Nonlocal Problem for Fractional Evolution …...evolution equations have attracted increasing...

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 784816 12 pageshttpdxdoiorg1011552013784816

Research ArticleNonlocal Problem for Fractional Evolution Equations ofMixed Type with the Measure of Noncompactness

Pengyu Chen and Yongxiang Li

Department of Mathematics Northwest Normal University Lanzhou 730070 China

Correspondence should be addressed to Pengyu Chen chpengyu123163com

Received 8 December 2012 Revised 14 March 2013 Accepted 18 March 2013

Academic Editor Xinan Hao

Copyright copy 2013 P Chen and Y Li This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A general class of semilinear fractional evolution equations of mixed type with nonlocal conditions on infinite dimensional Banachspaces is concerned Under more general conditions the existence of mild solutions and positive mild solutions is obtained byutilizing a new estimation technique of the measure of noncompactness and a new fixed point theorem with respect to convex-power condensing operator

1 Introduction

In this paper we use a new estimation technique of themeasure of noncompactness and fixed point theorem withrespect to convex-power condensing operator to discuss theexistence of mild solutions and positive mild solutions fornonlocal problem of fractional evolution equations (NPFEE)of mixed type with noncompact semigroup in Banach space119864

119862119863119902

119905119906 (119905) + 119860119906 (119905) = 119891 (119905 119906 (119905) 119866119906 (119905) 119878119906 (119905)) 119905 isin 119869

119906 (0) = 119892 (119906)

(1)

where 119862119863119902

119905is the Caputo fractional derivative of order 119902 0 lt

119902 lt 1 119860 119863(119860) sub 119864 rarr 119864 is a closed linear operator and minus119860generates a uniformly bounded 119862

0-semigroup 119879(119905) (119905 ge 0) in

119864 119891 119869 times 119864 times 119864 times 119864 rarr 119864 is a Caratheodory type function119869 = [0 119886] 119886 gt 0 is a constant 119892mapping from some space offunctions to be specified later and

119866119906 (119905) = int

119905

0

119870 (119905 119904) 119906 (119904) 119889119904 119905 isin 119869

119878119906 (119905) = int

119886

0

119867(119905 119904) 119906 (119904) 119889119904 119905 isin 119869

(2)

are integral operators with integral kernels 119870 isin 119862(ΔR)Δ = (119905 119904) | 0 le 119904 le 119905 le 119886 and119867 isin 119862(Δ 0R) Δ 0 = (119905 119904) |

0 le 119905 119904 le 119886In recent years fractional calculus has attracted many

physicists mathematicians and engineers and notable con-tributions have been made to both theory and applicationsof fractional differential equations It has been found that thedifferential equations involving fractional derivatives in timeare more realistic to describe many phenomena in practicalcases than those of integer order in time For more detailsabout fractional calculus and fractional differential equationswe refer to the books by Miller and Ross [1] Podlubny[2] and Kilbas et al [3] and the papers by Eidelman andKochubei [4] Lakshmikantham and Vatsala [5] Agarwalet al [6] Darwish and Ntouyas [7ndash10] and Darwish et al[11] One of the branches of fractional calculus is the theoryof fractional evolution equations Since fractional ordersemilinear evolution equations are abstract formulations formany problems arising in engineering and physics fractionalevolution equations have attracted increasing attention inrecent years see [12ndash26] and the references therein

The study of abstract nonlocal Cauchy problem wasinitiated by Byszewski and Lakshmikantham [27] Since itis demonstrated that the nonlocal problems have bettereffects in applications than the traditional Cauchy problemsdifferential equations with nonlocal conditions were studied

2 Abstract and Applied Analysis

bymany authors and some basic results on nonlocal problemshave been obtained see [17ndash32] and the references thereinfor more comments and citations In the past few years theexistence uniqueness and some other properties of mildsolutions to nonlocal problem of fractional evolution equa-tions (1) with 119891(119905 119906(119905) 119866119906(119905) 119878119906(119905)) = 119891(119905 119906(119905)) have beenextensively studied by using Banach contraction mappingprincipal Schauderrsquos fixed point theorem and Krasnoselskiirsquosfixed point theorem when 119879(119905) (119905 ge 0) is a compactsemigroup For more details on the basic theory of nonlocalproblem for fractional evolution equations one can see thepapers of Diagana et al [17] Wang et al [18] Li et al [19]Zhou and Jiao [20]Wang et al [21] Wang et al [22] Wang etal [23] Chang et al [24] Balachandran and Park [25] andBalachandran and Trujillo [26] However for the case thatthe semigroup 119879(119905) (119905 ge 0) is noncompact there are veryfew papers studied nonlocal problem of fractional evolutionequations that only Wang et al [14] discussed the existenceofmild solutions for nonlocal problem of fractional evolutionequations under the situation that 119879(119905) (119905 ge 0) is an analyticsemigroup of uniformly bounded linear operators

It is well known that the famous Sadovskiirsquos fixed pointtheorem is an important tool to study various differentialequations and integral equations on infinite dimensionalBanach spaces Early on Lakshmikantham and Leela [33]studied the following initial value problem (IVP) of ordinarydifferential equation in Banach space 119864

1199061015840(119905) = 119891 (119905 119906 (119905)) 119905 isin 119869

119906 (0) = 1199060

(3)

and they proved that if for any 119877 gt 0 119891 is uniformlycontinuous on 119869 times 119861

119877and satisfies the noncompactness

measure condition

120572 (119891 (119905 119863)) le 119871120572 (119863) forall119905 isin 119869 119863 sub 119861119877 (4)

where 119861119877= 119906 isin 119864 119906 le 119877 119871 is a positive constant

120572(sdot) denotes the Kuratowski measure of noncompactness in119864 then IVP (3) has a global solution provided that 119871 satisfiesthe condition

119871 lt1

119886 (5)

In fact there are a large amount of authors who studiedordinary differential equations in Banach spaces similar to(3) by using Sadovskiirsquos fixed point theorem and hypothesisanalogous to (4) they also required that the constants satisfya strong inequality similar to (5) Formore details on this factwe refer to Guo [34] Liu et al [35] and Liu et al [36]

It is easy to see that the inequality (5) is a strongrestrictive condition and it is difficult to be satisfied inapplications In order to remove the strong restriction on theconstant 119871 Sun and Zhang [37] generalized the definition ofcondensing operator to convex-power condensing operatorAnd based on the definition of this new kind of operator theyestablished a new fixed point theoremwith respect to convex-power condensing operator which generalizes the famousSchauderrsquos fixed point theorem and Sadovskiirsquos fixed point

theorem As an application they investigated the existenceof global mild solutions and positive mild solutions for theinitial value problem of evolution equations in 119864

1199061015840(119905) + 119860119906 (119905) = 119891 (119905 119906 (119905)) 119905 isin 119869

119906 (0) = 1199060

(6)

they assume that minus119860 generate a equicontinuous 1198620-

semigroup the nonlinear term 119891 is uniformly continuouson 119869 times 119861

119877and satisfies a suitable noncompactness

measure condition similar to (4) Recently Shi et al[38] developed the IVP (6) to the case that the nonlinearterm is 119891(119905 119906(119905) 119866119906(119905) 119878119906(119905)) and obtained the existence ofglobal mild solutions and positive mild solutions by usingthe new fixed point theorem with respect to convex-powercondensing operator established in [37] but they also requirethat the nonlinear term 119891 is uniformly continuous on119869 times 119861119877 times 119861119877 times 119861119877

We observed that in [33ndash38] the authors all demandthat the nonlinear term 119891 is uniformly continuous this isa very strong assumption As a matter of fact if 119891(119905 119906)is Lipschitz continuous on 119869 times 119861

119877with respect to 119906 then

the condition (4) is satisfied but 119891 may not be necessarilyuniformly continuous on 119869 times 119861

119877

Motivated by the above mentioned aspects in this paperwe studied the existence of mild solutions and positive mildsolutions for the NPFEE (1) by utilizing a new fixed pointtheorem with respect to convex-power condensing operatordue to Sun and Zhang [37] (see Lemma 8) Furthermorewe deleted the assumption that 119891 is uniformly continuousby using a new estimation technique of the measure ofnoncompactness (see Lemma 7)

2 Preliminaries

In this section we introduce some notations definitions andpreliminary facts which are used throughout this paper

Let 119864 be a real Banach space with the norm sdot Wedenote by 119862(119869 119864) the Banach space of all continuous 119864-value functions on interval 119869 with the supnorm 119906119862 =

sup119905isin119869119906(119905) and by 1198711(119869 119864) the Banach space of all 119864-value

Bochner integrable functions defined on 119869 with the norm1199061 = int1

0119906(119905)119889119905

Definition 1 (see [3]) The fractional integral of order 119902 gt 0

with the lower limit zero for a function 119906 isin 1198711(119869 119864) is definedas

119868119902

119905119906 (119905) =

1

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

119906 (119904) 119889119904 119905 gt 0 (7)

where Γ(sdot) is the Euler gamma function

Abstract and Applied Analysis 3

Definition 2 (see [3]) The Caputo fractional derivative oforder 119902 gt 0 with the lower limit zero for a function 119906 isdefined as

119862119863119902

119905119906 (119905) =

1

Γ (119899 minus 119902)int

119905

0

(119905 minus 119904)119899minus119902minus1

119906(119899)(119904) 119889119904

119905 gt 0 0 le 119899 minus 1 lt 119902 lt 119899

(8)

where the function 119906(119905) has absolutely continuous derivativesup to order 119899 minus 1

If 119906 is an abstract function with values in 119864 then theintegrals which appear in Definitions 1 and 2 are taken inBochnerrsquos sense

For 119906 isin 119864 define two operators T119902(119905) (119905 ge 0) andS119902(119905) (119905 ge 0) by

T119902 (119905) 119906 = int

infin

0

ℎ119902 (119904) 119879 (119905119902119904) 119906 119889119904

S119902 (119905) 119906 = 119902int

infin

0

119904ℎ119902 (119904) 119879 (119905

119902119904) 119906 119889119904

(9)

where

ℎ119902 (119904) =1

120587119902

infin

sum

119899=1

(minus119904)119899minus1

Γ (119899119902 + 1)

119899sin (119899120587119902) 119904 isin (0infin)

(10)

is the function of Wright type defined on (0infin) whichsatisfies

ℎ119902 (119904) ge 0 119904 isin (0infin) int

infin

0

ℎ119902 (119904) 119889119904 = 1

int

infin

0

119904Vℎ119902 (119904) 119889119904 =

Γ (1 + V)

Γ (1 + 119902V) V isin [0 1]

(11)

Let 119872 = sup119905isin[0+infin)

119879(119905)L(119864) where L(119864) stands forthe Banach space of all linear and bounded operators in 119864The following lemma follows from the results in [12 13 20]

Lemma 3 The operators T119902(119905) (119905 ge 0) and S119902(119905) (119905 ge 0)

have the following properties

(1) For any fixed 119905 ge 0 T119902(119905) and S

119902(119905) are linear and

bounded operators that is for any 119906 isin 11986410038171003817100381710038171003817T119902 (119905) 119906

10038171003817100381710038171003817le 119872119906

10038171003817100381710038171003817S119902 (119905) 119906

10038171003817100381710038171003817le

119902119872

Γ (1 + 119902)119906 =

119872

Γ (119902)119906

(12)

(2) For every 119906 isin 119864 119905 rarr T119902(119905)119906 and 119905 rarr S

119902(119905)119906 are

continuous functions from [0infin) into 119864(3) The operators T119902(119905) (119905 ge 0) and S119902(119905) (119905 ge 0) are

strongly continuous which means that for all 119906 isin 119864

and 0 le 1199051015840lt 11990510158401015840le 119886 one has

10038171003817100381710038171003817T119902(11990510158401015840) 119906 minusT

119902(1199051015840) 119906

10038171003817100381710038171003817997888rarr 0

10038171003817100381710038171003817S119902(11990510158401015840) 119906 minusS

119902(1199051015840) 119906

10038171003817100381710038171003817997888rarr 0 as 11990510158401015840 minus 1199051015840 997888rarr 0

(13)

Definition 4 A function 119906 isin 119862(119869 119864) is said to be a mildsolution of the NPFEE (1) if it satisfies

119906 (119905) = T119902 (119905) 119892 (119906) + int

119905

0

(119905 minus 119904)119902minus1

timesS119902 (119905 minus 119904) 119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904)) 119889119904

(14)

Next we recall some properties of the measure of non-compactness that will be used in the proof of our mainresults Since no confusion may occur we denote by 120572(sdot)

the Kuratowski measure of noncompactness on both thebounded sets of119864 and119862(119869 119864) For the details of the definitionand properties of themeasure of noncompactness we refer tothe monographs [39 40] For any 119863 sub 119862(119869 119864) and 119905 isin 119869 set119863(119905) = 119906(119905) | 119906 isin 119863 sub 119864 If 119863 sub 119862(119869 119864) is bounded then119863(119905) is bounded in 119864 and 120572(119863(119905)) le 120572(119863)

Lemma 5 (see [39]) Let 119864 be a Banach space let119863 sub 119862(119869 119864)

be bounded and equicontinuous Then 120572(119863(119905)) is continuouson 119869 and

120572 (119863) = max119905isin119869

120572 (119863 (119905)) = 120572 (119863 (119869)) (15)

Lemma 6 (see [41]) Let 119864 be a Banach space let 119863 =

119906119899 sub 119862(119869 119864) be a bounded and countable set Then 120572(119863(119905))is Lebesgue integral on 119869 and

120572(int119869

119906119899 (119905) 119889119905 | 119899 isin N) le 2int

119869

120572 (119863 (119905)) 119889119905 (16)

Lemma 7 (see [32 42]) Let 119864 be a Banach space let119863 sub 119864 bebounded Then there exists a countable set 119863

0sub 119863 such that

120572(119863) le 2120572(1198630)

Lemma 8 (fixed point theoremwith respect to convex-powercondensing operator (see [37])) Let 119864 be a Banach space let119863 sub 119864 be bounded closed and convex Suppose 119876 119863 rarr 119863

is a continuous operator and 119876(119863) is bounded For any 119878 sub 119863

and 1199090isin 119863 set

119876(11199090)

(119878) equiv 119876 (119878)

119876(1198991199090)

(119878) = 119876 (co 119876(119899minus11199090) (119878) 1199090) 119899 = 2 3

(17)

If there exist 1199090isin 119863 and positive integer 119899

0such that for any

bounded and nonprecompact subset 119878 sub 119863

120572 (119876(1198990 1199090)

(119878)) lt 120572 (119878) (18)

then 119876 has at least one fixed point in119863

Lemma 9 For 120590 isin (0 1] and 0 lt 119886 le 119887 one has1003816100381610038161003816119886120590minus 1198871205901003816100381610038161003816 le (119887 minus 119886)

120590 (19)

3 Existence of Mild Solutions

In this section we discuss the existence of mild solutions forthe NPFEE (1) We first make the following hypotheses


(H1) The function 119891 119869 times 119864 times 119864 times 119864 rarr 119864 satisfies theCaratheodory type conditions that is 119891(sdot 119906 119866119906 119878119906)is strongly measurable for all 119906 isin 119864 and 119891(119905 sdot sdot sdot) iscontinuous for ae 119905 isin 119869

(H2) For some 119903 gt 0 there exist constants 1199021isin [0 119902) 120588

1gt

0 and functions 120593119903isin 11987111199021 (119869R+) such that for ae

119905 isin 119869 and all 119906 isin 119864 satisfying 119906 le 1199031003817100381710038171003817119891 (119905 119906 119866119906 119878119906)

1003817100381710038171003817 le 120593119903 (119905)

lim inf119903rarr+infin

1003817100381710038171003817120593119903100381710038171003817100381711987111199021 [0119886]

119903= 1205881lt +infin

(20)

(H3) There exist positive constants 119871119894 (119894 = 1 2 3) such

that for any bounded and countable sets 119863119894 sub 119864 (119894 =

1 2 3) and ae 119905 isin 119869

120572 (119891 (119905 1198631 1198632 1198633)) le

3

sum

119894=1

119871119894120572 (119863119894) (21)

(H4) The nonlocal function 119892 119862(119869 119864) rarr 119864 is continuousand compact and there exist a constant 120588

2gt 0 and a

nondecreasing continuous function Φ R+ rarr R+

such that for some 119903 gt 0 and all 119906 isin Ω119903= 119906 isin

119862(119869 119864) 119906119862le 1199031003817100381710038171003817119892 (119906)

1003817100381710038171003817 le Φ (119903)


Φ (119903)

119903= 1205882 lt +infin

(22)

Theorem 10 Let 119864 be a real Banach space let 119860 119863(119860) sub

119864 rarr 119864 be a closed linear operator and minus119860 generatean equicontinuous 119862

0-semigroup 119879(119905) (119905 ge 0) of uniformly

bounded operators in119864 Assume that the hypotheses (H1) (H2)(H3) and (H4) are satisfied then the NPFEE (1) have at leastone mild solution in 119862(119869 119864) provided that

1198721205882 +1198721205881119886119902minus1199021

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus1199021

lt 1 (23)

Proof Consider the operator 119876 119862(119869 119864) rarr 119862(119869 119864) definedby

(119876119906) (119905) = T119902 (119905) 119892 (119906) + int119905

0

(119905 minus 119904)119902minus1

timesS119902 (119905 minus 119904) 119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904)) 119889119904 119905 isin 119869

(24)

By direct calculation we know that 119876 is well defined FromDefinition 4 it is easy to see that the mild solution of theNPFEE (1) is equivalent to the fixed point of the operator119876 In the following we will prove 119876 has a fixed point byapplying the fixed point theorem with respect to convex-power condensing operator

Firstly we prove that there exists a positive constant 119877such that 119876(Ω

119877) sub Ω

119877 If this is not true then for each

119903 gt 0 there would exist 119906119903isin Ω119903and 119905119903isin 119869 such that

(119876119906119903)(119905119903) gt 119903 It follows from Lemma 3 (1) the hypotheses

(H2) and (H4) and Holder inequality that

119903 lt1003817100381710038171003817(119876119906119903) (119905119903)

1003817100381710038171003817

le 1198721003817100381710038171003817119892 (119906119903)

1003817100381710038171003817

+119872

Γ (119902)int

119905119903

0

(119905119903minus 119904)119902minus1

times1003817100381710038171003817119891 (119904 119906119903 (119904) 119866119906119903 (119904) 119878119906119903 (119904))

1003817100381710038171003817 119889119904

le 119872Φ (119903) +119872

Γ (119902)(int

119905119903

0

(119905119903minus 119904)(119902minus1)(1minus1199021)

119889119904)

1minus1199021

times1003817100381710038171003817120593119903

100381710038171003817100381711987111199021 [0119905119903]

le 119872Φ (119903) +119872119886119902minus1199021

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus11990211003817100381710038171003817120593119903

100381710038171003817100381711987111199021 [0119886]

(25)

Dividing both sides of (25) by 119903 and taking the lower limit as119903 rarr +infin we get

1198721205882 +

1198721205881119886119902minus1199021

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus1199021

ge 1 (26)

which contradicts (23)Secondly we prove that 119876 is continuous in Ω

119877 To thisend let 119906

119899infin

119899=1sub Ω119877be a sequence such that lim

119899rarr+infin119906119899= 119906

inΩ119877 By the continuity of119892 and theCaratheodory continuity

of 119891 we have

lim119899rarr+infin

119892 (119906119899) = 119892 (119906)

lim119899rarr+infin

119891 (119904 119906119899 (119904) 119866119906119899 (119904) 119878119906119899 (119904))

= 119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904)) ae 119904 isin 119869

(27)

From the hypothesis (H2) we get for each 119905 isin 119869

(119905 minus 119904)119902minus1 1003817100381710038171003817119891 (119904 119906119899 (119904) 119866119906119899 (119904) 119878119906119899 (119904))

minus119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904))1003817100381710038171003817

le 2(119905 minus 119904)119902minus1

120593119877 (119904)

(28)

Using the fact that the function 119904 rarr 2(119905 minus 119904)119902minus1

120593119877(119904) isLebesgue integrable for 119904 isin [0 119905] 119905 isin 119869 by (27) and (28) andthe Lebesgue dominated convergence theorem we get that

1003817100381710038171003817(119876119906119899) (119905) minus (119876119906) (119905)1003817100381710038171003817

le 1198721003817100381710038171003817119892 (119906119899) minus 119892 (119906)

1003817100381710038171003817

+119872

Γ (119902)


times int

119905

0

(119905 minus 119904)119902minus1

times1003817100381710038171003817119891 (119904 119906119899 (119904) 119866119906119899 (119904) 119878119906119899 (119904))

minus119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904))1003817100381710038171003817 119889119904 997888rarr 0

as 119899 997888rarr infin

(29)

Therefore we know that

1003817100381710038171003817(119876119906119899) minus (119876119906)1003817100381710038171003817119862

997888rarr 0 as 119899 997888rarr infin (30)

which means that 119876 is continuousNow we demonstrate that the operator 119876 Ω

119877rarr Ω119877is

equicontinuous For any 119906 isin Ω119877and 0 le 119905

1lt 1199052le 119886 we get

that

(119876119906) (1199052) minus (119876119906) (1199051)

= T119902 (1199052) 119892 (119906) minusT119902 (1199051) 119892 (119906)

+ int

1199052

1199051

(1199052 minus 119904)119902minus1

S119902 (1199052 minus 119904)

times 119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904)) 119889119904

+ int

1199051

0

((1199052 minus 119904)119902minus1

minus (1199051 minus 119904)119902minus1

)

timesS119902 (1199052 minus 119904) 119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904)) 119889119904

+ int

1199051

0

(1199051minus 119904)119902minus1

(S119902(1199052minus 119904) minusS

119902(1199051minus 119904))

times 119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904)) 119889119904

= 1198681+ 1198682+ 1198683+ 1198684

(31)

where

1198681 = T119902 (1199052) 119892 (119906) minusT119902 (1199051) 119892 (119906)

1198682= int

1199052

1199051

(1199052minus 119904)119902minus1

S119902(1199052minus 119904)

times 119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904)) 119889119904

1198683 = int

1199051

0

((1199052 minus 119904)119902minus1


)

timesS119902(1199052minus 119904) 119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904)) 119889119904

1198684= int

1199051

0

(1199051minus 119904)119902minus1

(S119902(1199052minus 119904) minusS

119902(1199051minus 119904))

times 119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904)) 119889119904

(32)

It is obvious that

1003817100381710038171003817(119876119906) (1199052) minus (119876119906) (1199051)1003817100381710038171003817 le

4

sum

119894=1

10038171003817100381710038171198681198941003817100381710038171003817 (33)

Therefore we only need to check 119868119894 tend to 0 independently

of 119906 isin Ω119877when 119905

2minus 1199051rarr 0 119894 = 1 2 4

For 1198681 by Lemma 3(3) 119868

1 rarr 0 as 119905

2minus 1199051rarr 0

For 1198682 by the hypothesis (H2) Lemma 3(1) and Holder

inequality we have

100381710038171003817100381711986821003817100381710038171003817 le

119872

Γ (119902)int

1199052

1199051

(1199052minus 119904)119902minus1

120593119877 (119904) 119889119904

le119872

Γ (119902)(int

1199052

1199051


119889119904)

1minus1199021

times1003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [1199051 1199052]

le

1198721003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [0119886]

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus1199021

(1199052minus 1199051)119902minus1199021

997888rarr 0

as 1199052minus 1199051997888rarr 0

(34)

For 1198683 by the hypothesis (H2) Lemmas 3(1) and 9 and

Holder inequality we get that

100381710038171003817100381711986831003817100381710038171003817 le

119872

Γ (119902)int

1199051

0

((1199051 minus 119904)119902minus1


) 120593119877 (119904) 119889119904

le119872

Γ (119902)(int

1199051

0

((1199051minus 119904)119902minus1

minus(1199052minus 119904)119902minus1

)1(1minus1199021)

119889119904)

1minus1199021

times1003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [01199051]

le119872

Γ (119902)(int

1199051

0

((1199051minus 119904)(119902minus1)(1minus1199021)

minus(1199052minus 119904)(119902minus1)(1minus1199021)

) 119889119904)

1minus1199021

times1003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [0119886]

le

1198721003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [0119886]

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus1199021

times (119905(119902minus1199021)(1minus1199021)

1minus 119905(119902minus1199021)(1minus1199021)

2

+(1199052minus 1199051)(119902minus1199021)(1minus1199021)

)

1minus1199021

le

11987221minus11990211003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [0119886]

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus1199021

(1199052minus 1199051)119902minus1199021

997888rarr 0 as 1199052minus 1199051997888rarr 0

(35)


For 1199051= 0 0 lt 119905

2le 119886 it is easy to see that 119868

4 = 0 For 119905

1gt 0

and 120598 gt 0 small enough by the hypothesis (H2) Lemma 3and the equicontinuity of 119879(119905) we know that

100381710038171003817100381711986841003817100381710038171003817 le int

1199051minus120598

0

(1199051minus 119904)119902minus1

times10038171003817100381710038171003817S119902 (1199052 minus 119904) minus S119902 (1199051 minus 119904)

10038171003817100381710038171003817120593119877 (119904) 119889119904

+ int

1199051

1199051minus120598

(1199051minus 119904)119902minus1

times10038171003817100381710038171003817S119902(1199052minus 119904) minusS

119902(1199051minus 119904)

10038171003817100381710038171003817120593119877 (119904) 119889119904

le sup119904isin[01199051minus120598]

10038171003817100381710038171003817S119902(1199052minus 119904) minusS

119902(1199051minus 119904)

10038171003817100381710038171003817

times int

1199051minus120598

0

(1199051 minus 119904)119902minus1

120593119877 (119904) 119889119904

+2119872

Γ (119902)int

1199051

1199051minus120598

(1199051minus 119904)119902minus1

120593119877 (119904) 119889119904

le sup119904isin[01199051minus120598]

10038171003817100381710038171003817S119902(1199052minus 119904) minusS

119902(1199051minus 119904)

10038171003817100381710038171003817

sdot1003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [0119886](1 minus 1199021

119902 minus 1199021

)

1minus1199021

sdot (119905(119902minus1199021)(1minus1199021)

1minus 120598(119902minus1199021)(1minus1199021))

1minus1199021

+

21198721003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [0119886]

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus1199021

120598119902minus1199021 997888rarr 0

as 1199052 minus 1199051 997888rarr 0

(36)

As a result (119876119906)(1199052)minus(119876119906)(119905

1) tends to zero independently

of 119906 isin Ω119877as 1199052minus 1199051rarr 0 which means that 119876 Ω

119877rarr Ω119877

is equicontinuousLet 119865 = co 119876(Ω

119877) where co means the closure of convex

hull Then it is easy to verify that 119876 maps 119865 into itself and119865 sub 119862(119869 119864) is equicontinuous Now we are in the positionto prove that 119876 119865 rarr 119865 is a convex-power condensingoperator Take1199060 isin 119865 wewill prove that there exists a positiveinteger 1198990 such that for any bounded and nonprecompactsubset119863 sub 119865

120572 (119876(1198990 1199060)

(119863)) lt 120572 (119863) (37)

For any119863 sub 119865 and1199060isin 119865 by (17) and the equicontinuity of119865

we get that119876(1198991199060)(119863) sub Ω119877is also equicontinuousTherefore

we know from Lemma 5 that

120572 (119876(1198991199060)

(119863)) = max119905isin119869

120572 (119876(1198991199060)

(119863) (119905)) 119899 = 1 2

(38)

By Lemma 7 there exists a countable set1198631= 1199061

119899 sub 119863 such

that

120572 (119876 (119863) (119905)) le 2120572 (119876 (1198631) (119905)) (39)

Thanks to the fact that

int

119886

0

119906 (119904) 119889119904 isin 119886 co 119906 (119904) | 119904 isin 119869 119906 isin 119862 (119869 119864) (40)

we have

120572(int

119905

0

119870 (119905 119904) 119906 (119904) 119889119904 | 119906 isin 119863 119905 isin 119869)

le 1198861198700120572 (119906 (119905) | 119906 isin 119863 119905 isin 119869)

120572 (int

119886

0

119867(119905 119904) 119906 (119904) 119889119904 | 119906 isin 119863 119905 isin 119869)

le 1198861198670120572 (119906 (119905) | 119906 isin 119863 119905 isin 119869)

(41)

where 1198700= max

(119905119904)isinΔ|119870(119905 119904)| 119867

0= max

(119905119904)isinΔ 0|119867(119905 119904)|

Therefore by (24) (39) and (41) Lemma 6 and the hypothe-ses (H3) and (H4) we get that

120572 (119876(11199060)

(119863) (119905))

= 120572 (119876 (119863) (119905)) le 2120572 (119876 (1198631) (119905))

le 2120572 (T119902 (119905) 119892 (119906

1

119899)

+ int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904)

times119891 (119904 1199061

119899(119904) 119866119906

1

119899(119904) 119878119906

1

119899(119904)) 119889119904)

le4119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times 120572 (119891 (119904 1199061

119899(119904) 119866119906

1

119899(119904) 119878119906

1

119899(119904))) 119889119904

le4119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times [1198711120572 (1198631 (119904)) + 1198712120572 ((1198661198631) (119904))

+1198713120572 ((1198781198631) (119904))] 119889119904

le4119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times (1198711+ 11988611987001198712+ 11988611986701198713) 120572 (119863

1 (119904)) 119889119904

le4119872(119871

1 + 11988611987001198712 + 11988611986701198713) 119905119902

Γ (1 + 119902)120572 (119863)

(42)

Again by Lemma 7 there exists a countable set 1198632=

1199062

119899 sub co119876(11199060)(119863) 119906

0 such that

120572 (119876 (co 119876(11199060) (119863) 1199060) (119905)) le 2120572 (119876 (1198632) (119905)) (43)


Therefore by (24) (41) and (43) Lemma 6 and the hypothe-ses (H3) and (H4) we have that

120572 (119876(21199060)

(119863) (119905))

= 120572 (119876 (co 119876(11199060) (119863) 1199060) (119905))

le 2120572 (119876 (1198632) (119905))

le 2120572(int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904)

times119891 (119904 1199062

119899(119904) 119866119906

2

119899(119904) 119878119906

2

119899(119904)) 119889119904)

le4119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times 120572 (119891 (119904 1199062

119899(119904) 119866119906

2

119899(119904) 119878119906

2

119899(119904))) 119889119904

le4119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times (1198711+ 11988611987001198712+ 11988611986701198713) 120572 (119863

2 (119904)) 119889119904

le4119872(119871

1 + 11988611987001198712 + 11988611986701198713)

Γ (119902)

times int

119905

0

(119905 minus 119904)119902minus1

sdot 120572 (co 119876(11199060) (119863) 1199060 (119904)) 119889119904

le[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713)]2

Γ (119902) Γ (1 + 119902)

times int

119905

0

(119905 minus 119904)119902minus1

119904119902120572 (119863) 119889119904

=[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713) 119905119902]2

Γ (1 + 2119902)B (1 + 119902 119902)

times int

1

0

(1 minus 119904)119902minus1

119904119902119889119904 120572 (119863)

=[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713) 119905119902]2

Γ (1 + 2119902)120572 (119863)

(44)

where B(119901 119902) = int1

0119904119901minus1

(1 minus 119904)119902minus1

119889119904 is the Beta functionSuppose that

120572 (119876(1198961199060)

(119863) (119905))

le[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713) 119905119902]119896

Γ (1 + 119896119902)120572 (119863) forall119905 isin 119869

(45)

Then by Lemma 7 there exists a countable set 119863119896+1

=

119906119896+1

119899 sub co119876(1198961199060)(119863) 119906

0 such that

120572 (119876 (co 119876(1198961199060) (119863) 1199060) (119905)) le 2120572 (119876 (119863119896+1) (119905)) (46)

From (24) (41) and (46) using Lemma 6 and the hypotheses(H3) and (H4) we get that

120572 (119876(119896+11199060)

(119863) (119905))

= 120572 (119876 (co 119876(1198961199060) (119863) 1199060) (119905))

le 2120572 (119876 (119863119896+1) (119905))

le 2120572(int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904)

times119891 (119904 119906119896+1

119899(119904) 119866119906

119896+1

119899(119904) 119878119906

119896+1

119899(119904)) 119889119904)

le4119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times (1198711+ 11988611987001198712+ 11988611986701198713) 120572 (119863

119896+1 (119904)) 119889119904

le4119872(119871

1 + 11988611987001198712 + 11988611986701198713)

Γ (119902)

times int

119905

0

(119905 minus 119904)119902minus1

120572 (co 119876(1198961199060) (119863) 1199060 (119904)) 119889119904

le[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713)]119896+1

Γ (119902) Γ (1 + 119896119902)

times int

119905

0

(119905 minus 119904)119902minus1

119904119896119902120572 (119863) 119889119904

=[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713) 119905119902]119896+1

Γ (1 + (119896 + 1) 119902)B (1 + 119896119902 119902)

times int

1

0

(1 minus 119904)119902minus1

119904119896119902119889119904 120572 (119863)

=[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713) 119905119902]119896+1

Γ (1 + (119896 + 1) 119902)120572 (119863)

(47)

Hence by the method of mathematical induction for anypositive integer 119899 and 119905 isin 119869 we have

120572 (119876(1198991199060)

(119863) (119905)) le[4119872 (1198711 + 11988611987001198712 + 11988611986701198713) 119905

119902]119899

Γ (1 + 119899119902)120572 (119863)

(48)

Consequently from (38) and (48) we have

120572 (119876(1198991199060)

(119863)) = max119905isin119869

120572 (119876(1198991199060)

(119863) (119905))

le[4119872 (1198711 + 11988611987001198712 + 11988611986701198713) 119886

119902]119899

Γ (1 + 119899119902)120572 (119863)

(49)

By the well-known Stirlingrsquos formula we know that

Γ (1 + 119899119902) = radic2120587119899119902(119899119902

119890)

119899119902

119890]12119899119902

0 lt ] lt 1 (50)


Thus

[4119872 (1198711+ 11988611987001198712+ 11988611986701198713) 119886119902]119899

Γ (1 + 119899119902)

=[4119872 (1198711 + 11988611987001198712 + 11988611986701198713) 119886

119902]119899

radic2120587119899119902(119899119902119890)119899119902119890]12119899119902

997888rarr 0 as 119899 997888rarr infin

(51)

Therefore there exists a large enough positive integer 1198990 suchthat

[4119872 (1198711+ 11988611987001198712+ 11988611986701198713) 119886119902]1198990

Γ (1 + 1198990119902)

lt 1 (52)

Hence we get that

120572 (119876(1198990 1199060)

(119863)) lt 120572 (119863) (53)

Thus 119876 119865 rarr 119865 is a convex-power condensing operatorIt follows from Lemma 8 that 119876 has at least one fixed point119906 isin 119865 which is just a mild solution of the NPFEE (1) Thiscompletes the proof of Theorem 10

If we replace the hypotheses (H2) and (H4) by thefollowing hypotheses

(H2)1015840 there exist a function 120593 isin 11987111199021(119869R+) 119902

1isin [0 119902) and

a nondecreasing continuous function Ψ R+ rarr R+

such that

1003817100381710038171003817119891 (119905 119906 119866119906 119878119906)1003817100381710038171003817 le 120593 (119905) Ψ (119906) (54)

for all 119906 isin 119864 and ae 119905 isin 119869

(H4)1015840 the nonlocal function 119892 119862(119869 119864) rarr 119864 is continuousand compact there exist constants 0 lt 119887 lt 1119872 and119888 gt 0 such that for all 119906 isin 119862(119869 119864) 119892(119906) le 119887119906

119862+ 119888

we have the following existence result



0-semigroup 119879(119905)(119905 ge 0) of uniformly

bounded operators in 119864 Assume that the hypotheses (H1)(H2)1015840 (H3) and (H4)1015840 are satisfied then the NPFEE (1) hasat least one mild solution in 119862(119869 119864) provided that there existsa constant 119877 with

(119872119888 +Ψ (119877)119872119886

119902minus1199021

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus11990211003817100381710038171003817120593100381710038171003817100381711987111199021 [0119886]

)

times (1 minus119872119887)minus1le 119877

(55)

Proof From the proof of Theorem 10 we know that the mildsolution of the NPFEE (1) is equivalent to the fixed point ofthe operator119876 defined by (24) For any 119906 isin Ω

119877 by (24) (55)

and the hypotheses (H2)1015840 and (H4)1015840 we have

(119876119906) (119905) le 1198721003817100381710038171003817119892 (119906)

1003817100381710038171003817

+119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times1003817100381710038171003817119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904))

1003817100381710038171003817 119889119904

le 119872(119887119906119862 + 119888) +Ψ (119877)119872

Γ (119902)

times (int

119905

0

(119905 minus 119904)(119902minus1)(1minus1199021)119889119904)

1minus11990211003817100381710038171003817120593100381710038171003817100381711987111199021 [0119905]

le 119872(119887119877 + 119888) +Ψ (119877)119872119886

119902minus1199021

Γ (119902)

times (1 minus 1199021

119902 minus 1199021

)

1minus11990211003817100381710038171003817120593100381710038171003817100381711987111199021 [0119886]

le 119877

(56)

which implies 119876(Ω119877) sub Ω

119877 By adopting completely similar

method with the proof of Theorem 10 we can prove that theNPFEE (1) have at least one mild solution in 119862(119869 119864) Thiscompletes the proof of Theorem 11

Similar withTheorem 11 we have the following result

Corollary 12 Let 119864 be a real Banach space let 119860 119863(119860) sub

119864 rarr 119864 be a closed linear operator and minus119860 generate anequicontinuous 119862


bounded operators in 119864 Assume that the hypotheses (H1)(H2)1015840 (H3) and (H4)1015840 are satisfied then the NPFEE (1) haveat least one mild solution in 119862(119869 119864) provided that

(int

119886

0

12059311199021

(119904) 119889119904)

1199021

lt lim inf119903rarr+infin

(119903 (1 minus119872119887) +119872119888) Γ (119902)

Ψ (119903)119872119886119902minus1199021

times (119902 minus 1199021

1 minus 1199021

)

1minus1199021

(57)

4 Existence of Positive Mild Solutions

In this section we discuss the existence of positive mildsolutions for theNPFEE (1) we first introduce somenotationsand definitions which will be used in this section

Let 119864 be an ordered Banach space with the norm sdot

and let 119875 = 119906 isin 119864 | 119906 ge 120579 be a positive cone in 119864 whichdefines a partial ordering in119864 by119909 le 119910 if and only if119910minus119909 isin 119875where 120579denotes the zero element of119864119875 is said to be normal ifthere exists a positive constant119873 such that 120579 le 119909 le 119910 implies119909 le 119873119910 the infimum of all 119873 with the property aboveis called the normal constant of 119875 For more definitions anddetails of the cone 119875 we refer to the monographs [40 43]


Definition 13 A 1198620-semigroup 119879(119905) (119905 ge 0) in 119864 is called to

be positive if order inequality119879(119905)119906 ge 120579 holds for each 119906 ge 120579119906 isin 119864 and 119905 ge 0

For more details of the properties of positive 1198620-

semigroup we refer to the monograph [44] and the paper[45]

Lemma 14 If 119879(119905) (119905 ge 0) is a positive 1198620-semigroup in

119864 then T119902(119905) (119905 ge 0) and S

119902(119905) (119905 ge 0) are also positive

operators

Proof Since the semigroup 119879(119905) (119905 ge 0) and the functionℎ119902(119904) defined by (10) are positive by (9) we can easilyconclude that the operators T119902(119905) (119905 ge 0) and S

119902(119905) (119905 ge 0)

are also positive This completes the proof

Here we will obtain positive mild solutions under thefollowing assumptions

(A1) The function 119891 119869 times 119875 times 119875 times 119875 rarr 119875 satisfies theCaratheodory type conditions

(A2) There exist a constant 1199021 isin [0 119902) and a function 120593 isin

11987111199021(119869R+) such that for all 119906 isin 119875 and ae 119905 isin 119869119891(119905 119906 119866119906 119878119906) le 120593(119905)

(A3) There exist positive constants 119871119894(119894 = 1 2 3) such that

for any bounded and countable sets119863119894sub 119862(119869 119875) (119894 =

1 2 3) and ae 119905 isin 119869

120572 (119891 (119905 1198631 (119905) 1198632 (119905) 1198633 (119905))) le

3

sum

119894=1

119871119894120572 (119863119894 (119905)) (58)

(A4) The nonlocal function 119892 119862(119869 119875) rarr 119875 is continuousand compact and there exist constants 0 lt b lt

1(NM) and c gt 0 such that for all 119906 isin 119862(119869 119875)119892(119906) le 119887119906 + 119888 where119873 is the normal constant of thepositive cone 119875

Theorem 15 Let119864 be an ordered Banach space119875 be a normalpositive cone with normal constant N A D(A) sub E rarr

E be a closed linear operator and minusA generate a positiveand equicontinuous 1198620-semigroup 119879(119905) (119905 ge 0) of uniformlybounded operators in 119864 Assume that the assumptions (A1)(A2) (A3) and (A4) are satisfied then the NPFEE (1) have atleast one positive mild solution in 119862(119869 119875)

Proof From the proof of Theorem 10 we know that thepositive mild solution of the NPFEE (1) is equivalent to thefixed point of operator 119876 defined by (24) in 119862(119869 119875) Wechoose 119877 gt 0 big enough such that

119877 ge119873119872

1 minus 119873119872119887[119888 +

119886119902minus1199021

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus11990211003817100381710038171003817120593100381710038171003817100381711987111199021 [0119886]

] (59)

Then for any 119906 isin Ω119877(119875) = 119906 isin 119862(119869 119875) 119906

119862le 119877 by (24)

and the assumptions (A2) and (A4) we have that

120579 le (119876119906) (119905)

le T119902 (119905) (119887119906 + 119888)

+ int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904) 120593 (119904) 119889119904

(60)

By the normality of the cone 119875 (59) (60) the assumption(A4) and Holder inequality we get

(119876119906) (119905) le 119873119872(119887119906119862 + 119888)

+119873119872

Γ (119902)(int

119905

0


1minus1199021

times1003817100381710038171003817120593100381710038171003817100381711987111199021 [0119905]

le 119873119872(119887119877 + 119888) +119873119872119886

119902minus1199021

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus1199021

times1003817100381710038171003817120593100381710038171003817100381711987111199021 [0119886]

le 119877

(61)

Thus119876 Ω119877(119875) rarr Ω

119877(119875) Let 119865 = co 119876(Ω

119877(119875)) Similar to

the proof of Theorem 10 we can prove that 119876 119865 rarr 119865 is aconvex-power condensing operator It follows from Lemma 8that 119876 has at least one fixed point 119906 isin 119865 which is just apositive mild solution of the NPFEE (1) This completes theproof of Theorem 15

Theorem 16 Let119864 be an ordered Banach space119875 be a normalpositive cone with normal constant 119873 119860 119863(119860) sub 119864 rarr

119864 be a closed linear operator and minus119860 generate a positiveand equicontinuous 1198620-semigroup 119879(119905) (119905 ge 0) of uniformlybounded operators in 119864 Assume that the assumptions (A1)(A3) and the following assumptions

(A2)lowast There exist nonnegative continuous functions 1198921(119905)

1198922(119905) and abstract continuous function ℎ 119869 rarr 119875

such that119891 (119905 119906 V 119908) le 119892

1 (119905) 119906 + 1198922 (119905) V + ℎ (119905)

ae 119905 isin 119869 forall119906 V 119908 isin 119875

(62)

(A4)lowast The nonlocal function 119892 119862(119869 119875) rarr 119875 is continuousand compact and there exists a constant 119888 gt 0 suchthat for any 119906 isin 119862(119869 119875) 119892(119906) le 119888

are satisfied then the NPFEE (1) have at least one positive mildsolution in 119862(119869 119875)

Proof From the proof of Theorem 10 we know that thepositive mild solution of the NPFEE (1) is equivalent to thefixed point of operator 119876 defined by (24) in 119862(119869 119875) Let

(119861119906) (119905) = int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904)

times [1198921 (119904) 119906 (119904) + 1198922 (119904) 119866119906 (119904)] 119889119904 119905 isin 119869

(63)


We can prove that 119903(119861) = 0 where 119903(sdot) denotes the spectralradius of bounded linear operator In fact for any 119905 isin 119869 by(63) we get that

(119861119906) (119905)

le119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times

10038171003817100381710038171003817100381710038171003817

1198921 (119904) 119906 (119904)+1198922 (119904) int

119904

0

119870 (119904 120591) 119906 (120591) 119889120591

10038171003817100381710038171003817100381710038171003817

119889119904

le119872119866lowast(1 + 1198861198700) 119906119862

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

119889119904

=120573119905119902119906119862

Γ (1 + 119902)

(64)

where 119866lowast = maxmax119905isin1198691198921(119905)max119905isin1198691198922(119905) 120573 = 119872119866lowast(1 +

1198861198700) Further10038171003817100381710038171003817(1198612119906) (119905)

10038171003817100381710038171003817

le119872119866lowast

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times

10038171003817100381710038171003817100381710038171003817

(119861119906) (119904) + int

119904

0

119870 (119904 120591) (119861119906) (120591) 119889120591

10038171003817100381710038171003817100381710038171003817

119889119904

le119872119866lowast120573119906119862

Γ (119902) Γ (1 + 119902)

times int

119905

0

(119905 minus 119904)119902minus1

[119904119902+ int

119904

0

119870 (119904 120591) 120591119902119889120591] 119889119904

le1205732119906119862

Γ (1 + 2119902)B (1 + 119902 119902)

times int

119905

0

(119905 minus 119904)119902minus1

119904119902119889119904

=12057321199052119902119906119862

Γ (1 + 2119902)

(65)

By the method of mathematical induction for any positiveinteger 119899 and 119905 isin 119869 we obtain that

1003817100381710038171003817(119861119899119906) (119905)

1003817100381710038171003817 le120573119899119905119899119902119906119862

Γ (1 + 119899119902) (66)

Therefore we have

10038171003817100381710038171198611198991003817100381710038171003817119862

le120573119899119886119899119902

Γ (1 + 119899119902) (67)

Thus combining (50) we get that

119903 (119861) = lim119899rarrinfin

10038171003817100381710038171198611198991003817100381710038171003817

1119899

119862le lim119899rarrinfin

120573119886119902

2119899radic2120587119899119902(119899119902119890)

119902119890](12119899

2119902)= 0

(68)

Let 0 lt 120574 lt 1119873 From [46] we know that there exists anequivalent norm sdot

lowast in 119864 such that

119861lowastle 119903 (119861) + 120574 = 120574 (69)

where 119861lowast denotes the operator norm of 119861 with respect tothe norm sdot

lowastLet 119872

lowast= sup

119905isin[0+infin)119879(119905)

lowast

L(119864) and 119906lowast

119862=

sup119905isin119869119906(119905)

lowast Choose

119877lowast=119873119872lowast(Γ (1 + 119902) 119888 + 119886

119902ℎlowast

119862)

Γ (1 + 119902) (1 minus 119873120574) (70)

For any 119906 isin Ωlowast119877lowast(119875) = 119906 isin 119862(119869 119875) 119906

lowast

119862le 119877lowast by (24) and

the assumptions (A2)lowast and (A4)lowast we have that

120579 le (119876119906) (119905)

le T119902 (119905) 119892 (119906)

+ int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904)

times [1198921 (119904) 119906 (119904) + 1198922 (119904) 119866119906 (119904) + ℎ (119904)] 119889119904

le T119902 (119905) 119888 + 119861 (119906) (119905)

+ int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904) ℎ (119904) 119889119904

(71)

By the normality of the cone 119875 (69) (70) and (71) we have

(119876119906)(119905)lowastle 119873(

10038171003817100381710038171003817T119902(119905)119888

10038171003817100381710038171003817

lowast

+ (119861119906)(119905)lowast

+

10038171003817100381710038171003817100381710038171003817

int

119905

0

(119905 minus 119904)119902minus1

S119902(119905 minus 119904)ℎ(119904)119889119904

10038171003817100381710038171003817100381710038171003817

lowast

)

le 119873(119872lowast119888 + 119861

lowast119906lowast

119862+119872lowast119886119902ℎlowast

119862

Γ (1 + 119902))

le 119873119872lowast119888 + 119873120574119877

lowast+119873119872lowast119886119902ℎlowast

119862

Γ (1 + 119902)

= 119877lowast

(72)

Thus we have proved that 119876 Ωlowast

119877lowast(119875) rarr Ω

lowast

119877lowast(119875) Let

119865 = co 119876(Ωlowast

119877lowast(119875)) Similar to the proof of Theorem 10 we

can prove that 119876 119865 rarr 119865 is a convex-power condensingoperator It follows from Lemma 8 that 119876 has at least onefixed point 119906 isin 119865 which is just a positive mild solution ofthe NPFEE (1) This completes the proof of Theorem 16

Remark 17 Positive operator semigroups are widely appear-ing in heat conduction equations neutron transport equa-tions reaction diffusion equations and so on [47]Thereforeusing Theorems 15 and 16 to these partial differential equa-tions are very convenient

Remark 18 Analytic semigroup and differentiable semigroupare equicontinuous semigroup [48] In the application of


partial differential equations such as parabolic equationsand strongly damped wave equations the correspondingsolution semigroup is analytic semigroup Therefore theresults obtained in this paper have broad applicability

Acknowledgments

This research was supported by NNSF of China (11261053and 11061031) NSF of Gansu Province (1208RJZA129) andProject of NWNU-LKQN-11-3

References

[1] K S Miller and B Ross An Introduction to the FractionalCalculus and Differential Equations Wiley NewYork NY USA1993

[2] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[3] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204of North-Holland Mathematics Studies Elsevier Science BVAmsterdam The Netherlands 2006

[4] S D Eidelman and A N Kochubei ldquoCauchy problem forfractional diffusion equationsrdquo Journal of Differential Equationsvol 199 no 2 pp 211ndash255 2004

[5] V Lakshmikantham and A S Vatsala ldquoBasic theory of frac-tional differential equationsrdquo Nonlinear Analysis Theory Meth-ods amp Applications vol 69 no 8 pp 2677ndash2682 2008

[6] R P Agarwal V Lakshmikantham and J J Nieto ldquoOn theconcept of solution for fractional differential equations withuncertaintyrdquo Nonlinear Analysis Theory Methods amp Applica-tions vol 72 no 6 pp 2859ndash2862 2010

[7] M A Darwish and S K Ntouyas ldquoBoundary value problemsfor fractional functional differential equations of mixed typerdquoCommunications in Applied Analysis vol 13 no 1 pp 31ndash382009

[8] M A Darwish and S K Ntouyas ldquoSemilinear functionaldifferential equations of fractional order with state-dependentdelayrdquo Electronic Journal of Differential Equations vol 2009 no38 pp 1ndash10 2009

[9] M A Darwish and S K Ntouyas ldquoOn initial and boundaryvalue problems for fractional order mixed type functionaldifferential inclusionsrdquo Computers amp Mathematics with Appli-cations vol 59 no 3 pp 1253ndash1265 2010

[10] M A Darwish and S K Ntouyas ldquoOn a quadratic fractionalHammerstein-Volterra integral equation with linear modifica-tion of the argumentrdquo Nonlinear Analysis Theory Methods ampApplications vol 74 no 11 pp 3510ndash3517 2011

[11] M A Darwish J Henderson and D OrsquoRegan ldquoExistenceand asymptotic stability of solutions of a perturbed fractionalfunctional-integral equation with linear modification of theargumentrdquo Bulletin of the Korean Mathematical Society vol 48no 3 pp 539ndash553 2011

[12] M M El-Borai ldquoSome probability densities and fundamentalsolutions of fractional evolution equationsrdquo Chaos Solitons andFractals vol 14 no 3 pp 433ndash440 2002

[13] M M El-Borai ldquoSemigroups and some nonlinear fractionaldifferential equationsrdquo Applied Mathematics and Computationvol 149 no 3 pp 823ndash831 2004

[14] J Wang Y Zhou andM Feckan ldquoAbstract Cauchy problem forfractional differential equationsrdquo Nonlinear Dynamics vol 71no 4 pp 685ndash700 2013

[15] J Wang Y Zhou and M Medvedrsquo ldquoOn the solvability andoptimal controls of fractional integrodifferential evolution sys-tems with infinite delayrdquo Journal of Optimization Theory andApplications vol 152 no 1 pp 31ndash50 2012

[16] J Wang Y Zhou and W Wei ldquoOptimal feedback control forsemilinear fractional evolution equations in Banach spacesrdquoSystems amp Control Letters vol 61 no 4 pp 472ndash476 2012

[17] T Diagana G M Mophou and G M NrsquoGuerekata ldquoOnthe existence of mild solutions to some semilinear fractionalintegro-differential equationsrdquo Electronic Journal of QualitativeTheory of Differential Equations vol 2010 no 58 pp 1ndash17 2010

[18] R Wang T Xiao and J Liang ldquoA note on the fractionalCauchy problems with nonlocal initial conditionsrdquo AppliedMathematics Letters vol 24 no 8 pp 1435ndash1442 2011

[19] F Li J Liang and H K Xu ldquoExistence of mild solutions forfractional integrodifferential equations of Sobolev type withnonlocal conditionsrdquo Journal of Mathematical Analysis andApplications vol 391 no 2 pp 510ndash525 2012

[20] Y Zhou and F Jiao ldquoNonlocal Cauchy problem for fractionalevolution equationsrdquo Nonlinear Analysis Real World Applica-tions vol 11 no 5 pp 4465ndash4475 2010

[21] J Wang Y Zhou W Wei and H Xu ldquoNonlocal problems forfractional integrodifferential equations via fractional operatorsand optimal controlsrdquo Computers amp Mathematics with Applica-tions vol 62 no 3 pp 1427ndash1441 2011

[22] J Wang M Feckan and Y Zhou ldquoOn the new concept of solu-tions and existence results for impulsive fractional evolutionequationsrdquoDynamics of Partial Differential Equations vol 8 no4 pp 345ndash361 2011

[23] J Wang Z Fan and Y Zhou ldquoNonlocal controllability of semi-linear dynamic systems with fractional derivative in Banachspacesrdquo Journal of Optimization Theory and Applications vol154 no 1 pp 292ndash302 2012

[24] Y Chang V Kavitha and M Mallika Arjunan ldquoExistence anduniqueness of mild solutions to a semilinear integrodifferentialequation of fractional orderrdquo Nonlinear Analysis Theory Meth-ods amp Applications vol 71 no 11 pp 5551ndash5559 2009

[25] K Balachandran and J Y Park ldquoNonlocal Cauchy problem forabstract fractional semilinear evolution equationsrdquo NonlinearAnalysis Theory Methods amp Applications vol 71 no 10 pp4471ndash4475 2009

[26] K Balachandran and J J Trujillo ldquoThe nonlocal Cauchyproblem for nonlinear fractional integrodifferential equationsin Banach spacesrdquo Nonlinear Analysis Theory Methods ampApplications vol 72 no 12 pp 4587ndash4593 2010

[27] L Byszewski and V Lakshmikantham ldquoTheorem about theexistence and uniqueness of a solution of a nonlocal abstractCauchy problem in a Banach spacerdquoApplicable Analysis vol 40no 1 pp 11ndash19 1991

[28] L Byszewski ldquoTheorems about the existence and uniqueness ofsolutions of a semilinear evolution nonlocal Cauchy problemrdquoJournal of Mathematical Analysis and Applications vol 162 no2 pp 494ndash505 1991

[29] L Byszewski ldquoApplication of properties of the right-hand sidesof evolution equations to an investigation of nonlocal evolutionproblemsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 33 no 5 pp 413ndash426 1998


[30] H Liu and J-C Chang ldquoExistence for a class of partial differ-ential equations with nonlocal conditionsrdquo Nonlinear AnalysisTheory Methods amp Applications vol 70 no 9 pp 3076ndash30832009

[31] J Wang and W Wei ldquoA class of nonlocal impulsive problemsfor integrodifferential equations in Banach spacesrdquo Results inMathematics vol 58 no 3-4 pp 379ndash397 2010

[32] P Chen and Y Li ldquoMonotone iterative technique for a classof semilinear evolution equations with nonlocal conditionsrdquoResults in Mathematics 2012

[33] V Lakshmikantham and S Leela Nonlinear Differential Equa-tions in Abstract Spaces Pergamon Press New York NY USA1981

[34] DGuo ldquoSolutions of nonlinear integro-differential equations ofmixed type in Banach spacesrdquo Journal of Applied Mathematicsand Simulation vol 2 no 1 pp 1ndash11 1989

[35] L Liu C Wu and F Guo ldquoExistence theorems of global solu-tions of initial value problems for nonlinear integro-differentialequations of mixed type in Banach spaces and applicationsrdquoComputers amp Mathematics with Applications vol 47 no 1 pp13ndash22 2004

[36] L Liu F Guo C Wu and Y Wu ldquoExistence theorems ofglobal solutions for nonlinear Volterra type integral equationsin Banach spacesrdquo Journal of Mathematical Analysis and Appli-cations vol 309 no 2 pp 638ndash649 2005

[37] J X Sun and X Y Zhang ldquoThe fixed point theorem ofconvex-power condensing operator and applications to abstractsemilinear evolution equationsrdquo Acta Mathematica Sinica vol48 no 3 pp 439ndash446 2005 (Chinese)

[38] H B Shi W T Li and H R Sun ldquoExistence of mild solutionsfor abstractmixed type semilinear evolution equationsrdquoTurkishJournal of Mathematics vol 35 no 3 pp 457ndash472 2011

[39] J Banas and K GoebelMeasures of Noncompactness in BanachSpaces vol 60 ofLectureNotes in Pure andAppliedMathematicsMarcel Dekker New York NY USA 1980

[40] K Deimling Nonlinear Functional Analysis Springer NewYork NY USA 1985

[41] H P Heinz ldquoOn the behaviour of measures of noncompactnesswith respect to differentiation and integration of vector-valuedfunctionsrdquoNonlinear Analysis Theory Methods amp Applicationsvol 7 no 12 pp 1351ndash1371 1983

[42] Y X Li ldquoExistence of solutions to initial value problems forabstract semilinear evolution equationsrdquo Acta MathematicaSinica vol 48 no 6 pp 1089ndash1094 2005 (Chinese)

[43] D J Guo and V Lakshmikantham Nonlinear Problems inAbstract Cones Academic Press Orlando Fla USA 1988

[44] J Banasiak and L Arlotti Perturbations of Positive Semigroupswith Applications Springer London UK 2006

[45] Y X Li ldquoThe positive solutions of abstract semilinear evolutionequations and their applicationsrdquo Acta Mathematica Sinica vol39 no 5 pp 666ndash672 1996 (Chinese)

[46] I Vasile Fixed Point Theory D Reidel Publishing DordrechtThe Netherlands 1981

[47] R Nagel W Arendt A Grabosch et al One ParameterSemigroups of Positive Operators vol 1184 of Lecture Notes inMathematics Springer Berlin Germany 1986

[48] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer Berlin Germany 1983

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014


Stochastic AnalysisInternational Journal of


bymany authors and some basic results on nonlocal problemshave been obtained see [17ndash32] and the references thereinfor more comments and citations In the past few years theexistence uniqueness and some other properties of mildsolutions to nonlocal problem of fractional evolution equa-tions (1) with 119891(119905 119906(119905) 119866119906(119905) 119878119906(119905)) = 119891(119905 119906(119905)) have beenextensively studied by using Banach contraction mappingprincipal Schauderrsquos fixed point theorem and Krasnoselskiirsquosfixed point theorem when 119879(119905) (119905 ge 0) is a compactsemigroup For more details on the basic theory of nonlocalproblem for fractional evolution equations one can see thepapers of Diagana et al [17] Wang et al [18] Li et al [19]Zhou and Jiao [20]Wang et al [21] Wang et al [22] Wang etal [23] Chang et al [24] Balachandran and Park [25] andBalachandran and Trujillo [26] However for the case thatthe semigroup 119879(119905) (119905 ge 0) is noncompact there are veryfew papers studied nonlocal problem of fractional evolutionequations that only Wang et al [14] discussed the existenceofmild solutions for nonlocal problem of fractional evolutionequations under the situation that 119879(119905) (119905 ge 0) is an analyticsemigroup of uniformly bounded linear operators

It is well known that the famous Sadovskiirsquos fixed pointtheorem is an important tool to study various differentialequations and integral equations on infinite dimensionalBanach spaces Early on Lakshmikantham and Leela [33]studied the following initial value problem (IVP) of ordinarydifferential equation in Banach space 119864

1199061015840(119905) = 119891 (119905 119906 (119905)) 119905 isin 119869

119906 (0) = 1199060

(3)

and they proved that if for any 119877 gt 0 119891 is uniformlycontinuous on 119869 times 119861

119877and satisfies the noncompactness

measure condition

120572 (119891 (119905 119863)) le 119871120572 (119863) forall119905 isin 119869 119863 sub 119861119877 (4)

where 119861119877= 119906 isin 119864 119906 le 119877 119871 is a positive constant

120572(sdot) denotes the Kuratowski measure of noncompactness in119864 then IVP (3) has a global solution provided that 119871 satisfiesthe condition

119871 lt1

119886 (5)

In fact there are a large amount of authors who studiedordinary differential equations in Banach spaces similar to(3) by using Sadovskiirsquos fixed point theorem and hypothesisanalogous to (4) they also required that the constants satisfya strong inequality similar to (5) Formore details on this factwe refer to Guo [34] Liu et al [35] and Liu et al [36]

It is easy to see that the inequality (5) is a strongrestrictive condition and it is difficult to be satisfied inapplications In order to remove the strong restriction on theconstant 119871 Sun and Zhang [37] generalized the definition ofcondensing operator to convex-power condensing operatorAnd based on the definition of this new kind of operator theyestablished a new fixed point theoremwith respect to convex-power condensing operator which generalizes the famousSchauderrsquos fixed point theorem and Sadovskiirsquos fixed point

theorem As an application they investigated the existenceof global mild solutions and positive mild solutions for theinitial value problem of evolution equations in 119864

1199061015840(119905) + 119860119906 (119905) = 119891 (119905 119906 (119905)) 119905 isin 119869

119906 (0) = 1199060

(6)

they assume that minus119860 generate a equicontinuous 1198620-

semigroup the nonlinear term 119891 is uniformly continuouson 119869 times 119861

119877and satisfies a suitable noncompactness

measure condition similar to (4) Recently Shi et al[38] developed the IVP (6) to the case that the nonlinearterm is 119891(119905 119906(119905) 119866119906(119905) 119878119906(119905)) and obtained the existence ofglobal mild solutions and positive mild solutions by usingthe new fixed point theorem with respect to convex-powercondensing operator established in [37] but they also requirethat the nonlinear term 119891 is uniformly continuous on119869 times 119861119877 times 119861119877 times 119861119877

We observed that in [33ndash38] the authors all demandthat the nonlinear term 119891 is uniformly continuous this isa very strong assumption As a matter of fact if 119891(119905 119906)is Lipschitz continuous on 119869 times 119861

119877with respect to 119906 then

the condition (4) is satisfied but 119891 may not be necessarilyuniformly continuous on 119869 times 119861

119877

Motivated by the above mentioned aspects in this paperwe studied the existence of mild solutions and positive mildsolutions for the NPFEE (1) by utilizing a new fixed pointtheorem with respect to convex-power condensing operatordue to Sun and Zhang [37] (see Lemma 8) Furthermorewe deleted the assumption that 119891 is uniformly continuousby using a new estimation technique of the measure ofnoncompactness (see Lemma 7)

2 Preliminaries

In this section we introduce some notations definitions andpreliminary facts which are used throughout this paper

Let 119864 be a real Banach space with the norm sdot Wedenote by 119862(119869 119864) the Banach space of all continuous 119864-value functions on interval 119869 with the supnorm 119906119862 =

sup119905isin119869119906(119905) and by 1198711(119869 119864) the Banach space of all 119864-value

Bochner integrable functions defined on 119869 with the norm1199061 = int1

0119906(119905)119889119905

Definition 1 (see [3]) The fractional integral of order 119902 gt 0

with the lower limit zero for a function 119906 isin 1198711(119869 119864) is definedas

119868119902

119905119906 (119905) =

1

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

119906 (119904) 119889119904 119905 gt 0 (7)

where Γ(sdot) is the Euler gamma function



119862119863119902

119905119906 (119905) =

1

Γ (119899 minus 119902)int

119905

0


119906(119899)(119904) 119889119904

119905 gt 0 0 le 119899 minus 1 lt 119902 lt 119899

(8)




T119902 (119905) 119906 = int

infin

0

ℎ119902 (119904) 119879 (119905119902119904) 119906 119889119904

S119902 (119905) 119906 = 119902int

infin

0

119904ℎ119902 (119904) 119879 (119905

119902119904) 119906 119889119904

(9)

where

ℎ119902 (119904) =1

120587119902

infin

sum

119899=1


Γ (119899119902 + 1)

119899sin (119899120587119902) 119904 isin (0infin)

(10)



infin

0

ℎ119902 (119904) 119889119904 = 1

int

infin

0

119904Vℎ119902 (119904) 119889119904 =

Γ (1 + V)

Γ (1 + 119902V) V isin [0 1]

(11)








10038171003817100381710038171003817le 119872119906

10038171003817100381710038171003817S119902 (119905) 119906

10038171003817100381710038171003817le

119902119872

Γ (1 + 119902)119906 =

119872

Γ (119902)119906

(12)


119902(119905)119906 are




10038171003817100381710038171003817T119902(11990510158401015840) 119906 minusT

119902(1199051015840) 119906

10038171003817100381710038171003817997888rarr 0

10038171003817100381710038171003817S119902(11990510158401015840) 119906 minusS

119902(1199051015840) 119906


(13)


119906 (119905) = T119902 (119905) 119892 (119906) + int

119905

0

(119905 minus 119904)119902minus1

timesS119902 (119905 minus 119904) 119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904)) 119889119904

(14)





120572 (119863) = max119905isin119869

120572 (119863 (119905)) = 120572 (119863 (119869)) (15)



120572(int119869

119906119899 (119905) 119889119905 | 119899 isin N) le 2int

119869

120572 (119863 (119905)) 119889119905 (16)



120572(119863) le 2120572(1198630)



and 1199090isin 119863 set

119876(11199090)

(119878) equiv 119876 (119878)

119876(1198991199090)

(119878) = 119876 (co 119876(119899minus11199090) (119878) 1199090) 119899 = 2 3

(17)


0such that for any


120572 (119876(1198990 1199090)

(119878)) lt 120572 (119878) (18)



120590 (19)






1gt



1003817100381710038171003817 le 120593119903 (119905)


1003817100381710038171003817120593119903100381710038171003817100381711987111199021 [0119886]

119903= 1205881lt +infin

(20)



1 2 3) and ae 119905 isin 119869

120572 (119891 (119905 1198631 1198632 1198633)) le

3

sum

119894=1

119871119894120572 (119863119894) (21)


2gt 0 and a



119862(119869 119864) 119906119862le 1199031003817100381710038171003817119892 (119906)

1003817100381710038171003817 le Φ (119903)


Φ (119903)

119903= 1205882 lt +infin

(22)





1198721205882 +1198721205881119886119902minus1199021

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus1199021

lt 1 (23)


(119876119906) (119905) = T119902 (119905) 119892 (119906) + int119905

0

(119905 minus 119904)119902minus1


(24)



119877) sub Ω





119903 lt1003817100381710038171003817(119876119906119903) (119905119903)

1003817100381710038171003817

le 1198721003817100381710038171003817119892 (119906119903)

1003817100381710038171003817

+119872

Γ (119902)int

119905119903

0

(119905119903minus 119904)119902minus1

times1003817100381710038171003817119891 (119904 119906119903 (119904) 119866119906119903 (119904) 119878119906119903 (119904))

1003817100381710038171003817 119889119904

le 119872Φ (119903) +119872

Γ (119902)(int

119905119903

0


119889119904)

1minus1199021

times1003817100381710038171003817120593119903

100381710038171003817100381711987111199021 [0119905119903]

le 119872Φ (119903) +119872119886119902minus1199021

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus11990211003817100381710038171003817120593119903

100381710038171003817100381711987111199021 [0119886]

(25)


1198721205882 +

1198721205881119886119902minus1199021

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus1199021

ge 1 (26)



119899infin


119899rarr+infin119906119899= 119906


of 119891 we have

lim119899rarr+infin

119892 (119906119899) = 119892 (119906)

lim119899rarr+infin

119891 (119904 119906119899 (119904) 119866119906119899 (119904) 119878119906119899 (119904))

= 119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904)) ae 119904 isin 119869

(27)


(119905 minus 119904)119902minus1 1003817100381710038171003817119891 (119904 119906119899 (119904) 119866119906119899 (119904) 119878119906119899 (119904))

minus119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904))1003817100381710038171003817

le 2(119905 minus 119904)119902minus1

120593119877 (119904)

(28)



1003817100381710038171003817(119876119906119899) (119905) minus (119876119906) (119905)1003817100381710038171003817

le 1198721003817100381710038171003817119892 (119906119899) minus 119892 (119906)

1003817100381710038171003817

+119872

Γ (119902)


times int

119905

0

(119905 minus 119904)119902minus1

times1003817100381710038171003817119891 (119904 119906119899 (119904) 119866119906119899 (119904) 119878119906119899 (119904))

minus119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904))1003817100381710038171003817 119889119904 997888rarr 0


(29)


1003817100381710038171003817(119876119906119899) minus (119876119906)1003817100381710038171003817119862



119877rarr Ω119877is


1lt 1199052le 119886 we get

that

(119876119906) (1199052) minus (119876119906) (1199051)

= T119902 (1199052) 119892 (119906) minusT119902 (1199051) 119892 (119906)

+ int

1199052

1199051

(1199052 minus 119904)119902minus1

S119902 (1199052 minus 119904)

times 119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904)) 119889119904

+ int

1199051

0

((1199052 minus 119904)119902minus1


)

timesS119902 (1199052 minus 119904) 119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904)) 119889119904

+ int

1199051

0

(1199051minus 119904)119902minus1

(S119902(1199052minus 119904) minusS

119902(1199051minus 119904))

times 119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904)) 119889119904

= 1198681+ 1198682+ 1198683+ 1198684

(31)

where

1198681 = T119902 (1199052) 119892 (119906) minusT119902 (1199051) 119892 (119906)

1198682= int

1199052

1199051

(1199052minus 119904)119902minus1

S119902(1199052minus 119904)

times 119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904)) 119889119904

1198683 = int

1199051

0

((1199052 minus 119904)119902minus1


)

timesS119902(1199052minus 119904) 119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904)) 119889119904

1198684= int

1199051

0

(1199051minus 119904)119902minus1

(S119902(1199052minus 119904) minusS

119902(1199051minus 119904))

times 119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904)) 119889119904

(32)

It is obvious that

1003817100381710038171003817(119876119906) (1199052) minus (119876119906) (1199051)1003817100381710038171003817 le

4

sum

119894=1

10038171003817100381710038171198681198941003817100381710038171003817 (33)


of 119906 isin Ω119877when 119905

2minus 1199051rarr 0 119894 = 1 2 4

For 1198681 by Lemma 3(3) 119868

1 rarr 0 as 119905



inequality we have

100381710038171003817100381711986821003817100381710038171003817 le

119872

Γ (119902)int

1199052

1199051

(1199052minus 119904)119902minus1

120593119877 (119904) 119889119904

le119872

Γ (119902)(int

1199052

1199051


119889119904)

1minus1199021

times1003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [1199051 1199052]

le

1198721003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [0119886]

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus1199021

(1199052minus 1199051)119902minus1199021

997888rarr 0

as 1199052minus 1199051997888rarr 0

(34)



100381710038171003817100381711986831003817100381710038171003817 le

119872

Γ (119902)int

1199051

0

((1199051 minus 119904)119902minus1


) 120593119877 (119904) 119889119904

le119872

Γ (119902)(int

1199051

0

((1199051minus 119904)119902minus1


)1(1minus1199021)

119889119904)

1minus1199021

times1003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [01199051]

le119872

Γ (119902)(int

1199051

0



) 119889119904)

1minus1199021

times1003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [0119886]

le

1198721003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [0119886]

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus1199021



2


)

1minus1199021

le

11987221minus11990211003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [0119886]

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus1199021

(1199052minus 1199051)119902minus1199021


(35)


For 1199051= 0 0 lt 119905


4 = 0 For 119905

1gt 0


100381710038171003817100381711986841003817100381710038171003817 le int

1199051minus120598

0

(1199051minus 119904)119902minus1


10038171003817100381710038171003817120593119877 (119904) 119889119904

+ int

1199051

1199051minus120598

(1199051minus 119904)119902minus1


119902(1199051minus 119904)

10038171003817100381710038171003817120593119877 (119904) 119889119904

le sup119904isin[01199051minus120598]

10038171003817100381710038171003817S119902(1199052minus 119904) minusS

119902(1199051minus 119904)

10038171003817100381710038171003817

times int

1199051minus120598

0

(1199051 minus 119904)119902minus1

120593119877 (119904) 119889119904

+2119872

Γ (119902)int

1199051

1199051minus120598

(1199051minus 119904)119902minus1

120593119877 (119904) 119889119904

le sup119904isin[01199051minus120598]

10038171003817100381710038171003817S119902(1199052minus 119904) minusS

119902(1199051minus 119904)

10038171003817100381710038171003817

sdot1003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [0119886](1 minus 1199021

119902 minus 1199021

)

1minus1199021

sdot (119905(119902minus1199021)(1minus1199021)


1minus1199021

+

21198721003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [0119886]

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus1199021

120598119902minus1199021 997888rarr 0

as 1199052 minus 1199051 997888rarr 0

(36)

As a result (119876119906)(1199052)minus(119876119906)(119905



119877rarr Ω119877




120572 (119876(1198990 1199060)

(119863)) lt 120572 (119863) (37)




120572 (119876(1198991199060)

(119863)) = max119905isin119869

120572 (119876(1198991199060)

(119863) (119905)) 119899 = 1 2

(38)


119899 sub 119863 such

that

120572 (119876 (119863) (119905)) le 2120572 (119876 (1198631) (119905)) (39)


int

119886

0

119906 (119904) 119889119904 isin 119886 co 119906 (119904) | 119904 isin 119869 119906 isin 119862 (119869 119864) (40)

we have

120572(int

119905

0

119870 (119905 119904) 119906 (119904) 119889119904 | 119906 isin 119863 119905 isin 119869)

le 1198861198700120572 (119906 (119905) | 119906 isin 119863 119905 isin 119869)

120572 (int

119886

0

119867(119905 119904) 119906 (119904) 119889119904 | 119906 isin 119863 119905 isin 119869)

le 1198861198670120572 (119906 (119905) | 119906 isin 119863 119905 isin 119869)

(41)

where 1198700= max

(119905119904)isinΔ|119870(119905 119904)| 119867

0= max

(119905119904)isinΔ 0|119867(119905 119904)|


120572 (119876(11199060)

(119863) (119905))

= 120572 (119876 (119863) (119905)) le 2120572 (119876 (1198631) (119905))

le 2120572 (T119902 (119905) 119892 (119906

1

119899)

+ int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904)

times119891 (119904 1199061

119899(119904) 119866119906

1

119899(119904) 119878119906

1

119899(119904)) 119889119904)

le4119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times 120572 (119891 (119904 1199061

119899(119904) 119866119906

1

119899(119904) 119878119906

1

119899(119904))) 119889119904

le4119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times [1198711120572 (1198631 (119904)) + 1198712120572 ((1198661198631) (119904))

+1198713120572 ((1198781198631) (119904))] 119889119904

le4119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times (1198711+ 11988611987001198712+ 11988611986701198713) 120572 (119863

1 (119904)) 119889119904

le4119872(119871

1 + 11988611987001198712 + 11988611986701198713) 119905119902

Γ (1 + 119902)120572 (119863)

(42)


1199062

119899 sub co119876(11199060)(119863) 119906

0 such that

120572 (119876 (co 119876(11199060) (119863) 1199060) (119905)) le 2120572 (119876 (1198632) (119905)) (43)



120572 (119876(21199060)

(119863) (119905))

= 120572 (119876 (co 119876(11199060) (119863) 1199060) (119905))

le 2120572 (119876 (1198632) (119905))

le 2120572(int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904)

times119891 (119904 1199062

119899(119904) 119866119906

2

119899(119904) 119878119906

2

119899(119904)) 119889119904)

le4119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times 120572 (119891 (119904 1199062

119899(119904) 119866119906

2

119899(119904) 119878119906

2

119899(119904))) 119889119904

le4119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times (1198711+ 11988611987001198712+ 11988611986701198713) 120572 (119863

2 (119904)) 119889119904

le4119872(119871

1 + 11988611987001198712 + 11988611986701198713)

Γ (119902)

times int

119905

0

(119905 minus 119904)119902minus1

sdot 120572 (co 119876(11199060) (119863) 1199060 (119904)) 119889119904

le[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713)]2

Γ (119902) Γ (1 + 119902)

times int

119905

0

(119905 minus 119904)119902minus1

119904119902120572 (119863) 119889119904

=[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713) 119905119902]2

Γ (1 + 2119902)B (1 + 119902 119902)

times int

1

0

(1 minus 119904)119902minus1

119904119902119889119904 120572 (119863)

=[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713) 119905119902]2

Γ (1 + 2119902)120572 (119863)

(44)

where B(119901 119902) = int1

0119904119901minus1

(1 minus 119904)119902minus1


120572 (119876(1198961199060)

(119863) (119905))

le[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713) 119905119902]119896

Γ (1 + 119896119902)120572 (119863) forall119905 isin 119869

(45)


=

119906119896+1

119899 sub co119876(1198961199060)(119863) 119906

0 such that

120572 (119876 (co 119876(1198961199060) (119863) 1199060) (119905)) le 2120572 (119876 (119863119896+1) (119905)) (46)


120572 (119876(119896+11199060)

(119863) (119905))

= 120572 (119876 (co 119876(1198961199060) (119863) 1199060) (119905))

le 2120572 (119876 (119863119896+1) (119905))

le 2120572(int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904)

times119891 (119904 119906119896+1

119899(119904) 119866119906

119896+1

119899(119904) 119878119906

119896+1

119899(119904)) 119889119904)

le4119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times (1198711+ 11988611987001198712+ 11988611986701198713) 120572 (119863

119896+1 (119904)) 119889119904

le4119872(119871

1 + 11988611987001198712 + 11988611986701198713)

Γ (119902)

times int

119905

0

(119905 minus 119904)119902minus1

120572 (co 119876(1198961199060) (119863) 1199060 (119904)) 119889119904

le[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713)]119896+1

Γ (119902) Γ (1 + 119896119902)

times int

119905

0

(119905 minus 119904)119902minus1

119904119896119902120572 (119863) 119889119904

=[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713) 119905119902]119896+1

Γ (1 + (119896 + 1) 119902)B (1 + 119896119902 119902)

times int

1

0

(1 minus 119904)119902minus1

119904119896119902119889119904 120572 (119863)

=[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713) 119905119902]119896+1

Γ (1 + (119896 + 1) 119902)120572 (119863)

(47)


120572 (119876(1198991199060)

(119863) (119905)) le[4119872 (1198711 + 11988611987001198712 + 11988611986701198713) 119905

119902]119899

Γ (1 + 119899119902)120572 (119863)

(48)


120572 (119876(1198991199060)

(119863)) = max119905isin119869

120572 (119876(1198991199060)

(119863) (119905))

le[4119872 (1198711 + 11988611987001198712 + 11988611986701198713) 119886

119902]119899

Γ (1 + 119899119902)120572 (119863)

(49)


Γ (1 + 119899119902) = radic2120587119899119902(119899119902

119890)

119899119902

119890]12119899119902

0 lt ] lt 1 (50)


Thus

[4119872 (1198711+ 11988611987001198712+ 11988611986701198713) 119886119902]119899

Γ (1 + 119899119902)

=[4119872 (1198711 + 11988611987001198712 + 11988611986701198713) 119886

119902]119899

radic2120587119899119902(119899119902119890)119899119902119890]12119899119902


(51)


[4119872 (1198711+ 11988611987001198712+ 11988611986701198713) 119886119902]1198990

Γ (1 + 1198990119902)

lt 1 (52)

Hence we get that

120572 (119876(1198990 1199060)

(119863)) lt 120572 (119863) (53)




1isin [0 119902) and


such that

1003817100381710038171003817119891 (119905 119906 119866119906 119878119906)1003817100381710038171003817 le 120593 (119905) Ψ (119906) (54)



119862+ 119888






(119872119888 +Ψ (119877)119872119886

119902minus1199021

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus11990211003817100381710038171003817120593100381710038171003817100381711987111199021 [0119886]

)


(55)


119877 by (24) (55)


(119876119906) (119905) le 1198721003817100381710038171003817119892 (119906)

1003817100381710038171003817

+119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times1003817100381710038171003817119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904))

1003817100381710038171003817 119889119904

le 119872(119887119906119862 + 119888) +Ψ (119877)119872

Γ (119902)

times (int

119905

0


1minus11990211003817100381710038171003817120593100381710038171003817100381711987111199021 [0119905]

le 119872(119887119877 + 119888) +Ψ (119877)119872119886

119902minus1199021

Γ (119902)


119902 minus 1199021

)

1minus11990211003817100381710038171003817120593100381710038171003817100381711987111199021 [0119886]

le 119877

(56)









(int

119886

0

12059311199021

(119904) 119889119904)

1199021


(119903 (1 minus119872119887) +119872119888) Γ (119902)

Ψ (119903)119872119886119902minus1199021

times (119902 minus 1199021

1 minus 1199021

)

1minus1199021

(57)











119864 then T119902(119905) (119905 ge 0) and S


operators


119902(119905) (119905 ge 0)








1 2 3) and ae 119905 isin 119869

120572 (119891 (119905 1198631 (119905) 1198632 (119905) 1198633 (119905))) le

3

sum

119894=1

119871119894120572 (119863119894 (119905)) (58)






119877 ge119873119872

1 minus 119873119872119887[119888 +

119886119902minus1199021

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus11990211003817100381710038171003817120593100381710038171003817100381711987111199021 [0119886]

] (59)


119862le 119877 by (24)


120579 le (119876119906) (119905)

le T119902 (119905) (119887119906 + 119888)

+ int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904) 120593 (119904) 119889119904

(60)


(119876119906) (119905) le 119873119872(119887119906119862 + 119888)

+119873119872

Γ (119902)(int

119905

0


1minus1199021

times1003817100381710038171003817120593100381710038171003817100381711987111199021 [0119905]

le 119873119872(119887119877 + 119888) +119873119872119886

119902minus1199021

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus1199021

times1003817100381710038171003817120593100381710038171003817100381711987111199021 [0119886]

le 119877

(61)

Thus119876 Ω119877(119875) rarr Ω

119877(119875) Let 119865 = co 119876(Ω

119877(119875)) Similar to






such that119891 (119905 119906 V 119908) le 119892

1 (119905) 119906 + 1198922 (119905) V + ℎ (119905)


(62)




(119861119906) (119905) = int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904)

times [1198921 (119904) 119906 (119904) + 1198922 (119904) 119866119906 (119904)] 119889119904 119905 isin 119869

(63)



(119861119906) (119905)

le119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times

10038171003817100381710038171003817100381710038171003817

1198921 (119904) 119906 (119904)+1198922 (119904) int

119904

0

119870 (119904 120591) 119906 (120591) 119889120591

10038171003817100381710038171003817100381710038171003817

119889119904

le119872119866lowast(1 + 1198861198700) 119906119862

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

119889119904

=120573119905119902119906119862

Γ (1 + 119902)

(64)


1198861198700) Further10038171003817100381710038171003817(1198612119906) (119905)

10038171003817100381710038171003817


Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times

10038171003817100381710038171003817100381710038171003817

(119861119906) (119904) + int

119904

0

119870 (119904 120591) (119861119906) (120591) 119889120591

10038171003817100381710038171003817100381710038171003817

119889119904

le119872119866lowast120573119906119862

Γ (119902) Γ (1 + 119902)

times int

119905

0

(119905 minus 119904)119902minus1

[119904119902+ int

119904

0

119870 (119904 120591) 120591119902119889120591] 119889119904

le1205732119906119862

Γ (1 + 2119902)B (1 + 119902 119902)

times int

119905

0

(119905 minus 119904)119902minus1

119904119902119889119904

=12057321199052119902119906119862

Γ (1 + 2119902)

(65)


1003817100381710038171003817(119861119899119906) (119905)

1003817100381710038171003817 le120573119899119905119899119902119906119862

Γ (1 + 119899119902) (66)

Therefore we have

10038171003817100381710038171198611198991003817100381710038171003817119862

le120573119899119886119899119902

Γ (1 + 119899119902) (67)


119903 (119861) = lim119899rarrinfin

10038171003817100381710038171198611198991003817100381710038171003817

1119899


120573119886119902

2119899radic2120587119899119902(119899119902119890)

119902119890](12119899

2119902)= 0

(68)



119861lowastle 119903 (119861) + 120574 = 120574 (69)


lowastLet 119872

lowast= sup

119905isin[0+infin)119879(119905)

lowast

L(119864) and 119906lowast

119862=

sup119905isin119869119906(119905)

lowast Choose

119877lowast=119873119872lowast(Γ (1 + 119902) 119888 + 119886

119902ℎlowast

119862)

Γ (1 + 119902) (1 minus 119873120574) (70)


lowast

119862le 119877lowast by (24) and


120579 le (119876119906) (119905)

le T119902 (119905) 119892 (119906)

+ int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904)

times [1198921 (119904) 119906 (119904) + 1198922 (119904) 119866119906 (119904) + ℎ (119904)] 119889119904

le T119902 (119905) 119888 + 119861 (119906) (119905)

+ int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904) ℎ (119904) 119889119904

(71)


(119876119906)(119905)lowastle 119873(

10038171003817100381710038171003817T119902(119905)119888

10038171003817100381710038171003817

lowast

+ (119861119906)(119905)lowast

+

10038171003817100381710038171003817100381710038171003817

int

119905

0

(119905 minus 119904)119902minus1

S119902(119905 minus 119904)ℎ(119904)119889119904

10038171003817100381710038171003817100381710038171003817

lowast

)

le 119873(119872lowast119888 + 119861

lowast119906lowast

119862+119872lowast119886119902ℎlowast

119862

Γ (1 + 119902))

le 119873119872lowast119888 + 119873120574119877


119862

Γ (1 + 119902)

= 119877lowast

(72)



lowast

119877lowast(119875) Let

119865 = co 119876(Ωlowast







Acknowledgments


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014





119862119863119902

119905119906 (119905) =

1

Γ (119899 minus 119902)int

119905

0


119906(119899)(119904) 119889119904

119905 gt 0 0 le 119899 minus 1 lt 119902 lt 119899

(8)




T119902 (119905) 119906 = int

infin

0

ℎ119902 (119904) 119879 (119905119902119904) 119906 119889119904

S119902 (119905) 119906 = 119902int

infin

0

119904ℎ119902 (119904) 119879 (119905

119902119904) 119906 119889119904

(9)

where

ℎ119902 (119904) =1

120587119902

infin

sum

119899=1


Γ (119899119902 + 1)

119899sin (119899120587119902) 119904 isin (0infin)

(10)



infin

0

ℎ119902 (119904) 119889119904 = 1

int

infin

0

119904Vℎ119902 (119904) 119889119904 =

Γ (1 + V)

Γ (1 + 119902V) V isin [0 1]

(11)








10038171003817100381710038171003817le 119872119906

10038171003817100381710038171003817S119902 (119905) 119906

10038171003817100381710038171003817le

119902119872

Γ (1 + 119902)119906 =

119872

Γ (119902)119906

(12)


119902(119905)119906 are




10038171003817100381710038171003817T119902(11990510158401015840) 119906 minusT

119902(1199051015840) 119906

10038171003817100381710038171003817997888rarr 0

10038171003817100381710038171003817S119902(11990510158401015840) 119906 minusS

119902(1199051015840) 119906


(13)


119906 (119905) = T119902 (119905) 119892 (119906) + int

119905

0

(119905 minus 119904)119902minus1

timesS119902 (119905 minus 119904) 119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904)) 119889119904

(14)





120572 (119863) = max119905isin119869

120572 (119863 (119905)) = 120572 (119863 (119869)) (15)



120572(int119869

119906119899 (119905) 119889119905 | 119899 isin N) le 2int

119869

120572 (119863 (119905)) 119889119905 (16)



120572(119863) le 2120572(1198630)



and 1199090isin 119863 set

119876(11199090)

(119878) equiv 119876 (119878)

119876(1198991199090)

(119878) = 119876 (co 119876(119899minus11199090) (119878) 1199090) 119899 = 2 3

(17)


0such that for any


120572 (119876(1198990 1199090)

(119878)) lt 120572 (119878) (18)



120590 (19)






1gt



1003817100381710038171003817 le 120593119903 (119905)


1003817100381710038171003817120593119903100381710038171003817100381711987111199021 [0119886]

119903= 1205881lt +infin

(20)



1 2 3) and ae 119905 isin 119869

120572 (119891 (119905 1198631 1198632 1198633)) le

3

sum

119894=1

119871119894120572 (119863119894) (21)


2gt 0 and a



119862(119869 119864) 119906119862le 1199031003817100381710038171003817119892 (119906)

1003817100381710038171003817 le Φ (119903)


Φ (119903)

119903= 1205882 lt +infin

(22)





1198721205882 +1198721205881119886119902minus1199021

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus1199021

lt 1 (23)


(119876119906) (119905) = T119902 (119905) 119892 (119906) + int119905

0

(119905 minus 119904)119902minus1


(24)



119877) sub Ω





119903 lt1003817100381710038171003817(119876119906119903) (119905119903)

1003817100381710038171003817

le 1198721003817100381710038171003817119892 (119906119903)

1003817100381710038171003817

+119872

Γ (119902)int

119905119903

0

(119905119903minus 119904)119902minus1

times1003817100381710038171003817119891 (119904 119906119903 (119904) 119866119906119903 (119904) 119878119906119903 (119904))

1003817100381710038171003817 119889119904

le 119872Φ (119903) +119872

Γ (119902)(int

119905119903

0


119889119904)

1minus1199021

times1003817100381710038171003817120593119903

100381710038171003817100381711987111199021 [0119905119903]

le 119872Φ (119903) +119872119886119902minus1199021

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus11990211003817100381710038171003817120593119903

100381710038171003817100381711987111199021 [0119886]

(25)


1198721205882 +

1198721205881119886119902minus1199021

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus1199021

ge 1 (26)



119899infin


119899rarr+infin119906119899= 119906


of 119891 we have

lim119899rarr+infin

119892 (119906119899) = 119892 (119906)

lim119899rarr+infin

119891 (119904 119906119899 (119904) 119866119906119899 (119904) 119878119906119899 (119904))

= 119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904)) ae 119904 isin 119869

(27)


(119905 minus 119904)119902minus1 1003817100381710038171003817119891 (119904 119906119899 (119904) 119866119906119899 (119904) 119878119906119899 (119904))

minus119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904))1003817100381710038171003817

le 2(119905 minus 119904)119902minus1

120593119877 (119904)

(28)



1003817100381710038171003817(119876119906119899) (119905) minus (119876119906) (119905)1003817100381710038171003817

le 1198721003817100381710038171003817119892 (119906119899) minus 119892 (119906)

1003817100381710038171003817

+119872

Γ (119902)


times int

119905

0

(119905 minus 119904)119902minus1

times1003817100381710038171003817119891 (119904 119906119899 (119904) 119866119906119899 (119904) 119878119906119899 (119904))

minus119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904))1003817100381710038171003817 119889119904 997888rarr 0


(29)


1003817100381710038171003817(119876119906119899) minus (119876119906)1003817100381710038171003817119862



119877rarr Ω119877is


1lt 1199052le 119886 we get

that

(119876119906) (1199052) minus (119876119906) (1199051)

= T119902 (1199052) 119892 (119906) minusT119902 (1199051) 119892 (119906)

+ int

1199052

1199051

(1199052 minus 119904)119902minus1

S119902 (1199052 minus 119904)

times 119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904)) 119889119904

+ int

1199051

0

((1199052 minus 119904)119902minus1


)

timesS119902 (1199052 minus 119904) 119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904)) 119889119904

+ int

1199051

0

(1199051minus 119904)119902minus1

(S119902(1199052minus 119904) minusS

119902(1199051minus 119904))

times 119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904)) 119889119904

= 1198681+ 1198682+ 1198683+ 1198684

(31)

where

1198681 = T119902 (1199052) 119892 (119906) minusT119902 (1199051) 119892 (119906)

1198682= int

1199052

1199051

(1199052minus 119904)119902minus1

S119902(1199052minus 119904)

times 119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904)) 119889119904

1198683 = int

1199051

0

((1199052 minus 119904)119902minus1


)

timesS119902(1199052minus 119904) 119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904)) 119889119904

1198684= int

1199051

0

(1199051minus 119904)119902minus1

(S119902(1199052minus 119904) minusS

119902(1199051minus 119904))

times 119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904)) 119889119904

(32)

It is obvious that

1003817100381710038171003817(119876119906) (1199052) minus (119876119906) (1199051)1003817100381710038171003817 le

4

sum

119894=1

10038171003817100381710038171198681198941003817100381710038171003817 (33)


of 119906 isin Ω119877when 119905

2minus 1199051rarr 0 119894 = 1 2 4

For 1198681 by Lemma 3(3) 119868

1 rarr 0 as 119905



inequality we have

100381710038171003817100381711986821003817100381710038171003817 le

119872

Γ (119902)int

1199052

1199051

(1199052minus 119904)119902minus1

120593119877 (119904) 119889119904

le119872

Γ (119902)(int

1199052

1199051


119889119904)

1minus1199021

times1003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [1199051 1199052]

le

1198721003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [0119886]

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus1199021

(1199052minus 1199051)119902minus1199021

997888rarr 0

as 1199052minus 1199051997888rarr 0

(34)



100381710038171003817100381711986831003817100381710038171003817 le

119872

Γ (119902)int

1199051

0

((1199051 minus 119904)119902minus1


) 120593119877 (119904) 119889119904

le119872

Γ (119902)(int

1199051

0

((1199051minus 119904)119902minus1


)1(1minus1199021)

119889119904)

1minus1199021

times1003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [01199051]

le119872

Γ (119902)(int

1199051

0



) 119889119904)

1minus1199021

times1003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [0119886]

le

1198721003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [0119886]

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus1199021



2


)

1minus1199021

le

11987221minus11990211003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [0119886]

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus1199021

(1199052minus 1199051)119902minus1199021


(35)


For 1199051= 0 0 lt 119905


4 = 0 For 119905

1gt 0


100381710038171003817100381711986841003817100381710038171003817 le int

1199051minus120598

0

(1199051minus 119904)119902minus1


10038171003817100381710038171003817120593119877 (119904) 119889119904

+ int

1199051

1199051minus120598

(1199051minus 119904)119902minus1


119902(1199051minus 119904)

10038171003817100381710038171003817120593119877 (119904) 119889119904

le sup119904isin[01199051minus120598]

10038171003817100381710038171003817S119902(1199052minus 119904) minusS

119902(1199051minus 119904)

10038171003817100381710038171003817

times int

1199051minus120598

0

(1199051 minus 119904)119902minus1

120593119877 (119904) 119889119904

+2119872

Γ (119902)int

1199051

1199051minus120598

(1199051minus 119904)119902minus1

120593119877 (119904) 119889119904

le sup119904isin[01199051minus120598]

10038171003817100381710038171003817S119902(1199052minus 119904) minusS

119902(1199051minus 119904)

10038171003817100381710038171003817

sdot1003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [0119886](1 minus 1199021

119902 minus 1199021

)

1minus1199021

sdot (119905(119902minus1199021)(1minus1199021)


1minus1199021

+

21198721003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [0119886]

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus1199021

120598119902minus1199021 997888rarr 0

as 1199052 minus 1199051 997888rarr 0

(36)

As a result (119876119906)(1199052)minus(119876119906)(119905



119877rarr Ω119877




120572 (119876(1198990 1199060)

(119863)) lt 120572 (119863) (37)




120572 (119876(1198991199060)

(119863)) = max119905isin119869

120572 (119876(1198991199060)

(119863) (119905)) 119899 = 1 2

(38)


119899 sub 119863 such

that

120572 (119876 (119863) (119905)) le 2120572 (119876 (1198631) (119905)) (39)


int

119886

0

119906 (119904) 119889119904 isin 119886 co 119906 (119904) | 119904 isin 119869 119906 isin 119862 (119869 119864) (40)

we have

120572(int

119905

0

119870 (119905 119904) 119906 (119904) 119889119904 | 119906 isin 119863 119905 isin 119869)

le 1198861198700120572 (119906 (119905) | 119906 isin 119863 119905 isin 119869)

120572 (int

119886

0

119867(119905 119904) 119906 (119904) 119889119904 | 119906 isin 119863 119905 isin 119869)

le 1198861198670120572 (119906 (119905) | 119906 isin 119863 119905 isin 119869)

(41)

where 1198700= max

(119905119904)isinΔ|119870(119905 119904)| 119867

0= max

(119905119904)isinΔ 0|119867(119905 119904)|


120572 (119876(11199060)

(119863) (119905))

= 120572 (119876 (119863) (119905)) le 2120572 (119876 (1198631) (119905))

le 2120572 (T119902 (119905) 119892 (119906

1

119899)

+ int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904)

times119891 (119904 1199061

119899(119904) 119866119906

1

119899(119904) 119878119906

1

119899(119904)) 119889119904)

le4119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times 120572 (119891 (119904 1199061

119899(119904) 119866119906

1

119899(119904) 119878119906

1

119899(119904))) 119889119904

le4119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times [1198711120572 (1198631 (119904)) + 1198712120572 ((1198661198631) (119904))

+1198713120572 ((1198781198631) (119904))] 119889119904

le4119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times (1198711+ 11988611987001198712+ 11988611986701198713) 120572 (119863

1 (119904)) 119889119904

le4119872(119871

1 + 11988611987001198712 + 11988611986701198713) 119905119902

Γ (1 + 119902)120572 (119863)

(42)


1199062

119899 sub co119876(11199060)(119863) 119906

0 such that

120572 (119876 (co 119876(11199060) (119863) 1199060) (119905)) le 2120572 (119876 (1198632) (119905)) (43)



120572 (119876(21199060)

(119863) (119905))

= 120572 (119876 (co 119876(11199060) (119863) 1199060) (119905))

le 2120572 (119876 (1198632) (119905))

le 2120572(int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904)

times119891 (119904 1199062

119899(119904) 119866119906

2

119899(119904) 119878119906

2

119899(119904)) 119889119904)

le4119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times 120572 (119891 (119904 1199062

119899(119904) 119866119906

2

119899(119904) 119878119906

2

119899(119904))) 119889119904

le4119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times (1198711+ 11988611987001198712+ 11988611986701198713) 120572 (119863

2 (119904)) 119889119904

le4119872(119871

1 + 11988611987001198712 + 11988611986701198713)

Γ (119902)

times int

119905

0

(119905 minus 119904)119902minus1

sdot 120572 (co 119876(11199060) (119863) 1199060 (119904)) 119889119904

le[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713)]2

Γ (119902) Γ (1 + 119902)

times int

119905

0

(119905 minus 119904)119902minus1

119904119902120572 (119863) 119889119904

=[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713) 119905119902]2

Γ (1 + 2119902)B (1 + 119902 119902)

times int

1

0

(1 minus 119904)119902minus1

119904119902119889119904 120572 (119863)

=[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713) 119905119902]2

Γ (1 + 2119902)120572 (119863)

(44)

where B(119901 119902) = int1

0119904119901minus1

(1 minus 119904)119902minus1


120572 (119876(1198961199060)

(119863) (119905))

le[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713) 119905119902]119896

Γ (1 + 119896119902)120572 (119863) forall119905 isin 119869

(45)


=

119906119896+1

119899 sub co119876(1198961199060)(119863) 119906

0 such that

120572 (119876 (co 119876(1198961199060) (119863) 1199060) (119905)) le 2120572 (119876 (119863119896+1) (119905)) (46)


120572 (119876(119896+11199060)

(119863) (119905))

= 120572 (119876 (co 119876(1198961199060) (119863) 1199060) (119905))

le 2120572 (119876 (119863119896+1) (119905))

le 2120572(int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904)

times119891 (119904 119906119896+1

119899(119904) 119866119906

119896+1

119899(119904) 119878119906

119896+1

119899(119904)) 119889119904)

le4119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times (1198711+ 11988611987001198712+ 11988611986701198713) 120572 (119863

119896+1 (119904)) 119889119904

le4119872(119871

1 + 11988611987001198712 + 11988611986701198713)

Γ (119902)

times int

119905

0

(119905 minus 119904)119902minus1

120572 (co 119876(1198961199060) (119863) 1199060 (119904)) 119889119904

le[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713)]119896+1

Γ (119902) Γ (1 + 119896119902)

times int

119905

0

(119905 minus 119904)119902minus1

119904119896119902120572 (119863) 119889119904

=[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713) 119905119902]119896+1

Γ (1 + (119896 + 1) 119902)B (1 + 119896119902 119902)

times int

1

0

(1 minus 119904)119902minus1

119904119896119902119889119904 120572 (119863)

=[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713) 119905119902]119896+1

Γ (1 + (119896 + 1) 119902)120572 (119863)

(47)


120572 (119876(1198991199060)

(119863) (119905)) le[4119872 (1198711 + 11988611987001198712 + 11988611986701198713) 119905

119902]119899

Γ (1 + 119899119902)120572 (119863)

(48)


120572 (119876(1198991199060)

(119863)) = max119905isin119869

120572 (119876(1198991199060)

(119863) (119905))

le[4119872 (1198711 + 11988611987001198712 + 11988611986701198713) 119886

119902]119899

Γ (1 + 119899119902)120572 (119863)

(49)


Γ (1 + 119899119902) = radic2120587119899119902(119899119902

119890)

119899119902

119890]12119899119902

0 lt ] lt 1 (50)


Thus

[4119872 (1198711+ 11988611987001198712+ 11988611986701198713) 119886119902]119899

Γ (1 + 119899119902)

=[4119872 (1198711 + 11988611987001198712 + 11988611986701198713) 119886

119902]119899

radic2120587119899119902(119899119902119890)119899119902119890]12119899119902


(51)


[4119872 (1198711+ 11988611987001198712+ 11988611986701198713) 119886119902]1198990

Γ (1 + 1198990119902)

lt 1 (52)

Hence we get that

120572 (119876(1198990 1199060)

(119863)) lt 120572 (119863) (53)




1isin [0 119902) and


such that

1003817100381710038171003817119891 (119905 119906 119866119906 119878119906)1003817100381710038171003817 le 120593 (119905) Ψ (119906) (54)



119862+ 119888






(119872119888 +Ψ (119877)119872119886

119902minus1199021

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus11990211003817100381710038171003817120593100381710038171003817100381711987111199021 [0119886]

)


(55)


119877 by (24) (55)


(119876119906) (119905) le 1198721003817100381710038171003817119892 (119906)

1003817100381710038171003817

+119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times1003817100381710038171003817119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904))

1003817100381710038171003817 119889119904

le 119872(119887119906119862 + 119888) +Ψ (119877)119872

Γ (119902)

times (int

119905

0


1minus11990211003817100381710038171003817120593100381710038171003817100381711987111199021 [0119905]

le 119872(119887119877 + 119888) +Ψ (119877)119872119886

119902minus1199021

Γ (119902)


119902 minus 1199021

)

1minus11990211003817100381710038171003817120593100381710038171003817100381711987111199021 [0119886]

le 119877

(56)









(int

119886

0

12059311199021

(119904) 119889119904)

1199021


(119903 (1 minus119872119887) +119872119888) Γ (119902)

Ψ (119903)119872119886119902minus1199021

times (119902 minus 1199021

1 minus 1199021

)

1minus1199021

(57)











119864 then T119902(119905) (119905 ge 0) and S


operators


119902(119905) (119905 ge 0)








1 2 3) and ae 119905 isin 119869

120572 (119891 (119905 1198631 (119905) 1198632 (119905) 1198633 (119905))) le

3

sum

119894=1

119871119894120572 (119863119894 (119905)) (58)






119877 ge119873119872

1 minus 119873119872119887[119888 +

119886119902minus1199021

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus11990211003817100381710038171003817120593100381710038171003817100381711987111199021 [0119886]

] (59)


119862le 119877 by (24)


120579 le (119876119906) (119905)

le T119902 (119905) (119887119906 + 119888)

+ int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904) 120593 (119904) 119889119904

(60)


(119876119906) (119905) le 119873119872(119887119906119862 + 119888)

+119873119872

Γ (119902)(int

119905

0


1minus1199021

times1003817100381710038171003817120593100381710038171003817100381711987111199021 [0119905]

le 119873119872(119887119877 + 119888) +119873119872119886

119902minus1199021

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus1199021

times1003817100381710038171003817120593100381710038171003817100381711987111199021 [0119886]

le 119877

(61)

Thus119876 Ω119877(119875) rarr Ω

119877(119875) Let 119865 = co 119876(Ω

119877(119875)) Similar to






such that119891 (119905 119906 V 119908) le 119892

1 (119905) 119906 + 1198922 (119905) V + ℎ (119905)


(62)




(119861119906) (119905) = int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904)

times [1198921 (119904) 119906 (119904) + 1198922 (119904) 119866119906 (119904)] 119889119904 119905 isin 119869

(63)



(119861119906) (119905)

le119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times

10038171003817100381710038171003817100381710038171003817

1198921 (119904) 119906 (119904)+1198922 (119904) int

119904

0

119870 (119904 120591) 119906 (120591) 119889120591

10038171003817100381710038171003817100381710038171003817

119889119904

le119872119866lowast(1 + 1198861198700) 119906119862

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

119889119904

=120573119905119902119906119862

Γ (1 + 119902)

(64)


1198861198700) Further10038171003817100381710038171003817(1198612119906) (119905)

10038171003817100381710038171003817


Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times

10038171003817100381710038171003817100381710038171003817

(119861119906) (119904) + int

119904

0

119870 (119904 120591) (119861119906) (120591) 119889120591

10038171003817100381710038171003817100381710038171003817

119889119904

le119872119866lowast120573119906119862

Γ (119902) Γ (1 + 119902)

times int

119905

0

(119905 minus 119904)119902minus1

[119904119902+ int

119904

0

119870 (119904 120591) 120591119902119889120591] 119889119904

le1205732119906119862

Γ (1 + 2119902)B (1 + 119902 119902)

times int

119905

0

(119905 minus 119904)119902minus1

119904119902119889119904

=12057321199052119902119906119862

Γ (1 + 2119902)

(65)


1003817100381710038171003817(119861119899119906) (119905)

1003817100381710038171003817 le120573119899119905119899119902119906119862

Γ (1 + 119899119902) (66)

Therefore we have

10038171003817100381710038171198611198991003817100381710038171003817119862

le120573119899119886119899119902

Γ (1 + 119899119902) (67)


119903 (119861) = lim119899rarrinfin

10038171003817100381710038171198611198991003817100381710038171003817

1119899


120573119886119902

2119899radic2120587119899119902(119899119902119890)

119902119890](12119899

2119902)= 0

(68)



119861lowastle 119903 (119861) + 120574 = 120574 (69)


lowastLet 119872

lowast= sup

119905isin[0+infin)119879(119905)

lowast

L(119864) and 119906lowast

119862=

sup119905isin119869119906(119905)

lowast Choose

119877lowast=119873119872lowast(Γ (1 + 119902) 119888 + 119886

119902ℎlowast

119862)

Γ (1 + 119902) (1 minus 119873120574) (70)


lowast

119862le 119877lowast by (24) and


120579 le (119876119906) (119905)

le T119902 (119905) 119892 (119906)

+ int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904)

times [1198921 (119904) 119906 (119904) + 1198922 (119904) 119866119906 (119904) + ℎ (119904)] 119889119904

le T119902 (119905) 119888 + 119861 (119906) (119905)

+ int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904) ℎ (119904) 119889119904

(71)


(119876119906)(119905)lowastle 119873(

10038171003817100381710038171003817T119902(119905)119888

10038171003817100381710038171003817

lowast

+ (119861119906)(119905)lowast

+

10038171003817100381710038171003817100381710038171003817

int

119905

0

(119905 minus 119904)119902minus1

S119902(119905 minus 119904)ℎ(119904)119889119904

10038171003817100381710038171003817100381710038171003817

lowast

)

le 119873(119872lowast119888 + 119861

lowast119906lowast

119862+119872lowast119886119902ℎlowast

119862

Γ (1 + 119902))

le 119873119872lowast119888 + 119873120574119877


119862

Γ (1 + 119902)

= 119877lowast

(72)



lowast

119877lowast(119875) Let

119865 = co 119876(Ωlowast







Acknowledgments


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014






1gt



1003817100381710038171003817 le 120593119903 (119905)


1003817100381710038171003817120593119903100381710038171003817100381711987111199021 [0119886]

119903= 1205881lt +infin

(20)



1 2 3) and ae 119905 isin 119869

120572 (119891 (119905 1198631 1198632 1198633)) le

3

sum

119894=1

119871119894120572 (119863119894) (21)


2gt 0 and a



119862(119869 119864) 119906119862le 1199031003817100381710038171003817119892 (119906)

1003817100381710038171003817 le Φ (119903)


Φ (119903)

119903= 1205882 lt +infin

(22)





1198721205882 +1198721205881119886119902minus1199021

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus1199021

lt 1 (23)


(119876119906) (119905) = T119902 (119905) 119892 (119906) + int119905

0

(119905 minus 119904)119902minus1


(24)



119877) sub Ω





119903 lt1003817100381710038171003817(119876119906119903) (119905119903)

1003817100381710038171003817

le 1198721003817100381710038171003817119892 (119906119903)

1003817100381710038171003817

+119872

Γ (119902)int

119905119903

0

(119905119903minus 119904)119902minus1

times1003817100381710038171003817119891 (119904 119906119903 (119904) 119866119906119903 (119904) 119878119906119903 (119904))

1003817100381710038171003817 119889119904

le 119872Φ (119903) +119872

Γ (119902)(int

119905119903

0


119889119904)

1minus1199021

times1003817100381710038171003817120593119903

100381710038171003817100381711987111199021 [0119905119903]

le 119872Φ (119903) +119872119886119902minus1199021

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus11990211003817100381710038171003817120593119903

100381710038171003817100381711987111199021 [0119886]

(25)


1198721205882 +

1198721205881119886119902minus1199021

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus1199021

ge 1 (26)



119899infin


119899rarr+infin119906119899= 119906


of 119891 we have

lim119899rarr+infin

119892 (119906119899) = 119892 (119906)

lim119899rarr+infin

119891 (119904 119906119899 (119904) 119866119906119899 (119904) 119878119906119899 (119904))

= 119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904)) ae 119904 isin 119869

(27)


(119905 minus 119904)119902minus1 1003817100381710038171003817119891 (119904 119906119899 (119904) 119866119906119899 (119904) 119878119906119899 (119904))

minus119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904))1003817100381710038171003817

le 2(119905 minus 119904)119902minus1

120593119877 (119904)

(28)



1003817100381710038171003817(119876119906119899) (119905) minus (119876119906) (119905)1003817100381710038171003817

le 1198721003817100381710038171003817119892 (119906119899) minus 119892 (119906)

1003817100381710038171003817

+119872

Γ (119902)


times int

119905

0

(119905 minus 119904)119902minus1

times1003817100381710038171003817119891 (119904 119906119899 (119904) 119866119906119899 (119904) 119878119906119899 (119904))

minus119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904))1003817100381710038171003817 119889119904 997888rarr 0


(29)


1003817100381710038171003817(119876119906119899) minus (119876119906)1003817100381710038171003817119862



119877rarr Ω119877is


1lt 1199052le 119886 we get

that

(119876119906) (1199052) minus (119876119906) (1199051)

= T119902 (1199052) 119892 (119906) minusT119902 (1199051) 119892 (119906)

+ int

1199052

1199051

(1199052 minus 119904)119902minus1

S119902 (1199052 minus 119904)

times 119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904)) 119889119904

+ int

1199051

0

((1199052 minus 119904)119902minus1


)

timesS119902 (1199052 minus 119904) 119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904)) 119889119904

+ int

1199051

0

(1199051minus 119904)119902minus1

(S119902(1199052minus 119904) minusS

119902(1199051minus 119904))

times 119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904)) 119889119904

= 1198681+ 1198682+ 1198683+ 1198684

(31)

where

1198681 = T119902 (1199052) 119892 (119906) minusT119902 (1199051) 119892 (119906)

1198682= int

1199052

1199051

(1199052minus 119904)119902minus1

S119902(1199052minus 119904)

times 119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904)) 119889119904

1198683 = int

1199051

0

((1199052 minus 119904)119902minus1


)

timesS119902(1199052minus 119904) 119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904)) 119889119904

1198684= int

1199051

0

(1199051minus 119904)119902minus1

(S119902(1199052minus 119904) minusS

119902(1199051minus 119904))

times 119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904)) 119889119904

(32)

It is obvious that

1003817100381710038171003817(119876119906) (1199052) minus (119876119906) (1199051)1003817100381710038171003817 le

4

sum

119894=1

10038171003817100381710038171198681198941003817100381710038171003817 (33)


of 119906 isin Ω119877when 119905

2minus 1199051rarr 0 119894 = 1 2 4

For 1198681 by Lemma 3(3) 119868

1 rarr 0 as 119905



inequality we have

100381710038171003817100381711986821003817100381710038171003817 le

119872

Γ (119902)int

1199052

1199051

(1199052minus 119904)119902minus1

120593119877 (119904) 119889119904

le119872

Γ (119902)(int

1199052

1199051


119889119904)

1minus1199021

times1003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [1199051 1199052]

le

1198721003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [0119886]

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus1199021

(1199052minus 1199051)119902minus1199021

997888rarr 0

as 1199052minus 1199051997888rarr 0

(34)



100381710038171003817100381711986831003817100381710038171003817 le

119872

Γ (119902)int

1199051

0

((1199051 minus 119904)119902minus1


) 120593119877 (119904) 119889119904

le119872

Γ (119902)(int

1199051

0

((1199051minus 119904)119902minus1


)1(1minus1199021)

119889119904)

1minus1199021

times1003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [01199051]

le119872

Γ (119902)(int

1199051

0



) 119889119904)

1minus1199021

times1003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [0119886]

le

1198721003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [0119886]

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus1199021



2


)

1minus1199021

le

11987221minus11990211003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [0119886]

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus1199021

(1199052minus 1199051)119902minus1199021


(35)


For 1199051= 0 0 lt 119905


4 = 0 For 119905

1gt 0


100381710038171003817100381711986841003817100381710038171003817 le int

1199051minus120598

0

(1199051minus 119904)119902minus1


10038171003817100381710038171003817120593119877 (119904) 119889119904

+ int

1199051

1199051minus120598

(1199051minus 119904)119902minus1


119902(1199051minus 119904)

10038171003817100381710038171003817120593119877 (119904) 119889119904

le sup119904isin[01199051minus120598]

10038171003817100381710038171003817S119902(1199052minus 119904) minusS

119902(1199051minus 119904)

10038171003817100381710038171003817

times int

1199051minus120598

0

(1199051 minus 119904)119902minus1

120593119877 (119904) 119889119904

+2119872

Γ (119902)int

1199051

1199051minus120598

(1199051minus 119904)119902minus1

120593119877 (119904) 119889119904

le sup119904isin[01199051minus120598]

10038171003817100381710038171003817S119902(1199052minus 119904) minusS

119902(1199051minus 119904)

10038171003817100381710038171003817

sdot1003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [0119886](1 minus 1199021

119902 minus 1199021

)

1minus1199021

sdot (119905(119902minus1199021)(1minus1199021)


1minus1199021

+

21198721003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [0119886]

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus1199021

120598119902minus1199021 997888rarr 0

as 1199052 minus 1199051 997888rarr 0

(36)

As a result (119876119906)(1199052)minus(119876119906)(119905



119877rarr Ω119877




120572 (119876(1198990 1199060)

(119863)) lt 120572 (119863) (37)




120572 (119876(1198991199060)

(119863)) = max119905isin119869

120572 (119876(1198991199060)

(119863) (119905)) 119899 = 1 2

(38)


119899 sub 119863 such

that

120572 (119876 (119863) (119905)) le 2120572 (119876 (1198631) (119905)) (39)


int

119886

0

119906 (119904) 119889119904 isin 119886 co 119906 (119904) | 119904 isin 119869 119906 isin 119862 (119869 119864) (40)

we have

120572(int

119905

0

119870 (119905 119904) 119906 (119904) 119889119904 | 119906 isin 119863 119905 isin 119869)

le 1198861198700120572 (119906 (119905) | 119906 isin 119863 119905 isin 119869)

120572 (int

119886

0

119867(119905 119904) 119906 (119904) 119889119904 | 119906 isin 119863 119905 isin 119869)

le 1198861198670120572 (119906 (119905) | 119906 isin 119863 119905 isin 119869)

(41)

where 1198700= max

(119905119904)isinΔ|119870(119905 119904)| 119867

0= max

(119905119904)isinΔ 0|119867(119905 119904)|


120572 (119876(11199060)

(119863) (119905))

= 120572 (119876 (119863) (119905)) le 2120572 (119876 (1198631) (119905))

le 2120572 (T119902 (119905) 119892 (119906

1

119899)

+ int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904)

times119891 (119904 1199061

119899(119904) 119866119906

1

119899(119904) 119878119906

1

119899(119904)) 119889119904)

le4119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times 120572 (119891 (119904 1199061

119899(119904) 119866119906

1

119899(119904) 119878119906

1

119899(119904))) 119889119904

le4119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times [1198711120572 (1198631 (119904)) + 1198712120572 ((1198661198631) (119904))

+1198713120572 ((1198781198631) (119904))] 119889119904

le4119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times (1198711+ 11988611987001198712+ 11988611986701198713) 120572 (119863

1 (119904)) 119889119904

le4119872(119871

1 + 11988611987001198712 + 11988611986701198713) 119905119902

Γ (1 + 119902)120572 (119863)

(42)


1199062

119899 sub co119876(11199060)(119863) 119906

0 such that

120572 (119876 (co 119876(11199060) (119863) 1199060) (119905)) le 2120572 (119876 (1198632) (119905)) (43)



120572 (119876(21199060)

(119863) (119905))

= 120572 (119876 (co 119876(11199060) (119863) 1199060) (119905))

le 2120572 (119876 (1198632) (119905))

le 2120572(int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904)

times119891 (119904 1199062

119899(119904) 119866119906

2

119899(119904) 119878119906

2

119899(119904)) 119889119904)

le4119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times 120572 (119891 (119904 1199062

119899(119904) 119866119906

2

119899(119904) 119878119906

2

119899(119904))) 119889119904

le4119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times (1198711+ 11988611987001198712+ 11988611986701198713) 120572 (119863

2 (119904)) 119889119904

le4119872(119871

1 + 11988611987001198712 + 11988611986701198713)

Γ (119902)

times int

119905

0

(119905 minus 119904)119902minus1

sdot 120572 (co 119876(11199060) (119863) 1199060 (119904)) 119889119904

le[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713)]2

Γ (119902) Γ (1 + 119902)

times int

119905

0

(119905 minus 119904)119902minus1

119904119902120572 (119863) 119889119904

=[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713) 119905119902]2

Γ (1 + 2119902)B (1 + 119902 119902)

times int

1

0

(1 minus 119904)119902minus1

119904119902119889119904 120572 (119863)

=[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713) 119905119902]2

Γ (1 + 2119902)120572 (119863)

(44)

where B(119901 119902) = int1

0119904119901minus1

(1 minus 119904)119902minus1


120572 (119876(1198961199060)

(119863) (119905))

le[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713) 119905119902]119896

Γ (1 + 119896119902)120572 (119863) forall119905 isin 119869

(45)


=

119906119896+1

119899 sub co119876(1198961199060)(119863) 119906

0 such that

120572 (119876 (co 119876(1198961199060) (119863) 1199060) (119905)) le 2120572 (119876 (119863119896+1) (119905)) (46)


120572 (119876(119896+11199060)

(119863) (119905))

= 120572 (119876 (co 119876(1198961199060) (119863) 1199060) (119905))

le 2120572 (119876 (119863119896+1) (119905))

le 2120572(int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904)

times119891 (119904 119906119896+1

119899(119904) 119866119906

119896+1

119899(119904) 119878119906

119896+1

119899(119904)) 119889119904)

le4119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times (1198711+ 11988611987001198712+ 11988611986701198713) 120572 (119863

119896+1 (119904)) 119889119904

le4119872(119871

1 + 11988611987001198712 + 11988611986701198713)

Γ (119902)

times int

119905

0

(119905 minus 119904)119902minus1

120572 (co 119876(1198961199060) (119863) 1199060 (119904)) 119889119904

le[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713)]119896+1

Γ (119902) Γ (1 + 119896119902)

times int

119905

0

(119905 minus 119904)119902minus1

119904119896119902120572 (119863) 119889119904

=[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713) 119905119902]119896+1

Γ (1 + (119896 + 1) 119902)B (1 + 119896119902 119902)

times int

1

0

(1 minus 119904)119902minus1

119904119896119902119889119904 120572 (119863)

=[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713) 119905119902]119896+1

Γ (1 + (119896 + 1) 119902)120572 (119863)

(47)


120572 (119876(1198991199060)

(119863) (119905)) le[4119872 (1198711 + 11988611987001198712 + 11988611986701198713) 119905

119902]119899

Γ (1 + 119899119902)120572 (119863)

(48)


120572 (119876(1198991199060)

(119863)) = max119905isin119869

120572 (119876(1198991199060)

(119863) (119905))

le[4119872 (1198711 + 11988611987001198712 + 11988611986701198713) 119886

119902]119899

Γ (1 + 119899119902)120572 (119863)

(49)


Γ (1 + 119899119902) = radic2120587119899119902(119899119902

119890)

119899119902

119890]12119899119902

0 lt ] lt 1 (50)


Thus

[4119872 (1198711+ 11988611987001198712+ 11988611986701198713) 119886119902]119899

Γ (1 + 119899119902)

=[4119872 (1198711 + 11988611987001198712 + 11988611986701198713) 119886

119902]119899

radic2120587119899119902(119899119902119890)119899119902119890]12119899119902


(51)


[4119872 (1198711+ 11988611987001198712+ 11988611986701198713) 119886119902]1198990

Γ (1 + 1198990119902)

lt 1 (52)

Hence we get that

120572 (119876(1198990 1199060)

(119863)) lt 120572 (119863) (53)




1isin [0 119902) and


such that

1003817100381710038171003817119891 (119905 119906 119866119906 119878119906)1003817100381710038171003817 le 120593 (119905) Ψ (119906) (54)



119862+ 119888






(119872119888 +Ψ (119877)119872119886

119902minus1199021

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus11990211003817100381710038171003817120593100381710038171003817100381711987111199021 [0119886]

)


(55)


119877 by (24) (55)


(119876119906) (119905) le 1198721003817100381710038171003817119892 (119906)

1003817100381710038171003817

+119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times1003817100381710038171003817119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904))

1003817100381710038171003817 119889119904

le 119872(119887119906119862 + 119888) +Ψ (119877)119872

Γ (119902)

times (int

119905

0


1minus11990211003817100381710038171003817120593100381710038171003817100381711987111199021 [0119905]

le 119872(119887119877 + 119888) +Ψ (119877)119872119886

119902minus1199021

Γ (119902)


119902 minus 1199021

)

1minus11990211003817100381710038171003817120593100381710038171003817100381711987111199021 [0119886]

le 119877

(56)









(int

119886

0

12059311199021

(119904) 119889119904)

1199021


(119903 (1 minus119872119887) +119872119888) Γ (119902)

Ψ (119903)119872119886119902minus1199021

times (119902 minus 1199021

1 minus 1199021

)

1minus1199021

(57)











119864 then T119902(119905) (119905 ge 0) and S


operators


119902(119905) (119905 ge 0)








1 2 3) and ae 119905 isin 119869

120572 (119891 (119905 1198631 (119905) 1198632 (119905) 1198633 (119905))) le

3

sum

119894=1

119871119894120572 (119863119894 (119905)) (58)






119877 ge119873119872

1 minus 119873119872119887[119888 +

119886119902minus1199021

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus11990211003817100381710038171003817120593100381710038171003817100381711987111199021 [0119886]

] (59)


119862le 119877 by (24)


120579 le (119876119906) (119905)

le T119902 (119905) (119887119906 + 119888)

+ int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904) 120593 (119904) 119889119904

(60)


(119876119906) (119905) le 119873119872(119887119906119862 + 119888)

+119873119872

Γ (119902)(int

119905

0


1minus1199021

times1003817100381710038171003817120593100381710038171003817100381711987111199021 [0119905]

le 119873119872(119887119877 + 119888) +119873119872119886

119902minus1199021

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus1199021

times1003817100381710038171003817120593100381710038171003817100381711987111199021 [0119886]

le 119877

(61)

Thus119876 Ω119877(119875) rarr Ω

119877(119875) Let 119865 = co 119876(Ω

119877(119875)) Similar to






such that119891 (119905 119906 V 119908) le 119892

1 (119905) 119906 + 1198922 (119905) V + ℎ (119905)


(62)




(119861119906) (119905) = int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904)

times [1198921 (119904) 119906 (119904) + 1198922 (119904) 119866119906 (119904)] 119889119904 119905 isin 119869

(63)



(119861119906) (119905)

le119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times

10038171003817100381710038171003817100381710038171003817

1198921 (119904) 119906 (119904)+1198922 (119904) int

119904

0

119870 (119904 120591) 119906 (120591) 119889120591

10038171003817100381710038171003817100381710038171003817

119889119904

le119872119866lowast(1 + 1198861198700) 119906119862

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

119889119904

=120573119905119902119906119862

Γ (1 + 119902)

(64)


1198861198700) Further10038171003817100381710038171003817(1198612119906) (119905)

10038171003817100381710038171003817


Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times

10038171003817100381710038171003817100381710038171003817

(119861119906) (119904) + int

119904

0

119870 (119904 120591) (119861119906) (120591) 119889120591

10038171003817100381710038171003817100381710038171003817

119889119904

le119872119866lowast120573119906119862

Γ (119902) Γ (1 + 119902)

times int

119905

0

(119905 minus 119904)119902minus1

[119904119902+ int

119904

0

119870 (119904 120591) 120591119902119889120591] 119889119904

le1205732119906119862

Γ (1 + 2119902)B (1 + 119902 119902)

times int

119905

0

(119905 minus 119904)119902minus1

119904119902119889119904

=12057321199052119902119906119862

Γ (1 + 2119902)

(65)


1003817100381710038171003817(119861119899119906) (119905)

1003817100381710038171003817 le120573119899119905119899119902119906119862

Γ (1 + 119899119902) (66)

Therefore we have

10038171003817100381710038171198611198991003817100381710038171003817119862

le120573119899119886119899119902

Γ (1 + 119899119902) (67)


119903 (119861) = lim119899rarrinfin

10038171003817100381710038171198611198991003817100381710038171003817

1119899


120573119886119902

2119899radic2120587119899119902(119899119902119890)

119902119890](12119899

2119902)= 0

(68)



119861lowastle 119903 (119861) + 120574 = 120574 (69)


lowastLet 119872

lowast= sup

119905isin[0+infin)119879(119905)

lowast

L(119864) and 119906lowast

119862=

sup119905isin119869119906(119905)

lowast Choose

119877lowast=119873119872lowast(Γ (1 + 119902) 119888 + 119886

119902ℎlowast

119862)

Γ (1 + 119902) (1 minus 119873120574) (70)


lowast

119862le 119877lowast by (24) and


120579 le (119876119906) (119905)

le T119902 (119905) 119892 (119906)

+ int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904)

times [1198921 (119904) 119906 (119904) + 1198922 (119904) 119866119906 (119904) + ℎ (119904)] 119889119904

le T119902 (119905) 119888 + 119861 (119906) (119905)

+ int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904) ℎ (119904) 119889119904

(71)


(119876119906)(119905)lowastle 119873(

10038171003817100381710038171003817T119902(119905)119888

10038171003817100381710038171003817

lowast

+ (119861119906)(119905)lowast

+

10038171003817100381710038171003817100381710038171003817

int

119905

0

(119905 minus 119904)119902minus1

S119902(119905 minus 119904)ℎ(119904)119889119904

10038171003817100381710038171003817100381710038171003817

lowast

)

le 119873(119872lowast119888 + 119861

lowast119906lowast

119862+119872lowast119886119902ℎlowast

119862

Γ (1 + 119902))

le 119873119872lowast119888 + 119873120574119877


119862

Γ (1 + 119902)

= 119877lowast

(72)



lowast

119877lowast(119875) Let

119865 = co 119876(Ωlowast







Acknowledgments


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014




times int

119905

0

(119905 minus 119904)119902minus1

times1003817100381710038171003817119891 (119904 119906119899 (119904) 119866119906119899 (119904) 119878119906119899 (119904))

minus119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904))1003817100381710038171003817 119889119904 997888rarr 0


(29)


1003817100381710038171003817(119876119906119899) minus (119876119906)1003817100381710038171003817119862



119877rarr Ω119877is


1lt 1199052le 119886 we get

that

(119876119906) (1199052) minus (119876119906) (1199051)

= T119902 (1199052) 119892 (119906) minusT119902 (1199051) 119892 (119906)

+ int

1199052

1199051

(1199052 minus 119904)119902minus1

S119902 (1199052 minus 119904)

times 119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904)) 119889119904

+ int

1199051

0

((1199052 minus 119904)119902minus1


)

timesS119902 (1199052 minus 119904) 119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904)) 119889119904

+ int

1199051

0

(1199051minus 119904)119902minus1

(S119902(1199052minus 119904) minusS

119902(1199051minus 119904))

times 119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904)) 119889119904

= 1198681+ 1198682+ 1198683+ 1198684

(31)

where

1198681 = T119902 (1199052) 119892 (119906) minusT119902 (1199051) 119892 (119906)

1198682= int

1199052

1199051

(1199052minus 119904)119902minus1

S119902(1199052minus 119904)

times 119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904)) 119889119904

1198683 = int

1199051

0

((1199052 minus 119904)119902minus1


)

timesS119902(1199052minus 119904) 119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904)) 119889119904

1198684= int

1199051

0

(1199051minus 119904)119902minus1

(S119902(1199052minus 119904) minusS

119902(1199051minus 119904))

times 119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904)) 119889119904

(32)

It is obvious that

1003817100381710038171003817(119876119906) (1199052) minus (119876119906) (1199051)1003817100381710038171003817 le

4

sum

119894=1

10038171003817100381710038171198681198941003817100381710038171003817 (33)


of 119906 isin Ω119877when 119905

2minus 1199051rarr 0 119894 = 1 2 4

For 1198681 by Lemma 3(3) 119868

1 rarr 0 as 119905



inequality we have

100381710038171003817100381711986821003817100381710038171003817 le

119872

Γ (119902)int

1199052

1199051

(1199052minus 119904)119902minus1

120593119877 (119904) 119889119904

le119872

Γ (119902)(int

1199052

1199051


119889119904)

1minus1199021

times1003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [1199051 1199052]

le

1198721003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [0119886]

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus1199021

(1199052minus 1199051)119902minus1199021

997888rarr 0

as 1199052minus 1199051997888rarr 0

(34)



100381710038171003817100381711986831003817100381710038171003817 le

119872

Γ (119902)int

1199051

0

((1199051 minus 119904)119902minus1


) 120593119877 (119904) 119889119904

le119872

Γ (119902)(int

1199051

0

((1199051minus 119904)119902minus1


)1(1minus1199021)

119889119904)

1minus1199021

times1003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [01199051]

le119872

Γ (119902)(int

1199051

0



) 119889119904)

1minus1199021

times1003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [0119886]

le

1198721003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [0119886]

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus1199021



2


)

1minus1199021

le

11987221minus11990211003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [0119886]

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus1199021

(1199052minus 1199051)119902minus1199021


(35)


For 1199051= 0 0 lt 119905


4 = 0 For 119905

1gt 0


100381710038171003817100381711986841003817100381710038171003817 le int

1199051minus120598

0

(1199051minus 119904)119902minus1


10038171003817100381710038171003817120593119877 (119904) 119889119904

+ int

1199051

1199051minus120598

(1199051minus 119904)119902minus1


119902(1199051minus 119904)

10038171003817100381710038171003817120593119877 (119904) 119889119904

le sup119904isin[01199051minus120598]

10038171003817100381710038171003817S119902(1199052minus 119904) minusS

119902(1199051minus 119904)

10038171003817100381710038171003817

times int

1199051minus120598

0

(1199051 minus 119904)119902minus1

120593119877 (119904) 119889119904

+2119872

Γ (119902)int

1199051

1199051minus120598

(1199051minus 119904)119902minus1

120593119877 (119904) 119889119904

le sup119904isin[01199051minus120598]

10038171003817100381710038171003817S119902(1199052minus 119904) minusS

119902(1199051minus 119904)

10038171003817100381710038171003817

sdot1003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [0119886](1 minus 1199021

119902 minus 1199021

)

1minus1199021

sdot (119905(119902minus1199021)(1minus1199021)


1minus1199021

+

21198721003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [0119886]

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus1199021

120598119902minus1199021 997888rarr 0

as 1199052 minus 1199051 997888rarr 0

(36)

As a result (119876119906)(1199052)minus(119876119906)(119905



119877rarr Ω119877




120572 (119876(1198990 1199060)

(119863)) lt 120572 (119863) (37)




120572 (119876(1198991199060)

(119863)) = max119905isin119869

120572 (119876(1198991199060)

(119863) (119905)) 119899 = 1 2

(38)


119899 sub 119863 such

that

120572 (119876 (119863) (119905)) le 2120572 (119876 (1198631) (119905)) (39)


int

119886

0

119906 (119904) 119889119904 isin 119886 co 119906 (119904) | 119904 isin 119869 119906 isin 119862 (119869 119864) (40)

we have

120572(int

119905

0

119870 (119905 119904) 119906 (119904) 119889119904 | 119906 isin 119863 119905 isin 119869)

le 1198861198700120572 (119906 (119905) | 119906 isin 119863 119905 isin 119869)

120572 (int

119886

0

119867(119905 119904) 119906 (119904) 119889119904 | 119906 isin 119863 119905 isin 119869)

le 1198861198670120572 (119906 (119905) | 119906 isin 119863 119905 isin 119869)

(41)

where 1198700= max

(119905119904)isinΔ|119870(119905 119904)| 119867

0= max

(119905119904)isinΔ 0|119867(119905 119904)|


120572 (119876(11199060)

(119863) (119905))

= 120572 (119876 (119863) (119905)) le 2120572 (119876 (1198631) (119905))

le 2120572 (T119902 (119905) 119892 (119906

1

119899)

+ int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904)

times119891 (119904 1199061

119899(119904) 119866119906

1

119899(119904) 119878119906

1

119899(119904)) 119889119904)

le4119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times 120572 (119891 (119904 1199061

119899(119904) 119866119906

1

119899(119904) 119878119906

1

119899(119904))) 119889119904

le4119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times [1198711120572 (1198631 (119904)) + 1198712120572 ((1198661198631) (119904))

+1198713120572 ((1198781198631) (119904))] 119889119904

le4119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times (1198711+ 11988611987001198712+ 11988611986701198713) 120572 (119863

1 (119904)) 119889119904

le4119872(119871

1 + 11988611987001198712 + 11988611986701198713) 119905119902

Γ (1 + 119902)120572 (119863)

(42)


1199062

119899 sub co119876(11199060)(119863) 119906

0 such that

120572 (119876 (co 119876(11199060) (119863) 1199060) (119905)) le 2120572 (119876 (1198632) (119905)) (43)



120572 (119876(21199060)

(119863) (119905))

= 120572 (119876 (co 119876(11199060) (119863) 1199060) (119905))

le 2120572 (119876 (1198632) (119905))

le 2120572(int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904)

times119891 (119904 1199062

119899(119904) 119866119906

2

119899(119904) 119878119906

2

119899(119904)) 119889119904)

le4119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times 120572 (119891 (119904 1199062

119899(119904) 119866119906

2

119899(119904) 119878119906

2

119899(119904))) 119889119904

le4119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times (1198711+ 11988611987001198712+ 11988611986701198713) 120572 (119863

2 (119904)) 119889119904

le4119872(119871

1 + 11988611987001198712 + 11988611986701198713)

Γ (119902)

times int

119905

0

(119905 minus 119904)119902minus1

sdot 120572 (co 119876(11199060) (119863) 1199060 (119904)) 119889119904

le[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713)]2

Γ (119902) Γ (1 + 119902)

times int

119905

0

(119905 minus 119904)119902minus1

119904119902120572 (119863) 119889119904

=[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713) 119905119902]2

Γ (1 + 2119902)B (1 + 119902 119902)

times int

1

0

(1 minus 119904)119902minus1

119904119902119889119904 120572 (119863)

=[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713) 119905119902]2

Γ (1 + 2119902)120572 (119863)

(44)

where B(119901 119902) = int1

0119904119901minus1

(1 minus 119904)119902minus1


120572 (119876(1198961199060)

(119863) (119905))

le[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713) 119905119902]119896

Γ (1 + 119896119902)120572 (119863) forall119905 isin 119869

(45)


=

119906119896+1

119899 sub co119876(1198961199060)(119863) 119906

0 such that

120572 (119876 (co 119876(1198961199060) (119863) 1199060) (119905)) le 2120572 (119876 (119863119896+1) (119905)) (46)


120572 (119876(119896+11199060)

(119863) (119905))

= 120572 (119876 (co 119876(1198961199060) (119863) 1199060) (119905))

le 2120572 (119876 (119863119896+1) (119905))

le 2120572(int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904)

times119891 (119904 119906119896+1

119899(119904) 119866119906

119896+1

119899(119904) 119878119906

119896+1

119899(119904)) 119889119904)

le4119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times (1198711+ 11988611987001198712+ 11988611986701198713) 120572 (119863

119896+1 (119904)) 119889119904

le4119872(119871

1 + 11988611987001198712 + 11988611986701198713)

Γ (119902)

times int

119905

0

(119905 minus 119904)119902minus1

120572 (co 119876(1198961199060) (119863) 1199060 (119904)) 119889119904

le[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713)]119896+1

Γ (119902) Γ (1 + 119896119902)

times int

119905

0

(119905 minus 119904)119902minus1

119904119896119902120572 (119863) 119889119904

=[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713) 119905119902]119896+1

Γ (1 + (119896 + 1) 119902)B (1 + 119896119902 119902)

times int

1

0

(1 minus 119904)119902minus1

119904119896119902119889119904 120572 (119863)

=[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713) 119905119902]119896+1

Γ (1 + (119896 + 1) 119902)120572 (119863)

(47)


120572 (119876(1198991199060)

(119863) (119905)) le[4119872 (1198711 + 11988611987001198712 + 11988611986701198713) 119905

119902]119899

Γ (1 + 119899119902)120572 (119863)

(48)


120572 (119876(1198991199060)

(119863)) = max119905isin119869

120572 (119876(1198991199060)

(119863) (119905))

le[4119872 (1198711 + 11988611987001198712 + 11988611986701198713) 119886

119902]119899

Γ (1 + 119899119902)120572 (119863)

(49)


Γ (1 + 119899119902) = radic2120587119899119902(119899119902

119890)

119899119902

119890]12119899119902

0 lt ] lt 1 (50)


Thus

[4119872 (1198711+ 11988611987001198712+ 11988611986701198713) 119886119902]119899

Γ (1 + 119899119902)

=[4119872 (1198711 + 11988611987001198712 + 11988611986701198713) 119886

119902]119899

radic2120587119899119902(119899119902119890)119899119902119890]12119899119902


(51)


[4119872 (1198711+ 11988611987001198712+ 11988611986701198713) 119886119902]1198990

Γ (1 + 1198990119902)

lt 1 (52)

Hence we get that

120572 (119876(1198990 1199060)

(119863)) lt 120572 (119863) (53)




1isin [0 119902) and


such that

1003817100381710038171003817119891 (119905 119906 119866119906 119878119906)1003817100381710038171003817 le 120593 (119905) Ψ (119906) (54)



119862+ 119888






(119872119888 +Ψ (119877)119872119886

119902minus1199021

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus11990211003817100381710038171003817120593100381710038171003817100381711987111199021 [0119886]

)


(55)


119877 by (24) (55)


(119876119906) (119905) le 1198721003817100381710038171003817119892 (119906)

1003817100381710038171003817

+119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times1003817100381710038171003817119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904))

1003817100381710038171003817 119889119904

le 119872(119887119906119862 + 119888) +Ψ (119877)119872

Γ (119902)

times (int

119905

0


1minus11990211003817100381710038171003817120593100381710038171003817100381711987111199021 [0119905]

le 119872(119887119877 + 119888) +Ψ (119877)119872119886

119902minus1199021

Γ (119902)


119902 minus 1199021

)

1minus11990211003817100381710038171003817120593100381710038171003817100381711987111199021 [0119886]

le 119877

(56)









(int

119886

0

12059311199021

(119904) 119889119904)

1199021


(119903 (1 minus119872119887) +119872119888) Γ (119902)

Ψ (119903)119872119886119902minus1199021

times (119902 minus 1199021

1 minus 1199021

)

1minus1199021

(57)











119864 then T119902(119905) (119905 ge 0) and S


operators


119902(119905) (119905 ge 0)








1 2 3) and ae 119905 isin 119869

120572 (119891 (119905 1198631 (119905) 1198632 (119905) 1198633 (119905))) le

3

sum

119894=1

119871119894120572 (119863119894 (119905)) (58)






119877 ge119873119872

1 minus 119873119872119887[119888 +

119886119902minus1199021

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus11990211003817100381710038171003817120593100381710038171003817100381711987111199021 [0119886]

] (59)


119862le 119877 by (24)


120579 le (119876119906) (119905)

le T119902 (119905) (119887119906 + 119888)

+ int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904) 120593 (119904) 119889119904

(60)


(119876119906) (119905) le 119873119872(119887119906119862 + 119888)

+119873119872

Γ (119902)(int

119905

0


1minus1199021

times1003817100381710038171003817120593100381710038171003817100381711987111199021 [0119905]

le 119873119872(119887119877 + 119888) +119873119872119886

119902minus1199021

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus1199021

times1003817100381710038171003817120593100381710038171003817100381711987111199021 [0119886]

le 119877

(61)

Thus119876 Ω119877(119875) rarr Ω

119877(119875) Let 119865 = co 119876(Ω

119877(119875)) Similar to






such that119891 (119905 119906 V 119908) le 119892

1 (119905) 119906 + 1198922 (119905) V + ℎ (119905)


(62)




(119861119906) (119905) = int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904)

times [1198921 (119904) 119906 (119904) + 1198922 (119904) 119866119906 (119904)] 119889119904 119905 isin 119869

(63)



(119861119906) (119905)

le119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times

10038171003817100381710038171003817100381710038171003817

1198921 (119904) 119906 (119904)+1198922 (119904) int

119904

0

119870 (119904 120591) 119906 (120591) 119889120591

10038171003817100381710038171003817100381710038171003817

119889119904

le119872119866lowast(1 + 1198861198700) 119906119862

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

119889119904

=120573119905119902119906119862

Γ (1 + 119902)

(64)


1198861198700) Further10038171003817100381710038171003817(1198612119906) (119905)

10038171003817100381710038171003817


Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times

10038171003817100381710038171003817100381710038171003817

(119861119906) (119904) + int

119904

0

119870 (119904 120591) (119861119906) (120591) 119889120591

10038171003817100381710038171003817100381710038171003817

119889119904

le119872119866lowast120573119906119862

Γ (119902) Γ (1 + 119902)

times int

119905

0

(119905 minus 119904)119902minus1

[119904119902+ int

119904

0

119870 (119904 120591) 120591119902119889120591] 119889119904

le1205732119906119862

Γ (1 + 2119902)B (1 + 119902 119902)

times int

119905

0

(119905 minus 119904)119902minus1

119904119902119889119904

=12057321199052119902119906119862

Γ (1 + 2119902)

(65)


1003817100381710038171003817(119861119899119906) (119905)

1003817100381710038171003817 le120573119899119905119899119902119906119862

Γ (1 + 119899119902) (66)

Therefore we have

10038171003817100381710038171198611198991003817100381710038171003817119862

le120573119899119886119899119902

Γ (1 + 119899119902) (67)


119903 (119861) = lim119899rarrinfin

10038171003817100381710038171198611198991003817100381710038171003817

1119899


120573119886119902

2119899radic2120587119899119902(119899119902119890)

119902119890](12119899

2119902)= 0

(68)



119861lowastle 119903 (119861) + 120574 = 120574 (69)


lowastLet 119872

lowast= sup

119905isin[0+infin)119879(119905)

lowast

L(119864) and 119906lowast

119862=

sup119905isin119869119906(119905)

lowast Choose

119877lowast=119873119872lowast(Γ (1 + 119902) 119888 + 119886

119902ℎlowast

119862)

Γ (1 + 119902) (1 minus 119873120574) (70)


lowast

119862le 119877lowast by (24) and


120579 le (119876119906) (119905)

le T119902 (119905) 119892 (119906)

+ int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904)

times [1198921 (119904) 119906 (119904) + 1198922 (119904) 119866119906 (119904) + ℎ (119904)] 119889119904

le T119902 (119905) 119888 + 119861 (119906) (119905)

+ int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904) ℎ (119904) 119889119904

(71)


(119876119906)(119905)lowastle 119873(

10038171003817100381710038171003817T119902(119905)119888

10038171003817100381710038171003817

lowast

+ (119861119906)(119905)lowast

+

10038171003817100381710038171003817100381710038171003817

int

119905

0

(119905 minus 119904)119902minus1

S119902(119905 minus 119904)ℎ(119904)119889119904

10038171003817100381710038171003817100381710038171003817

lowast

)

le 119873(119872lowast119888 + 119861

lowast119906lowast

119862+119872lowast119886119902ℎlowast

119862

Γ (1 + 119902))

le 119873119872lowast119888 + 119873120574119877


119862

Γ (1 + 119902)

= 119877lowast

(72)



lowast

119877lowast(119875) Let

119865 = co 119876(Ωlowast







Acknowledgments


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014




For 1199051= 0 0 lt 119905


4 = 0 For 119905

1gt 0


100381710038171003817100381711986841003817100381710038171003817 le int

1199051minus120598

0

(1199051minus 119904)119902minus1


10038171003817100381710038171003817120593119877 (119904) 119889119904

+ int

1199051

1199051minus120598

(1199051minus 119904)119902minus1


119902(1199051minus 119904)

10038171003817100381710038171003817120593119877 (119904) 119889119904

le sup119904isin[01199051minus120598]

10038171003817100381710038171003817S119902(1199052minus 119904) minusS

119902(1199051minus 119904)

10038171003817100381710038171003817

times int

1199051minus120598

0

(1199051 minus 119904)119902minus1

120593119877 (119904) 119889119904

+2119872

Γ (119902)int

1199051

1199051minus120598

(1199051minus 119904)119902minus1

120593119877 (119904) 119889119904

le sup119904isin[01199051minus120598]

10038171003817100381710038171003817S119902(1199052minus 119904) minusS

119902(1199051minus 119904)

10038171003817100381710038171003817

sdot1003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [0119886](1 minus 1199021

119902 minus 1199021

)

1minus1199021

sdot (119905(119902minus1199021)(1minus1199021)


1minus1199021

+

21198721003817100381710038171003817120593119877

100381710038171003817100381711987111199021 [0119886]

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus1199021

120598119902minus1199021 997888rarr 0

as 1199052 minus 1199051 997888rarr 0

(36)

As a result (119876119906)(1199052)minus(119876119906)(119905



119877rarr Ω119877




120572 (119876(1198990 1199060)

(119863)) lt 120572 (119863) (37)




120572 (119876(1198991199060)

(119863)) = max119905isin119869

120572 (119876(1198991199060)

(119863) (119905)) 119899 = 1 2

(38)


119899 sub 119863 such

that

120572 (119876 (119863) (119905)) le 2120572 (119876 (1198631) (119905)) (39)


int

119886

0

119906 (119904) 119889119904 isin 119886 co 119906 (119904) | 119904 isin 119869 119906 isin 119862 (119869 119864) (40)

we have

120572(int

119905

0

119870 (119905 119904) 119906 (119904) 119889119904 | 119906 isin 119863 119905 isin 119869)

le 1198861198700120572 (119906 (119905) | 119906 isin 119863 119905 isin 119869)

120572 (int

119886

0

119867(119905 119904) 119906 (119904) 119889119904 | 119906 isin 119863 119905 isin 119869)

le 1198861198670120572 (119906 (119905) | 119906 isin 119863 119905 isin 119869)

(41)

where 1198700= max

(119905119904)isinΔ|119870(119905 119904)| 119867

0= max

(119905119904)isinΔ 0|119867(119905 119904)|


120572 (119876(11199060)

(119863) (119905))

= 120572 (119876 (119863) (119905)) le 2120572 (119876 (1198631) (119905))

le 2120572 (T119902 (119905) 119892 (119906

1

119899)

+ int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904)

times119891 (119904 1199061

119899(119904) 119866119906

1

119899(119904) 119878119906

1

119899(119904)) 119889119904)

le4119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times 120572 (119891 (119904 1199061

119899(119904) 119866119906

1

119899(119904) 119878119906

1

119899(119904))) 119889119904

le4119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times [1198711120572 (1198631 (119904)) + 1198712120572 ((1198661198631) (119904))

+1198713120572 ((1198781198631) (119904))] 119889119904

le4119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times (1198711+ 11988611987001198712+ 11988611986701198713) 120572 (119863

1 (119904)) 119889119904

le4119872(119871

1 + 11988611987001198712 + 11988611986701198713) 119905119902

Γ (1 + 119902)120572 (119863)

(42)


1199062

119899 sub co119876(11199060)(119863) 119906

0 such that

120572 (119876 (co 119876(11199060) (119863) 1199060) (119905)) le 2120572 (119876 (1198632) (119905)) (43)



120572 (119876(21199060)

(119863) (119905))

= 120572 (119876 (co 119876(11199060) (119863) 1199060) (119905))

le 2120572 (119876 (1198632) (119905))

le 2120572(int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904)

times119891 (119904 1199062

119899(119904) 119866119906

2

119899(119904) 119878119906

2

119899(119904)) 119889119904)

le4119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times 120572 (119891 (119904 1199062

119899(119904) 119866119906

2

119899(119904) 119878119906

2

119899(119904))) 119889119904

le4119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times (1198711+ 11988611987001198712+ 11988611986701198713) 120572 (119863

2 (119904)) 119889119904

le4119872(119871

1 + 11988611987001198712 + 11988611986701198713)

Γ (119902)

times int

119905

0

(119905 minus 119904)119902minus1

sdot 120572 (co 119876(11199060) (119863) 1199060 (119904)) 119889119904

le[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713)]2

Γ (119902) Γ (1 + 119902)

times int

119905

0

(119905 minus 119904)119902minus1

119904119902120572 (119863) 119889119904

=[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713) 119905119902]2

Γ (1 + 2119902)B (1 + 119902 119902)

times int

1

0

(1 minus 119904)119902minus1

119904119902119889119904 120572 (119863)

=[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713) 119905119902]2

Γ (1 + 2119902)120572 (119863)

(44)

where B(119901 119902) = int1

0119904119901minus1

(1 minus 119904)119902minus1


120572 (119876(1198961199060)

(119863) (119905))

le[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713) 119905119902]119896

Γ (1 + 119896119902)120572 (119863) forall119905 isin 119869

(45)


=

119906119896+1

119899 sub co119876(1198961199060)(119863) 119906

0 such that

120572 (119876 (co 119876(1198961199060) (119863) 1199060) (119905)) le 2120572 (119876 (119863119896+1) (119905)) (46)


120572 (119876(119896+11199060)

(119863) (119905))

= 120572 (119876 (co 119876(1198961199060) (119863) 1199060) (119905))

le 2120572 (119876 (119863119896+1) (119905))

le 2120572(int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904)

times119891 (119904 119906119896+1

119899(119904) 119866119906

119896+1

119899(119904) 119878119906

119896+1

119899(119904)) 119889119904)

le4119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times (1198711+ 11988611987001198712+ 11988611986701198713) 120572 (119863

119896+1 (119904)) 119889119904

le4119872(119871

1 + 11988611987001198712 + 11988611986701198713)

Γ (119902)

times int

119905

0

(119905 minus 119904)119902minus1

120572 (co 119876(1198961199060) (119863) 1199060 (119904)) 119889119904

le[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713)]119896+1

Γ (119902) Γ (1 + 119896119902)

times int

119905

0

(119905 minus 119904)119902minus1

119904119896119902120572 (119863) 119889119904

=[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713) 119905119902]119896+1

Γ (1 + (119896 + 1) 119902)B (1 + 119896119902 119902)

times int

1

0

(1 minus 119904)119902minus1

119904119896119902119889119904 120572 (119863)

=[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713) 119905119902]119896+1

Γ (1 + (119896 + 1) 119902)120572 (119863)

(47)


120572 (119876(1198991199060)

(119863) (119905)) le[4119872 (1198711 + 11988611987001198712 + 11988611986701198713) 119905

119902]119899

Γ (1 + 119899119902)120572 (119863)

(48)


120572 (119876(1198991199060)

(119863)) = max119905isin119869

120572 (119876(1198991199060)

(119863) (119905))

le[4119872 (1198711 + 11988611987001198712 + 11988611986701198713) 119886

119902]119899

Γ (1 + 119899119902)120572 (119863)

(49)


Γ (1 + 119899119902) = radic2120587119899119902(119899119902

119890)

119899119902

119890]12119899119902

0 lt ] lt 1 (50)


Thus

[4119872 (1198711+ 11988611987001198712+ 11988611986701198713) 119886119902]119899

Γ (1 + 119899119902)

=[4119872 (1198711 + 11988611987001198712 + 11988611986701198713) 119886

119902]119899

radic2120587119899119902(119899119902119890)119899119902119890]12119899119902


(51)


[4119872 (1198711+ 11988611987001198712+ 11988611986701198713) 119886119902]1198990

Γ (1 + 1198990119902)

lt 1 (52)

Hence we get that

120572 (119876(1198990 1199060)

(119863)) lt 120572 (119863) (53)




1isin [0 119902) and


such that

1003817100381710038171003817119891 (119905 119906 119866119906 119878119906)1003817100381710038171003817 le 120593 (119905) Ψ (119906) (54)



119862+ 119888






(119872119888 +Ψ (119877)119872119886

119902minus1199021

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus11990211003817100381710038171003817120593100381710038171003817100381711987111199021 [0119886]

)


(55)


119877 by (24) (55)


(119876119906) (119905) le 1198721003817100381710038171003817119892 (119906)

1003817100381710038171003817

+119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times1003817100381710038171003817119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904))

1003817100381710038171003817 119889119904

le 119872(119887119906119862 + 119888) +Ψ (119877)119872

Γ (119902)

times (int

119905

0


1minus11990211003817100381710038171003817120593100381710038171003817100381711987111199021 [0119905]

le 119872(119887119877 + 119888) +Ψ (119877)119872119886

119902minus1199021

Γ (119902)


119902 minus 1199021

)

1minus11990211003817100381710038171003817120593100381710038171003817100381711987111199021 [0119886]

le 119877

(56)









(int

119886

0

12059311199021

(119904) 119889119904)

1199021


(119903 (1 minus119872119887) +119872119888) Γ (119902)

Ψ (119903)119872119886119902minus1199021

times (119902 minus 1199021

1 minus 1199021

)

1minus1199021

(57)











119864 then T119902(119905) (119905 ge 0) and S


operators


119902(119905) (119905 ge 0)








1 2 3) and ae 119905 isin 119869

120572 (119891 (119905 1198631 (119905) 1198632 (119905) 1198633 (119905))) le

3

sum

119894=1

119871119894120572 (119863119894 (119905)) (58)






119877 ge119873119872

1 minus 119873119872119887[119888 +

119886119902minus1199021

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus11990211003817100381710038171003817120593100381710038171003817100381711987111199021 [0119886]

] (59)


119862le 119877 by (24)


120579 le (119876119906) (119905)

le T119902 (119905) (119887119906 + 119888)

+ int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904) 120593 (119904) 119889119904

(60)


(119876119906) (119905) le 119873119872(119887119906119862 + 119888)

+119873119872

Γ (119902)(int

119905

0


1minus1199021

times1003817100381710038171003817120593100381710038171003817100381711987111199021 [0119905]

le 119873119872(119887119877 + 119888) +119873119872119886

119902minus1199021

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus1199021

times1003817100381710038171003817120593100381710038171003817100381711987111199021 [0119886]

le 119877

(61)

Thus119876 Ω119877(119875) rarr Ω

119877(119875) Let 119865 = co 119876(Ω

119877(119875)) Similar to






such that119891 (119905 119906 V 119908) le 119892

1 (119905) 119906 + 1198922 (119905) V + ℎ (119905)


(62)




(119861119906) (119905) = int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904)

times [1198921 (119904) 119906 (119904) + 1198922 (119904) 119866119906 (119904)] 119889119904 119905 isin 119869

(63)



(119861119906) (119905)

le119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times

10038171003817100381710038171003817100381710038171003817

1198921 (119904) 119906 (119904)+1198922 (119904) int

119904

0

119870 (119904 120591) 119906 (120591) 119889120591

10038171003817100381710038171003817100381710038171003817

119889119904

le119872119866lowast(1 + 1198861198700) 119906119862

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

119889119904

=120573119905119902119906119862

Γ (1 + 119902)

(64)


1198861198700) Further10038171003817100381710038171003817(1198612119906) (119905)

10038171003817100381710038171003817


Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times

10038171003817100381710038171003817100381710038171003817

(119861119906) (119904) + int

119904

0

119870 (119904 120591) (119861119906) (120591) 119889120591

10038171003817100381710038171003817100381710038171003817

119889119904

le119872119866lowast120573119906119862

Γ (119902) Γ (1 + 119902)

times int

119905

0

(119905 minus 119904)119902minus1

[119904119902+ int

119904

0

119870 (119904 120591) 120591119902119889120591] 119889119904

le1205732119906119862

Γ (1 + 2119902)B (1 + 119902 119902)

times int

119905

0

(119905 minus 119904)119902minus1

119904119902119889119904

=12057321199052119902119906119862

Γ (1 + 2119902)

(65)


1003817100381710038171003817(119861119899119906) (119905)

1003817100381710038171003817 le120573119899119905119899119902119906119862

Γ (1 + 119899119902) (66)

Therefore we have

10038171003817100381710038171198611198991003817100381710038171003817119862

le120573119899119886119899119902

Γ (1 + 119899119902) (67)


119903 (119861) = lim119899rarrinfin

10038171003817100381710038171198611198991003817100381710038171003817

1119899


120573119886119902

2119899radic2120587119899119902(119899119902119890)

119902119890](12119899

2119902)= 0

(68)



119861lowastle 119903 (119861) + 120574 = 120574 (69)


lowastLet 119872

lowast= sup

119905isin[0+infin)119879(119905)

lowast

L(119864) and 119906lowast

119862=

sup119905isin119869119906(119905)

lowast Choose

119877lowast=119873119872lowast(Γ (1 + 119902) 119888 + 119886

119902ℎlowast

119862)

Γ (1 + 119902) (1 minus 119873120574) (70)


lowast

119862le 119877lowast by (24) and


120579 le (119876119906) (119905)

le T119902 (119905) 119892 (119906)

+ int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904)

times [1198921 (119904) 119906 (119904) + 1198922 (119904) 119866119906 (119904) + ℎ (119904)] 119889119904

le T119902 (119905) 119888 + 119861 (119906) (119905)

+ int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904) ℎ (119904) 119889119904

(71)


(119876119906)(119905)lowastle 119873(

10038171003817100381710038171003817T119902(119905)119888

10038171003817100381710038171003817

lowast

+ (119861119906)(119905)lowast

+

10038171003817100381710038171003817100381710038171003817

int

119905

0

(119905 minus 119904)119902minus1

S119902(119905 minus 119904)ℎ(119904)119889119904

10038171003817100381710038171003817100381710038171003817

lowast

)

le 119873(119872lowast119888 + 119861

lowast119906lowast

119862+119872lowast119886119902ℎlowast

119862

Γ (1 + 119902))

le 119873119872lowast119888 + 119873120574119877


119862

Γ (1 + 119902)

= 119877lowast

(72)



lowast

119877lowast(119875) Let

119865 = co 119876(Ωlowast







Acknowledgments


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014





120572 (119876(21199060)

(119863) (119905))

= 120572 (119876 (co 119876(11199060) (119863) 1199060) (119905))

le 2120572 (119876 (1198632) (119905))

le 2120572(int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904)

times119891 (119904 1199062

119899(119904) 119866119906

2

119899(119904) 119878119906

2

119899(119904)) 119889119904)

le4119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times 120572 (119891 (119904 1199062

119899(119904) 119866119906

2

119899(119904) 119878119906

2

119899(119904))) 119889119904

le4119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times (1198711+ 11988611987001198712+ 11988611986701198713) 120572 (119863

2 (119904)) 119889119904

le4119872(119871

1 + 11988611987001198712 + 11988611986701198713)

Γ (119902)

times int

119905

0

(119905 minus 119904)119902minus1

sdot 120572 (co 119876(11199060) (119863) 1199060 (119904)) 119889119904

le[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713)]2

Γ (119902) Γ (1 + 119902)

times int

119905

0

(119905 minus 119904)119902minus1

119904119902120572 (119863) 119889119904

=[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713) 119905119902]2

Γ (1 + 2119902)B (1 + 119902 119902)

times int

1

0

(1 minus 119904)119902minus1

119904119902119889119904 120572 (119863)

=[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713) 119905119902]2

Γ (1 + 2119902)120572 (119863)

(44)

where B(119901 119902) = int1

0119904119901minus1

(1 minus 119904)119902minus1


120572 (119876(1198961199060)

(119863) (119905))

le[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713) 119905119902]119896

Γ (1 + 119896119902)120572 (119863) forall119905 isin 119869

(45)


=

119906119896+1

119899 sub co119876(1198961199060)(119863) 119906

0 such that

120572 (119876 (co 119876(1198961199060) (119863) 1199060) (119905)) le 2120572 (119876 (119863119896+1) (119905)) (46)


120572 (119876(119896+11199060)

(119863) (119905))

= 120572 (119876 (co 119876(1198961199060) (119863) 1199060) (119905))

le 2120572 (119876 (119863119896+1) (119905))

le 2120572(int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904)

times119891 (119904 119906119896+1

119899(119904) 119866119906

119896+1

119899(119904) 119878119906

119896+1

119899(119904)) 119889119904)

le4119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times (1198711+ 11988611987001198712+ 11988611986701198713) 120572 (119863

119896+1 (119904)) 119889119904

le4119872(119871

1 + 11988611987001198712 + 11988611986701198713)

Γ (119902)

times int

119905

0

(119905 minus 119904)119902minus1

120572 (co 119876(1198961199060) (119863) 1199060 (119904)) 119889119904

le[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713)]119896+1

Γ (119902) Γ (1 + 119896119902)

times int

119905

0

(119905 minus 119904)119902minus1

119904119896119902120572 (119863) 119889119904

=[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713) 119905119902]119896+1

Γ (1 + (119896 + 1) 119902)B (1 + 119896119902 119902)

times int

1

0

(1 minus 119904)119902minus1

119904119896119902119889119904 120572 (119863)

=[4119872 (119871

1+ 11988611987001198712+ 11988611986701198713) 119905119902]119896+1

Γ (1 + (119896 + 1) 119902)120572 (119863)

(47)


120572 (119876(1198991199060)

(119863) (119905)) le[4119872 (1198711 + 11988611987001198712 + 11988611986701198713) 119905

119902]119899

Γ (1 + 119899119902)120572 (119863)

(48)


120572 (119876(1198991199060)

(119863)) = max119905isin119869

120572 (119876(1198991199060)

(119863) (119905))

le[4119872 (1198711 + 11988611987001198712 + 11988611986701198713) 119886

119902]119899

Γ (1 + 119899119902)120572 (119863)

(49)


Γ (1 + 119899119902) = radic2120587119899119902(119899119902

119890)

119899119902

119890]12119899119902

0 lt ] lt 1 (50)


Thus

[4119872 (1198711+ 11988611987001198712+ 11988611986701198713) 119886119902]119899

Γ (1 + 119899119902)

=[4119872 (1198711 + 11988611987001198712 + 11988611986701198713) 119886

119902]119899

radic2120587119899119902(119899119902119890)119899119902119890]12119899119902


(51)


[4119872 (1198711+ 11988611987001198712+ 11988611986701198713) 119886119902]1198990

Γ (1 + 1198990119902)

lt 1 (52)

Hence we get that

120572 (119876(1198990 1199060)

(119863)) lt 120572 (119863) (53)




1isin [0 119902) and


such that

1003817100381710038171003817119891 (119905 119906 119866119906 119878119906)1003817100381710038171003817 le 120593 (119905) Ψ (119906) (54)



119862+ 119888






(119872119888 +Ψ (119877)119872119886

119902minus1199021

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus11990211003817100381710038171003817120593100381710038171003817100381711987111199021 [0119886]

)


(55)


119877 by (24) (55)


(119876119906) (119905) le 1198721003817100381710038171003817119892 (119906)

1003817100381710038171003817

+119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times1003817100381710038171003817119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904))

1003817100381710038171003817 119889119904

le 119872(119887119906119862 + 119888) +Ψ (119877)119872

Γ (119902)

times (int

119905

0


1minus11990211003817100381710038171003817120593100381710038171003817100381711987111199021 [0119905]

le 119872(119887119877 + 119888) +Ψ (119877)119872119886

119902minus1199021

Γ (119902)


119902 minus 1199021

)

1minus11990211003817100381710038171003817120593100381710038171003817100381711987111199021 [0119886]

le 119877

(56)









(int

119886

0

12059311199021

(119904) 119889119904)

1199021


(119903 (1 minus119872119887) +119872119888) Γ (119902)

Ψ (119903)119872119886119902minus1199021

times (119902 minus 1199021

1 minus 1199021

)

1minus1199021

(57)











119864 then T119902(119905) (119905 ge 0) and S


operators


119902(119905) (119905 ge 0)








1 2 3) and ae 119905 isin 119869

120572 (119891 (119905 1198631 (119905) 1198632 (119905) 1198633 (119905))) le

3

sum

119894=1

119871119894120572 (119863119894 (119905)) (58)






119877 ge119873119872

1 minus 119873119872119887[119888 +

119886119902minus1199021

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus11990211003817100381710038171003817120593100381710038171003817100381711987111199021 [0119886]

] (59)


119862le 119877 by (24)


120579 le (119876119906) (119905)

le T119902 (119905) (119887119906 + 119888)

+ int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904) 120593 (119904) 119889119904

(60)


(119876119906) (119905) le 119873119872(119887119906119862 + 119888)

+119873119872

Γ (119902)(int

119905

0


1minus1199021

times1003817100381710038171003817120593100381710038171003817100381711987111199021 [0119905]

le 119873119872(119887119877 + 119888) +119873119872119886

119902minus1199021

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus1199021

times1003817100381710038171003817120593100381710038171003817100381711987111199021 [0119886]

le 119877

(61)

Thus119876 Ω119877(119875) rarr Ω

119877(119875) Let 119865 = co 119876(Ω

119877(119875)) Similar to






such that119891 (119905 119906 V 119908) le 119892

1 (119905) 119906 + 1198922 (119905) V + ℎ (119905)


(62)




(119861119906) (119905) = int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904)

times [1198921 (119904) 119906 (119904) + 1198922 (119904) 119866119906 (119904)] 119889119904 119905 isin 119869

(63)



(119861119906) (119905)

le119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times

10038171003817100381710038171003817100381710038171003817

1198921 (119904) 119906 (119904)+1198922 (119904) int

119904

0

119870 (119904 120591) 119906 (120591) 119889120591

10038171003817100381710038171003817100381710038171003817

119889119904

le119872119866lowast(1 + 1198861198700) 119906119862

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

119889119904

=120573119905119902119906119862

Γ (1 + 119902)

(64)


1198861198700) Further10038171003817100381710038171003817(1198612119906) (119905)

10038171003817100381710038171003817


Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times

10038171003817100381710038171003817100381710038171003817

(119861119906) (119904) + int

119904

0

119870 (119904 120591) (119861119906) (120591) 119889120591

10038171003817100381710038171003817100381710038171003817

119889119904

le119872119866lowast120573119906119862

Γ (119902) Γ (1 + 119902)

times int

119905

0

(119905 minus 119904)119902minus1

[119904119902+ int

119904

0

119870 (119904 120591) 120591119902119889120591] 119889119904

le1205732119906119862

Γ (1 + 2119902)B (1 + 119902 119902)

times int

119905

0

(119905 minus 119904)119902minus1

119904119902119889119904

=12057321199052119902119906119862

Γ (1 + 2119902)

(65)


1003817100381710038171003817(119861119899119906) (119905)

1003817100381710038171003817 le120573119899119905119899119902119906119862

Γ (1 + 119899119902) (66)

Therefore we have

10038171003817100381710038171198611198991003817100381710038171003817119862

le120573119899119886119899119902

Γ (1 + 119899119902) (67)


119903 (119861) = lim119899rarrinfin

10038171003817100381710038171198611198991003817100381710038171003817

1119899


120573119886119902

2119899radic2120587119899119902(119899119902119890)

119902119890](12119899

2119902)= 0

(68)



119861lowastle 119903 (119861) + 120574 = 120574 (69)


lowastLet 119872

lowast= sup

119905isin[0+infin)119879(119905)

lowast

L(119864) and 119906lowast

119862=

sup119905isin119869119906(119905)

lowast Choose

119877lowast=119873119872lowast(Γ (1 + 119902) 119888 + 119886

119902ℎlowast

119862)

Γ (1 + 119902) (1 minus 119873120574) (70)


lowast

119862le 119877lowast by (24) and


120579 le (119876119906) (119905)

le T119902 (119905) 119892 (119906)

+ int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904)

times [1198921 (119904) 119906 (119904) + 1198922 (119904) 119866119906 (119904) + ℎ (119904)] 119889119904

le T119902 (119905) 119888 + 119861 (119906) (119905)

+ int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904) ℎ (119904) 119889119904

(71)


(119876119906)(119905)lowastle 119873(

10038171003817100381710038171003817T119902(119905)119888

10038171003817100381710038171003817

lowast

+ (119861119906)(119905)lowast

+

10038171003817100381710038171003817100381710038171003817

int

119905

0

(119905 minus 119904)119902minus1

S119902(119905 minus 119904)ℎ(119904)119889119904

10038171003817100381710038171003817100381710038171003817

lowast

)

le 119873(119872lowast119888 + 119861

lowast119906lowast

119862+119872lowast119886119902ℎlowast

119862

Γ (1 + 119902))

le 119873119872lowast119888 + 119873120574119877


119862

Γ (1 + 119902)

= 119877lowast

(72)



lowast

119877lowast(119875) Let

119865 = co 119876(Ωlowast







Acknowledgments


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014




Thus

[4119872 (1198711+ 11988611987001198712+ 11988611986701198713) 119886119902]119899

Γ (1 + 119899119902)

=[4119872 (1198711 + 11988611987001198712 + 11988611986701198713) 119886

119902]119899

radic2120587119899119902(119899119902119890)119899119902119890]12119899119902


(51)


[4119872 (1198711+ 11988611987001198712+ 11988611986701198713) 119886119902]1198990

Γ (1 + 1198990119902)

lt 1 (52)

Hence we get that

120572 (119876(1198990 1199060)

(119863)) lt 120572 (119863) (53)




1isin [0 119902) and


such that

1003817100381710038171003817119891 (119905 119906 119866119906 119878119906)1003817100381710038171003817 le 120593 (119905) Ψ (119906) (54)



119862+ 119888






(119872119888 +Ψ (119877)119872119886

119902minus1199021

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus11990211003817100381710038171003817120593100381710038171003817100381711987111199021 [0119886]

)


(55)


119877 by (24) (55)


(119876119906) (119905) le 1198721003817100381710038171003817119892 (119906)

1003817100381710038171003817

+119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times1003817100381710038171003817119891 (119904 119906 (119904) 119866119906 (119904) 119878119906 (119904))

1003817100381710038171003817 119889119904

le 119872(119887119906119862 + 119888) +Ψ (119877)119872

Γ (119902)

times (int

119905

0


1minus11990211003817100381710038171003817120593100381710038171003817100381711987111199021 [0119905]

le 119872(119887119877 + 119888) +Ψ (119877)119872119886

119902minus1199021

Γ (119902)


119902 minus 1199021

)

1minus11990211003817100381710038171003817120593100381710038171003817100381711987111199021 [0119886]

le 119877

(56)









(int

119886

0

12059311199021

(119904) 119889119904)

1199021


(119903 (1 minus119872119887) +119872119888) Γ (119902)

Ψ (119903)119872119886119902minus1199021

times (119902 minus 1199021

1 minus 1199021

)

1minus1199021

(57)











119864 then T119902(119905) (119905 ge 0) and S


operators


119902(119905) (119905 ge 0)








1 2 3) and ae 119905 isin 119869

120572 (119891 (119905 1198631 (119905) 1198632 (119905) 1198633 (119905))) le

3

sum

119894=1

119871119894120572 (119863119894 (119905)) (58)






119877 ge119873119872

1 minus 119873119872119887[119888 +

119886119902minus1199021

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus11990211003817100381710038171003817120593100381710038171003817100381711987111199021 [0119886]

] (59)


119862le 119877 by (24)


120579 le (119876119906) (119905)

le T119902 (119905) (119887119906 + 119888)

+ int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904) 120593 (119904) 119889119904

(60)


(119876119906) (119905) le 119873119872(119887119906119862 + 119888)

+119873119872

Γ (119902)(int

119905

0


1minus1199021

times1003817100381710038171003817120593100381710038171003817100381711987111199021 [0119905]

le 119873119872(119887119877 + 119888) +119873119872119886

119902minus1199021

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus1199021

times1003817100381710038171003817120593100381710038171003817100381711987111199021 [0119886]

le 119877

(61)

Thus119876 Ω119877(119875) rarr Ω

119877(119875) Let 119865 = co 119876(Ω

119877(119875)) Similar to






such that119891 (119905 119906 V 119908) le 119892

1 (119905) 119906 + 1198922 (119905) V + ℎ (119905)


(62)




(119861119906) (119905) = int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904)

times [1198921 (119904) 119906 (119904) + 1198922 (119904) 119866119906 (119904)] 119889119904 119905 isin 119869

(63)



(119861119906) (119905)

le119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times

10038171003817100381710038171003817100381710038171003817

1198921 (119904) 119906 (119904)+1198922 (119904) int

119904

0

119870 (119904 120591) 119906 (120591) 119889120591

10038171003817100381710038171003817100381710038171003817

119889119904

le119872119866lowast(1 + 1198861198700) 119906119862

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

119889119904

=120573119905119902119906119862

Γ (1 + 119902)

(64)


1198861198700) Further10038171003817100381710038171003817(1198612119906) (119905)

10038171003817100381710038171003817


Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times

10038171003817100381710038171003817100381710038171003817

(119861119906) (119904) + int

119904

0

119870 (119904 120591) (119861119906) (120591) 119889120591

10038171003817100381710038171003817100381710038171003817

119889119904

le119872119866lowast120573119906119862

Γ (119902) Γ (1 + 119902)

times int

119905

0

(119905 minus 119904)119902minus1

[119904119902+ int

119904

0

119870 (119904 120591) 120591119902119889120591] 119889119904

le1205732119906119862

Γ (1 + 2119902)B (1 + 119902 119902)

times int

119905

0

(119905 minus 119904)119902minus1

119904119902119889119904

=12057321199052119902119906119862

Γ (1 + 2119902)

(65)


1003817100381710038171003817(119861119899119906) (119905)

1003817100381710038171003817 le120573119899119905119899119902119906119862

Γ (1 + 119899119902) (66)

Therefore we have

10038171003817100381710038171198611198991003817100381710038171003817119862

le120573119899119886119899119902

Γ (1 + 119899119902) (67)


119903 (119861) = lim119899rarrinfin

10038171003817100381710038171198611198991003817100381710038171003817

1119899


120573119886119902

2119899radic2120587119899119902(119899119902119890)

119902119890](12119899

2119902)= 0

(68)



119861lowastle 119903 (119861) + 120574 = 120574 (69)


lowastLet 119872

lowast= sup

119905isin[0+infin)119879(119905)

lowast

L(119864) and 119906lowast

119862=

sup119905isin119869119906(119905)

lowast Choose

119877lowast=119873119872lowast(Γ (1 + 119902) 119888 + 119886

119902ℎlowast

119862)

Γ (1 + 119902) (1 minus 119873120574) (70)


lowast

119862le 119877lowast by (24) and


120579 le (119876119906) (119905)

le T119902 (119905) 119892 (119906)

+ int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904)

times [1198921 (119904) 119906 (119904) + 1198922 (119904) 119866119906 (119904) + ℎ (119904)] 119889119904

le T119902 (119905) 119888 + 119861 (119906) (119905)

+ int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904) ℎ (119904) 119889119904

(71)


(119876119906)(119905)lowastle 119873(

10038171003817100381710038171003817T119902(119905)119888

10038171003817100381710038171003817

lowast

+ (119861119906)(119905)lowast

+

10038171003817100381710038171003817100381710038171003817

int

119905

0

(119905 minus 119904)119902minus1

S119902(119905 minus 119904)ℎ(119904)119889119904

10038171003817100381710038171003817100381710038171003817

lowast

)

le 119873(119872lowast119888 + 119861

lowast119906lowast

119862+119872lowast119886119902ℎlowast

119862

Γ (1 + 119902))

le 119873119872lowast119888 + 119873120574119877


119862

Γ (1 + 119902)

= 119877lowast

(72)



lowast

119877lowast(119875) Let

119865 = co 119876(Ωlowast







Acknowledgments


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014









119864 then T119902(119905) (119905 ge 0) and S


operators


119902(119905) (119905 ge 0)








1 2 3) and ae 119905 isin 119869

120572 (119891 (119905 1198631 (119905) 1198632 (119905) 1198633 (119905))) le

3

sum

119894=1

119871119894120572 (119863119894 (119905)) (58)






119877 ge119873119872

1 minus 119873119872119887[119888 +

119886119902minus1199021

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus11990211003817100381710038171003817120593100381710038171003817100381711987111199021 [0119886]

] (59)


119862le 119877 by (24)


120579 le (119876119906) (119905)

le T119902 (119905) (119887119906 + 119888)

+ int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904) 120593 (119904) 119889119904

(60)


(119876119906) (119905) le 119873119872(119887119906119862 + 119888)

+119873119872

Γ (119902)(int

119905

0


1minus1199021

times1003817100381710038171003817120593100381710038171003817100381711987111199021 [0119905]

le 119873119872(119887119877 + 119888) +119873119872119886

119902minus1199021

Γ (119902)(1 minus 1199021

119902 minus 1199021

)

1minus1199021

times1003817100381710038171003817120593100381710038171003817100381711987111199021 [0119886]

le 119877

(61)

Thus119876 Ω119877(119875) rarr Ω

119877(119875) Let 119865 = co 119876(Ω

119877(119875)) Similar to






such that119891 (119905 119906 V 119908) le 119892

1 (119905) 119906 + 1198922 (119905) V + ℎ (119905)


(62)




(119861119906) (119905) = int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904)

times [1198921 (119904) 119906 (119904) + 1198922 (119904) 119866119906 (119904)] 119889119904 119905 isin 119869

(63)



(119861119906) (119905)

le119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times

10038171003817100381710038171003817100381710038171003817

1198921 (119904) 119906 (119904)+1198922 (119904) int

119904

0

119870 (119904 120591) 119906 (120591) 119889120591

10038171003817100381710038171003817100381710038171003817

119889119904

le119872119866lowast(1 + 1198861198700) 119906119862

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

119889119904

=120573119905119902119906119862

Γ (1 + 119902)

(64)


1198861198700) Further10038171003817100381710038171003817(1198612119906) (119905)

10038171003817100381710038171003817


Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times

10038171003817100381710038171003817100381710038171003817

(119861119906) (119904) + int

119904

0

119870 (119904 120591) (119861119906) (120591) 119889120591

10038171003817100381710038171003817100381710038171003817

119889119904

le119872119866lowast120573119906119862

Γ (119902) Γ (1 + 119902)

times int

119905

0

(119905 minus 119904)119902minus1

[119904119902+ int

119904

0

119870 (119904 120591) 120591119902119889120591] 119889119904

le1205732119906119862

Γ (1 + 2119902)B (1 + 119902 119902)

times int

119905

0

(119905 minus 119904)119902minus1

119904119902119889119904

=12057321199052119902119906119862

Γ (1 + 2119902)

(65)


1003817100381710038171003817(119861119899119906) (119905)

1003817100381710038171003817 le120573119899119905119899119902119906119862

Γ (1 + 119899119902) (66)

Therefore we have

10038171003817100381710038171198611198991003817100381710038171003817119862

le120573119899119886119899119902

Γ (1 + 119899119902) (67)


119903 (119861) = lim119899rarrinfin

10038171003817100381710038171198611198991003817100381710038171003817

1119899


120573119886119902

2119899radic2120587119899119902(119899119902119890)

119902119890](12119899

2119902)= 0

(68)



119861lowastle 119903 (119861) + 120574 = 120574 (69)


lowastLet 119872

lowast= sup

119905isin[0+infin)119879(119905)

lowast

L(119864) and 119906lowast

119862=

sup119905isin119869119906(119905)

lowast Choose

119877lowast=119873119872lowast(Γ (1 + 119902) 119888 + 119886

119902ℎlowast

119862)

Γ (1 + 119902) (1 minus 119873120574) (70)


lowast

119862le 119877lowast by (24) and


120579 le (119876119906) (119905)

le T119902 (119905) 119892 (119906)

+ int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904)

times [1198921 (119904) 119906 (119904) + 1198922 (119904) 119866119906 (119904) + ℎ (119904)] 119889119904

le T119902 (119905) 119888 + 119861 (119906) (119905)

+ int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904) ℎ (119904) 119889119904

(71)


(119876119906)(119905)lowastle 119873(

10038171003817100381710038171003817T119902(119905)119888

10038171003817100381710038171003817

lowast

+ (119861119906)(119905)lowast

+

10038171003817100381710038171003817100381710038171003817

int

119905

0

(119905 minus 119904)119902minus1

S119902(119905 minus 119904)ℎ(119904)119889119904

10038171003817100381710038171003817100381710038171003817

lowast

)

le 119873(119872lowast119888 + 119861

lowast119906lowast

119862+119872lowast119886119902ℎlowast

119862

Γ (1 + 119902))

le 119873119872lowast119888 + 119873120574119877


119862

Γ (1 + 119902)

= 119877lowast

(72)



lowast

119877lowast(119875) Let

119865 = co 119876(Ωlowast







Acknowledgments


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014





(119861119906) (119905)

le119872

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times

10038171003817100381710038171003817100381710038171003817

1198921 (119904) 119906 (119904)+1198922 (119904) int

119904

0

119870 (119904 120591) 119906 (120591) 119889120591

10038171003817100381710038171003817100381710038171003817

119889119904

le119872119866lowast(1 + 1198861198700) 119906119862

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

119889119904

=120573119905119902119906119862

Γ (1 + 119902)

(64)


1198861198700) Further10038171003817100381710038171003817(1198612119906) (119905)

10038171003817100381710038171003817


Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times

10038171003817100381710038171003817100381710038171003817

(119861119906) (119904) + int

119904

0

119870 (119904 120591) (119861119906) (120591) 119889120591

10038171003817100381710038171003817100381710038171003817

119889119904

le119872119866lowast120573119906119862

Γ (119902) Γ (1 + 119902)

times int

119905

0

(119905 minus 119904)119902minus1

[119904119902+ int

119904

0

119870 (119904 120591) 120591119902119889120591] 119889119904

le1205732119906119862

Γ (1 + 2119902)B (1 + 119902 119902)

times int

119905

0

(119905 minus 119904)119902minus1

119904119902119889119904

=12057321199052119902119906119862

Γ (1 + 2119902)

(65)


1003817100381710038171003817(119861119899119906) (119905)

1003817100381710038171003817 le120573119899119905119899119902119906119862

Γ (1 + 119899119902) (66)

Therefore we have

10038171003817100381710038171198611198991003817100381710038171003817119862

le120573119899119886119899119902

Γ (1 + 119899119902) (67)


119903 (119861) = lim119899rarrinfin

10038171003817100381710038171198611198991003817100381710038171003817

1119899


120573119886119902

2119899radic2120587119899119902(119899119902119890)

119902119890](12119899

2119902)= 0

(68)



119861lowastle 119903 (119861) + 120574 = 120574 (69)


lowastLet 119872

lowast= sup

119905isin[0+infin)119879(119905)

lowast

L(119864) and 119906lowast

119862=

sup119905isin119869119906(119905)

lowast Choose

119877lowast=119873119872lowast(Γ (1 + 119902) 119888 + 119886

119902ℎlowast

119862)

Γ (1 + 119902) (1 minus 119873120574) (70)


lowast

119862le 119877lowast by (24) and


120579 le (119876119906) (119905)

le T119902 (119905) 119892 (119906)

+ int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904)

times [1198921 (119904) 119906 (119904) + 1198922 (119904) 119866119906 (119904) + ℎ (119904)] 119889119904

le T119902 (119905) 119888 + 119861 (119906) (119905)

+ int

119905

0

(119905 minus 119904)119902minus1

S119902 (119905 minus 119904) ℎ (119904) 119889119904

(71)


(119876119906)(119905)lowastle 119873(

10038171003817100381710038171003817T119902(119905)119888

10038171003817100381710038171003817

lowast

+ (119861119906)(119905)lowast

+

10038171003817100381710038171003817100381710038171003817

int

119905

0

(119905 minus 119904)119902minus1

S119902(119905 minus 119904)ℎ(119904)119889119904

10038171003817100381710038171003817100381710038171003817

lowast

)

le 119873(119872lowast119888 + 119861

lowast119906lowast

119862+119872lowast119886119902ℎlowast

119862

Γ (1 + 119902))

le 119873119872lowast119888 + 119873120574119877


119862

Γ (1 + 119902)

= 119877lowast

(72)



lowast

119877lowast(119875) Let

119865 = co 119876(Ωlowast







Acknowledgments


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014





Acknowledgments


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014



Nonlocal Problem for Fractional Evolution …...evolution equations have attracted increasing...

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