 (119911) 119909

119899+1minus 119911⟩

+ (1 minus 120572119899]) 120573119899 119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

(68)


minus 1199111003817100381710038171003817

2

le1 minus 120572119899

(] minus 120588120591)

1 + 120572119899

(] minus 120588120591)

1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

2

+2120572119899

1 + 120572119899

(] minus 120588120591)⟨120588119880 (119909

119899) minus 120583119865 (119911) 119909

119899+1minus 119911⟩

+2 (1 minus 120572

119899]) 120573119899

1 + 120572119899

(] minus 120588120591)119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

le (1 minus 120572119899

(] minus 120588120591))1003817100381710038171003817119909119899

minus 1199111003817100381710038171003817

2

+2120572119899

(] minus 120588120591)

1 + 120572119899

(] minus 120588120591)

times 1

] minus 120588120591⟨120588119880 (119911) minus 120583119865 (119911) 119909

119899+1minus 119911⟩

+(1 minus 120572

119899]) 120573119899

120572119899

(] minus 120588120591)119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817

(69)

Let 120574119899

= 120572119899(] minus 120588120591) and 120575

119899= (2120572

119899(] minus 120588120591)(1 +


119899+1minus 119911⟩ + ((1 minus

120572119899])120573119899120572119899(] minus 120588120591))119878119911 minus 119911119909

119899+1minus 119911

Sinceinfin

sum

119899=1

120572119899

= infin


1

] minus 120588120591⟨120588119880 (119911) minus 120583119865 (119911) 119909

119899+1minus 119911⟩

+(1 minus 120572

119899]) 120573119899

120572119899

(] minus 120588120591)119878119911 minus 119911

1003817100381710038171003817119909119899+1

minus 1199111003817100381710038171003817 le 0

(70)


sum

119899=1

120574119899


120575119899

120574119899

le 0 (71)







4 Conclusions




Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













4 Conclusions




Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Strong Convergence Algorithm for...

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