Research Article On the Classical Paranormed Sequence Spaces...

Research ArticleOn the Classical Paranormed Sequence Spaces andRelated Duals over the Non-Newtonian Complex Field

ULur Kadak1 Murat KiriGci2 and Ahmet Faruk Ccedilakmak3

1Department of Mathematics Faculty of Sciences and Arts Bozok University 66200 Yozgat Turkey2Department of Mathematical Education Hasan Ali Yucel Education Faculty Istanbul University 34470 Istanbul Turkey3Department of Mathematical Engineering Yıldız Technical University 80750 Istanbul Turkey

Correspondence should be addressed to Ugur Kadak ugurkadakgmailcom

Received 20 October 2014 Accepted 7 May 2015

Academic Editor Jeff Connor

Copyright copy 2015 Ugur Kadak et al 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

The studies on sequence spaces were extended by using the notion of associated multiplier sequences A multiplier sequence canbe used to accelerate the convergence of the sequences in some spaces In some sense it can be viewed as a catalyst which is usedto accelerate the process of chemical reaction Sometimes the associated multiplier sequence delays the rate of convergence of asequence In the present paper the classical paranormed sequence spaces have been introduced and proved that the spaces are⋆-complete By using the notion of multiplier sequence the 120572- 120573- and 120574-duals of certain paranormed spaces have been computedand their basis has been constructed

1 Introduction

The theory of sequence spaces is the fundamental of summa-bility Summability is a wide field of mathematics mainly inanalysis and functional analysis and has many applicationsfor instance in numerical analysis to speed up the rate ofconvergence operator theory the theory of orthogonal seriesand approximation theory The classical summability theorydeals with the generalization of the convergence of sequencesor series of real or complex numbers Besides this the studieson paranormed sequence spaces were initiated by Nakano [1]and Simons [2] at the initial stage Later on it was furtherstudied by Maddox [3] Lascarides [4] and Lascarides andMaddox [5] In recent years Mursaleen et al [6ndash8] haveinvestigated some matrix transformations of paranormedsequence spaces Also Kirisci and Basar [9 10] motivatedthe notion of generalized difference matrix and Demirizand Cakan [11] determined some new paranormed sequencespaces

In the period from 1967 till 1972 Grossman and Katz[12] introduced the non-Newtonian calculus consisting of thebranches of geometric bigeometric quadratic biquadratic

calculus and so forth Also Grossman extended this notionto the other fields in [13 14] All these calculi can be describedsimultaneously within the framework of a general theoryWeprefer to use the name non-Newtonian to indicate any of thecalculi other than the classical calculus Every property inclassical calculus has an analogue in non-Newtonian calculuswhich is a methodology that allows one to have a differentlook at problems which can be investigated via calculus Insome cases for example for wage-rate (in dollars euro etc)related problems the use of bigeometric calculus which isa kind of non-Newtonian calculus is advocated instead of atraditional Newtonian one

Bashirov et al [15 16] have recently concentrated on non-Newtonian calculus and gave the results with applicationscorresponding to the well-known properties of derivativesand integrals in classical calculus Further Misirli and Gurefehave introduced multiplicative Adams Bashforth-Moultonmethod for numerical solution of differential equationsin [17] Also some authors have also worked on classicalsequence spaces and related topics by using non-Newtoniancalculus [18 19] Further Kadak [20] and Kadak et al[21ndash23] have determined Kothe-Toeplitz duals and matrix

Hindawi Publishing CorporationJournal of Function SpacesVolume 2015 Article ID 416906 11 pageshttpdxdoiorg1011552015416906

2 Journal of Function Spaces

transformations between certain sequence spaces over thenon-Newtonian complex field and have generalized Runge-Kutta method with respect to the non-Newtonian calculus

2 Preliminaries Background and Notations

A generator is a one-to-one function whose domain is R

and whose range is a subset R120572of R where R

120572= 120572119909

119909 isin R Each generator generates exactly one arithmeticand conversely each arithmetic is generated by exactly onegenerator For example the identity function 119868 generatesclassical arithmetic and exponential function generates geo-metric (multiplicative) arithmetic As a generator we choosethe function 120572 such that those basic algebraic operations aredefined as follows

120572 - addition 119909

+ 119910 = 120572 120572minus1(119909) + 120572

minus1(119910)

120572 - subtraction 119909

minus 119910 = 120572 120572minus1(119909) minus 120572

minus1(119910)

120572 -multiplication 119909

times 119910 = 120572 120572minus1(119909) times 120572

minus1(119910)

120572 - division 119909

119910

= 120572 120572minus1(119909) divide 120572

minus1(119910)

120572 - order 119909

lt 119910 lArrrArr 120572minus1(119909) lt 120572

minus1(119910)

(1)

for all119909 119910 isin R120572sube R As an example if we choose the function

120572 = exp

120572 R 997888rarr Rexp sube R

119909 997891997888rarr 119910 = 120572 (119909) = 119890119909

(2)

120572-arithmetic can be interpreted as geometric arithmetic

120572 - addition 119909

+ 119910 = 119890ln119909+ln119910

= 119909 sdot 119910


minus 119910 = 119890ln119909minusln119910

= 119909divide119910


times 119910 = 119890ln119909 ln119910

= 119909ln119910

= 119910ln119909

120572 - division 119909

119910

= 119890ln119909ln119910

= 1199091 ln119910

(3)

By an arithmetic we mean a complete ordered field whoserealm is a subset of R There are infinitely many arithmeticsall of which are isomorphic that is structurally equivalentThe 120572-positive real numbers denoted byR+

120572 are the numbers

119909 in R120572such that

0

lt 119909 the 120572-negative real numbersdenoted by Rminus

120572 are those for which 119909

lt

0 The 120572-zero

0 and the 119886119897119901ℎ119886-one

1 turn out to be 120572(0) and 120572(1) Also120572(minus119901) = 120572minus120572

minus1(

119901 ) =

minus

119901 holds for all 119901 isin Z+ Thusthe set of all 120572-integers can be given by

Z120572= 120572 (minus2) 120572 (minus1) 120572 (0) 120572 (1) 120572 (2)

=

minus

2

minus

1

0

1

2 (4)

One can immediately conclude that the set of exp-integer canbe written as

Zexp =

11198902

1119890

1 119890 1198902 (5)

Besides the 120572-summation is defined by

120572

infin

sum

119896=0119909119896= 120572

infin

sum

119896=0120572minus1(119909119896)

= 120572 120572minus1(1199090) + sdot sdot sdot + 120572

minus1(119909119896) + sdot sdot sdot

(6)

for all 119909119896isin R120572sube R

Definition 1 (see [18]) Let119883 be a nonempty set and 119889120572 119883 times

119883 rarr R120572sube R be a function such that for all 119909 119910 119911 isin 119883 the

following axioms hold

(NM1) 119889120572(119909 119910) =

0 if and only if 119909 = 119910(NM2) 119889

120572(119909 119910) = 119889

120572(119910 119909)

(NM3) 119889120572(119909 119910)

le 119889120572(119909 119911)

+ 119889120572(119911 119910)

Then the pair (119883 119889120572) and 119889

120572are called an 120572-metric space and

an 120572-metric on119883 respectively

Throughout this paper we define the 119901th 120572-exponent 119909119901120572and 119902th 120572-root 119909(1119902)120572 of 119909 isin R

120572sube R as

1199092120572

= 119909


minus1(119909) = 120572 [120572

minus1(119909)]

2

1199093120572

= 1199092120572

times 119909

= 120572 120572minus1120572 [120572minus1(119909) times 120572

minus1(119909)] times 120572

minus1(119909)

= 120572 [120572minus1(119909)]

3

119909119901120572

= 119909(119901minus1)

120572

times 119909 = 120572 [120572minus1(119909)]

119901

(7)

Hence 120572radic119909 = 119909(12)120572= 119910 provided there exists an 119910 isin R

120572sube R

such that 1199102120572= 119909 For each 120572-nonnegative number 119909 the

symbol 120572radic119909 will be used to denote 120572[120572minus1(119909)]12 which is theunique 120572-nonnegative number 119910 whose 120572-square is equal to119909 For each number 119909 isin R

120572 120572radic1199092120572 = |119909|

120572= 120572(|120572

minus1(119909)|)where

the absolute value |119909|120572of 119909 isin R

120572is defined by

|119909|120572=

119909 119909

gt

0

0 119909 =

0

minus 119909 119909

lt

0

(8)

Definition 2 (see [18]) Let119883 = (119883 119889120572) be an 120572-metric space

Then the basic notions can be defined as follows

(a) A sequence 119909 = (119909119896) is a function from the set N into

the setR120572 The 120572-real number 119909

119896denotes the value of

the function at 119896 isin N and is called the 119896th term of thesequence

(b) A sequence (119909119899) in 119883 = (119883 119889

120572) is said to be 120572-

convergent if for every given 120576

gt

0 (120576 isin R120572) there

Journal of Function Spaces 3

exist an 1198990 = 1198990(120576) isin N and 119909 isin 119883 such that119889120572(119909119899 119909) = |119909

119899

minus 119909|120572

lt 120576 for all 119899 gt 1198990 which isdenoted by 120572lim

119899rarrinfin119909119899= 119909 or 119909

119899

120572

997888rarr 119909 as 119899 rarr infin

(c) A sequence (119909119899) in119883 = (119883 119889

120572) is said to be 120572-Cauchy

if for every 120576

gt

0 there is an 1198990 = 1198990(120576) isin N such that119889120572(119909119899 119909119898)

lt 120576 for all119898 119899 gt 1198990

Following [12] we give a new type of calculus by usingthe notion of non-Newtonian complex numbers denoted by⋆-calculus (ldquostar-rdquo) which is a branch of non-Newtoniancalculus From now on we will use the notation ⋆-calculuscorresponding calculus which is based on two arbitrarilyselected generator functions

21 ⋆-Arithmetic (ldquoStarrdquo-Arithmetic) Suppose that 120572 and 120573

are two arbitrarily selected generators and (ldquostar-rdquo) also isthe ordered pair of arithmetics that is 120573-arith-metic and 120572-arithmetic The sets (R

120573

+

minus

times

) and (R120572

+

minus

times

)

are complete ordered fields (see [19]) and beta- (alpha-)generator generates beta- (alpha-) arithmetics respectivelyDefinitions given for 120573-arithmetic are also valid for 120572-arithmetic The important point to note here is that 120572-arithmetic is used for arguments and 120573-arithmetic is usedfor values in particular changes in arguments and values aremeasured by 120572-differences and 120573-differences respectively

Definition 3 (see [13]) The ⋆-limit of a function 119891 at anelement 119886 in R

120572is if it exists the unique number 119887 in R

120573

such that

⋆ lim119909rarr119886

119891 (119909) = 119887

lArrrArr forall120576

gt

0 exist120575

gt

0 ni

10038161003816100381610038161003816119891 (119909)

minus 119887

10038161003816100381610038161003816120573

lt 120576 forall119909 isin R120572 10038161003816100381610038161003816119909

minus 119886

10038161003816100381610038161003816120572

lt 120575

(9)

and is denoted by ⋆lim119909rarr119886

119891(119909) = 119887 Also we can give thedefinition for every sequence (119909

119899) of arguments of 119891 distinct

from 119886 if (119909119899) is 120572-convergent to 119886 then 119891(119909

119899) 120573-converges

to 119887

A function119891 is⋆-continuous at a point 119886 inR120572if and only

if 119886 is an argument of 119891 and ⋆lim119909rarr119886

119891(119909) = 119891(119886) When120572 and 120573 are the identity function 119868 the concepts of ⋆-limitand⋆-continuity are identical with those of classical limit andclassical continuity

The isomorphism from 120572-arithmetic to 120573-arithmetic isthe unique function 120580 (iota) that possesses the following threeproperties

(i) 120580 is one to one

(ii) 120580 is from R120572to R120573

(iii) For any numbers 119906 and V in R120572

120580 (119906

+ V) = 120580 (119906)

+ 120580 (V)

120580 (119906

minus V) = 120580 (119906)

minus 120580 (V)

120580 (119906

times V) = 120580 (119906)

times 120580 (V)

120580 (119906

V) = 120580 (119906)

120580 (V)

V =

0 119906 le V lArrrArr 120580 (119906)

le 120580 (V)

(10)

It turns out that 120580(119909) = 120573120572minus1(119909) for every 119909 in R

120572and that

120580(

119899 ) =

119899 for every integer 119899 Since for example 119906

+ V =

120580minus1120580(119906)

+ 120580(V) it should be clear that any statement in 120572-arithmetic can readily be transformed into a statement in 120573-arithmetic

22 Non-Newtonian Complex Field Let

119886 isin (R120572

+

minus

times

) and

119887 isin (R120573

+

minus

times

) be arbitrarily chosenelements from corresponding arithmetics Then the orderedpair (

119886

119887) is called a ⋆-point and the set of all ⋆-points iscalled the set of ⋆-complex numbers which is denoted byC⋆that is

C⋆

= 119911⋆

= (

119886

119887) |

119886 isinR120572subeR

119887 isinR120573subeR (11)

Define the binary operations addition (oplus) and multiplication(⊙) of ⋆-complex numbers 119911⋆

1= (

119886 1

119887 1) and 119911⋆

2= (

119886 2

119887 2)as

oplus C⋆ timesC⋆ 997888rarr C⋆

(119911⋆

1 119911⋆

2) 997891997888rarr 119911

⋆

1oplus 119911⋆

2= (120572 1198861 + 1198862 120573 1198871 + 1198872)

= (

119886 1

+

119886 2

119887 1

+

119887 2)

⊙ C⋆ timesC⋆ 997888rarr C⋆

(119911⋆

1 119911⋆

2) 997891997888rarr 119911

⋆

1⊙ 119911⋆

2

= (120572 11988611198862 minus 11988711198872 120573 11988611198872 + 11988711198862)

(12)

where

119886 1

119886 2 isin R120572and

119887 1

119887 2 isin R120573

Theorem 4 (see [19]) (C⋆ oplus ⊙) is a field

Following Grossman and Katz [12] we can give the def-inition of ⋆-distance regarding ⋆-calculus

Definition 5 (see [19]) The ⋆-distance 119889⋆ between two

arbitrarily elements 119911⋆1= (

119886 1

119887 1) and 119911⋆

2= (

119886 2

119887 2) of theset C⋆ is defined by

119889⋆ C⋆ timesC⋆ 997888rarr [

0 infin) = 1198611015840

sub R120573

(119911⋆

1 119911⋆

2) 997891997888rarr 119889

⋆

(119911⋆

1 119911⋆

2)

= (120580 (

119886 1

minus

119886 2)2120572

+ (

119887 1

minus

119887 2)2120573

)

(12)120573

= 120573radic(1198861 minus 1198862)2+ (1198871 minus 1198872)

2

(13)


Definition 6 (see [20]) Given a sequence (119911⋆119896) = (

119909119896

119910119896) of

⋆-complex numbers the formal notation

⋆

infin

sum

119896=0119911⋆

119896= 119911⋆

0 oplus 119911⋆

1 oplus 119911⋆

2 oplus sdot sdot sdot oplus 119911⋆

119896oplus sdot sdot sdot

= (120572

infin

sum

119896=0

119909119896120573

infin

sum

119896=0

119910119896)

= (120572

infin

sum

119896=0120572minus1119909119896 120573

infin

sum

119896=0120573minus1119909119896)

isin C⋆

(14)

for all 119896 isin N is called an infinite series with ⋆-complex termsor simply complex ⋆-series Also for integers 119899 isin N thefinite ⋆-sums 119904⋆

119899=⋆sum119899

119896=0 119911⋆

119896are called the partial sums of

complex ⋆-series If the sequence ⋆-converges to a complexnumber 119904⋆ then we say that the series ⋆-converges and write119904⋆

=⋆suminfin

119899=0 119911⋆

119899The number 119904⋆ is then called the⋆-sumof this

series If (119904119899) ⋆-diverges we say that the series ⋆-diverges or

that it is ⋆-divergent

Definition 7 (see [22]) Let119883 be a real or complex linear spaceand let

sdot

be a function from119883 to the setR+120573of nonnegative

120573-real numbersThen the pair (119883

sdot

) is called a ⋆-normedspace and

sdot

is a ⋆-norm for119883 if the following axioms aresatisfied for all elements 119909 119910 isin 119883 and for all scalars 120582

(NN1)

119909

=

0 hArr 119909 = 120579⋆ (120579⋆ = (

0

0 ))

(NN2)

120582 ⊙ 119909

=

| 120582

|

times

119909

(NN3)

119909 oplus 119910

le

119909

+

119910

It is trivial that a ⋆-norm

sdot

on 119883 defines a ⋆-metric 119889⋆

on119883 which is given by 119889⋆(119909 119910) =

119909 ⊖ 119910

(119909 119910 isin 119883) andis called the ⋆-metric induced by the ⋆-norm

Let 119911⋆ isin C⋆ be an arbitrary element The distancefunction 119889⋆(119911⋆ 120579⋆) is called ⋆-norm of 119911⋆ In other words

119911⋆

= 119889⋆

(119911⋆

120579⋆

) = (120580 (

119886

minus

0 )2120572

+ (

119887

minus

0 )2120573

)

(12)120573

= 120573 radic1198862+ 119887

2

(15)

where 119911⋆ = (

119886

119887 ) and 120579⋆ = (

0

0 )In particular in multiplicative calculus by taking 120572 = 119868

the identity function and 120573 = exp the exponential functionand the axioms of ⋆-normed space turn into

(N(MC)1)

119909

= 1 hArr 119909 = 120579⋆ (120579⋆ = (0 1))

(N(MC)2)

120582 ⊙ 119909

=

119909

|120582|

(N(MC)3)

119909 oplus 119910

le

119909

119910

Then we say that (119883

sdot

) is multiplicative normed space

Definition 8 (see [21]) Let 119911⋆ = (

119886

119887 ) isin C⋆ We define the⋆-complex conjugate 119911⋆ of 119911⋆ by 119911⋆ = (120572119886 120573minus120573

minus1(

119887 )) =

(

119886

minus

119887 ) Conjugation changes the sign of the imaginarypart of 119911⋆ but leaves the real part the same Thus

Re (119911⋆) = Re (119911⋆) = (119911⋆

oplus 119911⋆

)

2 =

119886

Im (119911⋆

) =

minus Im (119911⋆

) = (119911⋆

⊖ 119911⋆

)

2 =

119887

(16)

Remark 9 (see [21]) The following conditions hold

(i) Let 119911⋆1 = (

119886 1

119887 1) 119911⋆

2 = (

119886 2

119887 2) isin C⋆ We can givethe ⋆-division of two ⋆-complex numbers 119911⋆1 and 119911⋆2as

119911⋆

1 ⊘ 119911⋆

2

= (120572

(11988611198862 + 11988711198872)

(1198862

2+ 1198872

2)

120573

(11988711198862 minus 11988611198872)

(1198862

2+ 1198872

2)

)

(17)

(ii) Let 120572 and 120573 be the same generators and let 119911⋆ =

(

119886

119887 ) isin C⋆ Then the relation 119911⋆

⊙ 119911⋆=

119911⋆

2120573

holds Really

119911⋆

⊙ 119911⋆= (

119886

119887 ) ⊙ (

119886

minus

119887 ) = (120572 1198862+ 119887

2 120573 (0))

= 120573 1198862+ 119887

2 = 120573 (120573

minus1120573radic1198862+ 119887

2)

2

=

119911⋆

2120573

(18)

Theorem 10 (see [19]) (C⋆ 119889⋆) is a complete metric spacewhere 119889⋆ is defined by (13)

Corollary 11 (see [19]) C⋆ is a Banach space with the ⋆-norm

sdot

defined by

119911⋆

= (120580(

119886 )2120572

+

119887

2120573

)(12)120573 119911⋆ = (

119886

119887 ) isin

C⋆

Following Tekin and Basar [19] we can give someexamples of ⋆-normed sequence spaces First consider thefollowing relationswhich are derived from the correspondingmetrics given in (13) by putting as usual

119911⋆

= 119889⋆

(119911⋆

120579⋆

)

Theorem 12 (see [19]) The following statements hold

(a) The spaces ℓ⋆infin 119888⋆ and 119888⋆0 are Banach spaces with the

norm sdot ⋆

infindefined by

119911⋆

infin= sup119896isinN

119911⋆

119896

119911 = (119911⋆

119896) isin 120582⋆

120582 isin ℓinfin 119888 1198880 (19)

(b) The space ℓ⋆119901is Banach spaces with the norm sdot

⋆

119901

defined by

119911⋆

119901= (⋆sum

119896

119911⋆

119896

119901120573

)

(1119901)120573

119901 ge 1 119911 = (119911⋆

119896) isin ℓ⋆

119901 (20)


Theorem 13 (see [20]) (a) The spaces 119887119904⋆ 119888119904⋆ and 119888119904⋆

0 areBanach spaces with the norm sdot

⋆

119887119904defined by

119909⋆

119887119904= 119909

⋆

119888119904= sup119899isinN

⋆

119899

sum

119896=0119909119896

119909 = (119909119896) isin 120583⋆

120583 isin 119887119904 119888119904 1198881199040

(21)

(b) The spaces 119887V⋆ 119887V⋆119901(119901 ge 1) and 119887V⋆

infinare Banach

spaces with the corresponding norms defined by

119909⋆

119887V = ⋆sum119896

(Δ1015840

119909)119896

119909⋆

119887V119901

= (⋆sum

119896

(Δ119909)119896

119901120573

)

(1119901)120573

119909⋆

119887Vinfin

= sup119896isinN

(Δ119909)119896

(22)

where (Δ1015840119909)119896= (119909119896⊖119909119896+1) and (Δ119909)119896 = (119909

119896⊖119909119896minus1) 119909minus1 = 120579

⋆

for all 119896 isin N

Analogous to classical analysis a sequence space 120583⋆ witha linear ⋆-metric topology (cf [19]) is called a ⋆119870-spaceprovided that each of the maps 119901

119894 120583⋆

rarr C⋆ defined by119901119894(119909) = 119909

119894is ⋆-continuous by (9) for all 119894 isin N Additionally

a ⋆119870-space 120583⋆ is called an ⋆FK-space provided that 120583⋆ isa complete linear non-Newtonian metric space denoted by⋆-linear (see [20]) An ⋆FK-space whose non-Newtoniantopology is normable and is called a ⋆BK-space

3 Some Inequalities and Inclusion Relations

Definition 14 (Schauder basis) If a ⋆-normed sequence space120582⋆ contains a sequence (119887

119899) with the property that for every

119909 isin 120582⋆ there is a unique sequence of scalars (120585

119899) such that

⋆ lim119899rarrinfin

1003817100381710038171003817119909 ⊖ (1205850 ⊙ 1198870 oplus 1205851 ⊙ 1198871 oplus sdot sdot sdot oplus 120585119899 ⊙ 119887119899)

1003817100381710038171003817

⋆

= 120579⋆ (23)

with corresponding norm then (119887119899) is called a Schauder basis

(in non-Newtonian sense) briefly ⋆-basis for 120582⋆ The series⋆sum119896120585119896⊙ 119887119896which has the sum 119909 is then called the expansion

of 119909with respect to (119887119899) and is written as 119909 =

⋆sum119896120585119896⊙119887119896The

concepts of Schauder and algebraic⋆-bases coincide for finitedimensional spaces Nevertheless there are ⋆-linear spaceswithout a Schauder ⋆-basis

Let 119890 = (119890119896) and 119890(119899) = (119890

(119899)

119896) (119899 isin N) be the sequences

with 119890119896= 1⋆ for all 119896 isin N and 119890(119899)

119896= 120575⋆

119899119896 where 120575⋆

119899119896denotes

the non-Newtonian Kronecker delta defined by

120575⋆

119899119896=

1⋆ 119899 = 119896

120579⋆

119899 = 119896

(24)

Example 15 The sequence 119890 119890(0) 119890(1) 119896isinN is a Schauder

⋆-basis for the space 119888⋆ and any 119909 = (119909119896) in 119888⋆ has a unique

representation of the form

119909 = 120585 ⊙ 119890 oplus⋆sum

119896

(119909119896⊖ 120585) ⊙ 119890

(119896)

where ⋆ lim119896rarrinfin

119909119896= 120585

(25)

Theorem 16 The space 119887119904⋆ is norm isomorphic to the spaceℓ⋆

infin that is 119887119904⋆ cong ℓ

⋆

infin

Proof To prove this we should show the existence of a ⋆-norm preserving linear bijection between the spaces 119887119904⋆ andℓ⋆

infinConsider the transformation119879 defined from 119887119904

⋆ to ℓ⋆infinby

119879119909 = (⋆sum119896

119895=0 119909119895) By using the corresponding operations oplusand ⊙ the ⋆-linearity of 119879 is obvious Further it is trivial that119909 = 120579

⋆ whenever 119879119909 = 120579⋆ and hence 119879 is injective Let 119910 =

(119910119896) isin ℓ⋆

infinand define the sequence 119909 = (119909

119896) by 119909

119896= 119910119896⊖119910119896minus1

for all 119896 isin N with 119910minus1 = 120579

⋆ Then we obtain that

sup119896isinN

⋆

119896

sum

119895=0119909119895

= sup119896isinN

⋆

119896

sum

119895=0(119910119895⊖119910119895minus1)

= sup119896isinN

119910119896

=

1003817100381710038171003817119910

1003817100381710038171003817

⋆

infinlt infin

(26)

Thus we observe that 119909⋆119887119904

lt infin and hence 119909 isin 119887119904⋆

Consequently 119879 is surjective and is norm preserving Hence119879 is a linear bijection which therefore says that the spaces 119887119904⋆and ℓ⋆infin

are norm isomorphic as desired

Theorem 17 Then the following relations are satisfied

(i) 120583 sube 120583⋆ holds for each 120583 isin ℓ

infin 119888 1198880 ℓ119901 119887119904 119888119904 1198881199040

119887V 119887Vinfin

(ii) ℓ⋆1 sube 119888119904⋆

sube 119888⋆

0 sube 119888⋆

sube ℓ⋆

infinsube 120596⋆ and ℓ⋆1 sube 119887V⋆0 sube 119887V⋆ sube

119888⋆ where 119887V⋆0 = 119887V⋆ cap 119888⋆0

(iii) If the inverse function 120573minus1 is bounded in classical meanthen 1198880 sube 119888

⋆

0 sube 119888 sube 119888⋆

sube ℓinfinsube ℓ⋆

infinsube 120596 holds

Proof Since the proof is trivial for the conditions (i) and (ii)we prove only (iii)

(iii) Using (i) and (ii) we need only to show ℓ⋆

infinsube 120596 119888⋆ sube

ℓinfin and 119888⋆0 sube 119888 Now consider 119911 = (119911

119896) isin 119888⋆ is givenThen for

every 120576

gt

0 there exist an 1198990 = 1198990(120576) isin N and 119897 isin C⋆ such that119889⋆

(119911119896 ℓ)

lt 120576 for all 119899 gt 1198990 Since 120573minus1 is a bounded function

there exists an element 119872 gt 0 such that |120573minus1(119909)| lt 119872 forall 119909 isin R On the other hand by applying the well-knowninequality

119911119896

le

119911119896⊖ ℓ

+

ℓ

le 120598

+

ℓ

(27)

which implies that |119911119896| = 120573

minus1 119911119896

le 120573minus1(120598

+

ℓ

)Therefore by taking into account the boundedness of 120573minus1

there exists 1198720 gt 0 such that |120573minus1(120598

+

ℓ

)| lt 1198720 weobtain that (119911

119896) is bounded in classical mean Thus 119911 isin

ℓinfin Hence 119888⋆ sube ℓ

infin The remaining part can be obtained

similarly


Corollary 18 The spaces ℓ⋆infin 119888⋆ 119888⋆0 119887119904

⋆ 119888119904⋆ 119887V⋆119901 and ℓ⋆

119901are

⋆-norm isomorphic to the spaces ℓinfin 119888 1198880 119887119904 119888119904 119887V119901 and ℓ119901

respectively

Now we give some well-known inequalities in the non-Newtonian sense which are essential in the study

Lemma 19 (Youngrsquos inequality) Let 119901 and 119902 be conjugate realnumbers Then

119906

times V

le

119906119901120572

119901

+

V119902120572

119902

(28)

holds for all 119906 V isin R+120572and 119901 gt 1

Proof For any generator function 120572 we must show that thefollowing inequality holds

120572minus1(119906) 120572minus1(V) le

(120572minus1(119906))

119901

119901

+

(120572minus1(V))119902

119902

(29)

It is trivial that (29) holds for 119906 =

0 or V =

0 Let 119906 V benonzero 120572-real numbers Consider the function 119891 [0infin] sube

R120572rarr R120573sube R defined by

119891 (119905) = (120580 (119905))120582120573

minus

120582

times 120580 (119905) = 120573 (120572minus1(119905))

120582

minus120582120572minus1(119905) (30)

where 120580 = 120573 ∘ 120572minus1 and 0 lt 120582 lt 1 Then the ⋆-derivative of 119891

(see [23]) can be written as

119891⋆

(119905) = 120573

(120573minus1∘ 119891)

1015840

(119905)

(120572minus1)

1015840

(119905)

= 120573

((120572minus1)

120582

minus 120582120572minus1)

1015840

(119905)

(120572minus1)

1015840

(119905)

= 120573120582 (120572minus1(119905))

120582minus1minus120582

(31)

From the first derivative test in non-Newtonian sense thecondition 119891

⋆

(119905) =

0 holds and 119905 =

1 is a critial point of119891 Besides this

119891⋆⋆

(119905) = 120573

(120573minus1119891⋆

(119905))

1015840

(120572minus1)

1015840

(119905)

= 120573120582 (120582 minus 1) (120572minus1 (119905))120582

(32)

and by using the second derivative test in non-Newtoniansense we have 119891⋆⋆(

1 ) = 120573120582(120582 minus 1)

lt

0 which implies that119891 has a maximum at

1 that is 119891(

1 ) = 1205731 minus 120582 In otherwords we say that

[120572minus1(119905)]

120582

minus120582120572minus1(119905) le 1minus120582 forall119905 isin [0infin] sube R

120572 (33)

Now taking 119905 = (119906)119901120572

(V)119902120572 = 120572(120572minus1(119906))119901

(120572minus1(V))119902 and

120582 = 1119901 in (33) we get

(

(120572minus1(119906))

119901

(120572minus1(V))119902

)

1119901

minus

1119901

(120572minus1(119906))

119901

(120572minus1(V))119902

le 1minus 1119901

120572minus1(119906)

120572minus1(V)119902119901

(120572minus1(V))119902

le (1minus 1119901

+

1119901

120572minus1(119906)119901

120572minus1(V)119902

)(120572minus1(V))119902

(34)

Hence the inclusion (29) holdsThis step completes the proof

Theorem 20 (Holderrsquos inequality) Let 119901 and 119902 be conjugatepositive real numbers and 119906⋆

119896 V⋆119896isin C⋆ for 119896 isin 0 1 2 119899

Then the following inequality holds

120573

119899

sum

119896=0

119906⋆

119896⊙ V⋆119896

le (120573

119899

sum

119896=0

119906⋆

119896

119901120573

)

(1119901)120573

times (120573

119899

sum

119896=0

V⋆119896

119902120573

)

(1119902)120573

(35)

Proof The inequality clearly holds when 119906 = 120579⋆

= (

0

0 ) orV = (

0

0 ) We may assume 119906 V = 120579⋆ in the following proof

Let

120576 = (120573

119899

sum

119896=0

119906⋆

119896

119901120573

)

(1119901)120573

120575 = (120573

119899

sum

119896=0

V⋆119896

119902120573

)

(1119902)120573

(36)

and 119904⋆119896= 119906⋆

119896⊘ 120576 119905⋆119896= V⋆119896⊘ 120575 where 120576 = (

0

120576 ) isin R120573sube C⋆

and 120575 = (

0

120575 ) isin R120573

sube C⋆ By taking into accountLemma 19 for each 119896 isin 0 1 2 119899 we obtain

119904⋆

119896⊙ 119905⋆

119896

=

119904⋆

119896

times

119905⋆

119896

le

119904⋆

119896

119901120573

119901

+

119905⋆

119896

119902120573

119902

(37)

which implies that

120573

119899

sum

119896=0

119904⋆

119896⊙ 119905⋆

119896

le120573

119899

sum

119896=0

119904⋆

119896

119901120573

119901

+120573

119899

sum

119896=0

119905⋆

119896

119902120573

119902

(38)

Then as is easy to see

120573

119899

sum

119896=0

(119906⋆

119896⊘ 120576) ⊙ (V⋆

119896⊘ 120575)

le120573

119899

sum

119896=0

119906⋆

119896

119901120573

120576

times

119901

+120573

119899

sum

119896=0

V⋆119896

119902120573

120575

times

119902

=

1

119901

+

1

119902

=

1

(39)

Therefore we deduce by combining this with the inclusion(39) that (35) holds for every 119896 isin 0 1 2 119899


In particular for 119901 = 2 the inequality (35) turns out to be

120573

119899

sum

119896=0

119906⋆

119896⊙ V⋆119896

le (120573

119899

sum

119896=0

119906⋆

119896

2120573

times120573

119899

sum

119896=0

V⋆119896

2120573

)

(12)120573

(40)

denoted by Cauchy-Schwartz inequality in non-Newtoniansense

Theorem21 (Minkowskirsquos inequality) Let 119901 ge 1 and 119906⋆119896 V⋆119896isin

C⋆ for all 119896 isin 0 1 2 119899 Then

(120573

119899

sum

119896=0

119906⋆

119896oplus V⋆119896

119901120573

)

(1119901)120573

le (120573

119899

sum

119896=0

119906⋆

119896

119901120573

)

(1119901)120573

+ (120573

119899

sum

119896=0

V⋆119896

119901120573

)

(1119901)120573

(41)

Proof The case 119901 = 1 is trivial Let 119901 gt 1 and 119906⋆119896 V⋆119896isin C⋆

One can immediately conclude that

120573

119899

sum

119896=0

119906⋆

119896oplus V⋆119896

119901120573

le120573

119899

sum

119896=0

119906⋆

119896

times

119906⋆

119896oplus V⋆119896

(119901minus1)120573

+120573

119899

sum

119896=0

V⋆119896

times

119906⋆

119896oplus V⋆119896

(119901minus1)120573

(42)

This leads us withTheorem 20 to the consequence that

120573

119899

sum

119896=0

119906⋆

119896oplus V⋆119896

119901120573

le[

[

(120573

119899

sum

119896=0

119906⋆

119896

119901120573

)

(1119901)120573

+ (120573

119899

sum

119896=0

V⋆119896

119901120573

)

(1119901)120573

]

]

times (120573

119899

sum

119896=0

119906⋆

119896oplus V⋆119896

(119902119901minus119902)120573

)

(1119902)120573

(43)

This concludes the proof

4 Non-Newtonian ParanormedSequence Spaces

Firstly we give the definition of non-Newtonian paranormbriefly ⋆-paranorm

Definition 22 Let 119883 be a real or complex ⋆-linear space andlet 119892⋆ be a subadditive function from 119883 to the subset R+

120573sube

R+Then the pair (119883 119892⋆) is called a ⋆-paranormed space and119892⋆ is a⋆-paranorm for119883 if the following axioms are satisfied

for all elements 119909 119910 isin 119883 and for all scalars 120582

(N(PN)1) 119892⋆

(119909) =

0 if 119909 = 120579⋆ (120579⋆ = (0⋆ 0⋆ ))

(N(PN)2) 119892⋆

(⊖119909) = 119892⋆

(119909) (⊖119909 is opposite⋆-vectorof 119909)(N(PN)3) 119892

⋆

(119909 oplus 119910)

le 119892⋆

(119909)

+ 119892⋆

(119910)

(N(PN)4) If (120582⋆119899) is a sequence of complex scalars

that is 120582⋆ = (

120582

120582 ) with 120582⋆

119899

119889⋆

997888997888rarr 120582⋆ as 119899 rarr infin

and 119909119899 119909 isin 119883 for all 119899 isin N with 119909

119899

119892⋆

997888997888rarr 119909 then

120582⋆

119899⊙ 119909119899

119892⋆

997888997888rarr 120582⋆

⊙ 119909 as 119899 rarr infin

In particular in bigeometric calculus case that is120572 = 120573 =

exp the conditions (N(PN)1) (N(PN)2) and (N(PN)4) alsohold with zero⋆-vector 120579⋆ = ((1 1) (1 1) ) and (N(PN)3)turns into

(BG(PN)3) 119892⋆

(119909 oplus 119910) le 119892⋆

(119909)119892⋆

(119910)

Assume hereafter that 119901 = (119901119896) is a bounded sequence of

strictly positive real numbers so that 0 lt 119901119896le sup119901

119896=

119867 lt infin and 119872 = max1 119867 We will assume throughoutthat 119901

119896times 1199011015840

119896= 119901119896+ 1199011015840

119896provided that 1 lt inf 119901

119896le 119867 lt infin for

all 119896 isin NQuite recently Tekin and Basar [19] have introduced

the sets ℓ⋆infin 119888⋆

119888⋆

0 and ℓ⋆

119901of sequences over the complex

field C⋆ which correspond to the sets ℓinfin 119888 1198880 and ℓ

119901over

the complex field C respectively It is natural to expectthat the Banach spaces ℓ⋆

infin 119888⋆

119888⋆

0 and ℓ⋆

119901can be extended

to the complete ⋆-paranormed sequence spaces so as theMaddoxrsquos spaces are derived on the real or complex field fromthe classical sequence spaces Now we may give the spacesℓ⋆

infin(119901) 119888⋆

(119901) 119888⋆

0 (119901) ℓ⋆

(119901) 119887119904⋆

(119901) 119888119904⋆

(119901) 119888119904⋆

0 (119901) 119887V⋆

(119901) and119887V⋆infin(119901) in non-Newtonian sense which correspond to the

well-known examples of the paranormed sequence spaces in(CC)

ℓ⋆

infin(119901) = 119909= (119909

119896) isin 120596⋆

sup119896isinN

119909119896

(119901119896)120573

ltinfin

= 119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆

sup119896isinN

120573radic1205762

119896+ 120575

2119896

119901119896

ltinfin

119888⋆

(119901) = 119909= (119909119896) isin 120596⋆

exist 119897 isinC⋆

ni⋆ lim119896rarrinfin

119909119896⊖ 119897

(119901119896)120573

= 120579⋆

= 119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆

exist 119897 = (

ℓ

ℓ ) isinC⋆

ni120573 lim119896rarrinfin

120573(ℓradic2 (1205762119896+ 120575

2119896))

119901119896

=

0

119888⋆

0 (119901) = 119909= (119909119896) isin 120596⋆


119909119896

(119901119896)120573

= 120579⋆

= 119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆

120573 lim119896rarrinfin

120573radic1205762

119896+ 120575

2119896

119901119896

ltinfin


ℓ⋆

(119901) = 119909= (119909119896) isin 120596⋆

120573sum

119896

119909119896

(119901119896)120573

ltinfin

(0 lt 119901119896lt infin)

= 119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆

120573sum

119896

radic1205762

119896+ 120575

2119896

119901119896

ltinfin

(44)

Following Kadak [20] we define the several sets 119887119904⋆(119901)119888119904⋆

(119901) 119888119904⋆

0 (119901) 119887V⋆

(119901) and 119887V⋆infin(119901) of sequences in the sense

of non-Newtonian calculus as follows

119887119904⋆

(119901) =

119909= (119909119896) isin 120596⋆

(⋆

119896

sum

119895=0119909119895)isinℓ⋆

infin(119901)

=

119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆

sup119896isinN

120573

radic(

119896

sum

119895=0120576119895)

2

+ (

119896

sum

119895=0120575119895)

2119901119896

ltinfin

119888119904⋆

(119901) =

119909= (119909119896) isin 120596⋆

(⋆

119896

sum

119895=0119909119895)isin 119888⋆

(119901)

=

119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆


120573

radic(

119896

sum

119895=0120576119895minus ℓ)

2

+ (

119896

sum

119895=0120575119895minus ℓ)

2119901119896

ltinfin

(ℓ = (

ℓ

ℓ ) isin C⋆

)

119888119904⋆

0 (119901) = 119909= (119909119896) isin 120596⋆

(⋆

119899

sum

119896=0119909119896)isin 119888⋆

0 (119901)

119887V⋆ (119901) = 119909= (119909119896) isin 120596⋆

120573

infin

sum

119896=0

Δ119909119896

(119901119896)120573

ltinfin

(Δ119909119896= 119909119896⊖ 119909119896minus1 119909minus1 = 120579

⋆

)

= 119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆

120573

infin

sum

119896=0

radic(120576119896minus 120576119896minus1)

2+ (120575119896minus 120575119896minus1)

2119901119896

ltinfin

119887V⋆infin(119901) = 119909= (119909

119896) isin 120596⋆

sup119896isinN

Δ119909119896

(119901119896)120573

ltinfin

(45)

It is a routine verification that each of the sets ℓ⋆infin(119901) 119888⋆

(119901)

119888⋆

0 (119901) ℓ⋆

(119901) 119887119904⋆

(119901) 119888119904⋆

(119901) 119888119904⋆

0 (119901) 119887V⋆

(119901) and 119887V⋆infin(119901) is a

⋆-linear space

Theorem 23 The following statements hold

(i) Define the functions 119892⋆ and 119892⋆ by

119892⋆

(119909) = sup119896isinN

119909119896

(119901119896119872)120573

119892⋆

(119909) = (120573sum

119896

119909119896

(119901119896)120573

)

(1119872)120573

(46)

Then 119888⋆

(119901) and 119888⋆

0 (119901) are complete ⋆-paranormedspaces by 119892⋆ if 119901

119896isin ℓ⋆

infin Also the spaces ℓ⋆

infin(119901) and

ℓ⋆

(119901) are complete ⋆-paranormed spaces paranormedby 119892⋆ and 119892⋆ respectively if and only if inf119901

119896gt 0

(ii) The sets 119887119904⋆(119901) 119888119904⋆(119901) and 119888119904⋆0 (119901) of sequences are thecomplete ⋆-paranormed spaces paranormed by 119892⋆1 by

119892⋆

1 (119909) = sup119896isinN

⋆

119896

sum

119895=0119909119895

(119901119896119872)120573

iff inf 119901119896gt 0 (47)

(iii) The sets 119887V⋆(119901) and 119887V⋆infin(119901) are the complete ⋆-

paranormed spaces by 119892⋆2 and 119892⋆3 defined by

119892⋆

2 (119909) = (120573sum

119896

Δ119909119896

(119901119896)120573

)

(1119872)120573

119892⋆

3 (119909) = sup119896isinN

Δ119909119896

(1119872)120573

iff inf 119901119896gt 0

(48)

respectively where Δ119909119896= 119909119896⊖ 119909119896minus1 119909minus1 = 120579

⋆ for all119896 isin N

Proof To avoid repetition of similar statements we give theproof only for the space 119887V⋆(119901) in case (iii) The remainingparts can be obtained similarly

The ⋆-linearity of 119887V⋆(119901) with respect to coordinatewiseaddition and scalar multiplication follows from the following


inequalities which are satisfied for 119906 119909 isin 119887V⋆(119901) (seeTheorem 21)

(⋆sum

119896

Δ (119906119896oplus119909119896)

(119901119896)120573

)

(1119872)120573

le (⋆sum

119896

Δ119906119896

(119901119896)120573

)

(1119872)120573

+ (⋆sum

119896

Δ119909119896

(119901119896)120573

)

(1119872)120573

(49)

and the condition

120582⋆

(119901119896)120573

le max

1

120582⋆

119872120573

(50)

holds for any scalar 120582⋆

= (

120582

120582 ) isin C⋆ (cf [3]) Itis clear that 119892⋆(120579⋆) =

0 and 119892⋆

(⊖119909) = 119892⋆

(119909) for all119909 isin 119887V⋆(119901) Hence by combining the inclusions (49) and(50) with subadditivity of 119892⋆ we get the inequality 119892⋆(120582⋆ ⊙119909)

lemax

1

120582⋆

times 119892⋆

(119909)Let (119909119899) be any sequence of the points of the space

119887V⋆(119901) such that 119892⋆(119909119899 ⊖ 119909) rarr 120579⋆ and let (120582

119899) be any

sequence of ⋆-complex scalars such that 120582⋆119899

rarr 120582⋆ with

corresponding⋆-metricThen since the⋆-triangle inequality119892⋆

(119909119899

)

le 119892⋆

(119909)

+ 119892⋆

(119909119899

⊖ 119909) holds the sequence 119892⋆(119909119899) is120573-bounded and we thus have

119892⋆

(120582⋆

119899⊙119909119899

⊖120582⋆

⊙119909)

= (120573sum

119896

Δ (120582⋆

119899⊙119909(119899)

119896⊖120582⋆

⊙119909119896)

(119901119896)120573

)

(1119872)120573

le

120582⋆

119899⊖120582⋆

times 119892⋆

(119909119899

)

+

120582⋆

times 119892⋆

(119909119899

⊖119909)

(51)

which tends to

0 as 119899 rarr infin That is to say that the scalarmultiplication is ⋆-continuous Hence 119892⋆ is a ⋆-paranormon the space 119887V⋆(119901)

It remains to prove the ⋆-completeness of the space119887V⋆(119901) Let 119909119894 be any Cauchy sequence in the space 119887V⋆(119901)where 119909

119894= 119909(119894)

0 119909(119894)

1 119909(119894)

2 Then for a given 120598

gt

0 thereexists a positive integer 1198990(120598) such that

119892⋆

(119909119894

⊖119909119895

)

lt

120598 forall119894 119895 ge 1198990 (120598) (52)

By taking into account the definition of 119892⋆ for each fixed 119896 isinN we have that

Δ [(119909119894

)119896

⊖ (119909119895

)119896

]

le (120573sum

119896

Δ [(119909119894

)119896

⊖ (119909119895

)119896

]

(119901119896)120573

)

(1119872)120573

lt 120598

forall119894 119895 ge 1198990 (120598)

(53)

which leads us to the fact that (Δ1199090)119896 (Δ119909

1)119896 (Δ119909

2)119896 is a

Cauchy sequence for every fixed 119896 isin N Since C⋆ is complete(see [19]) it ⋆-converges that is (Δ119909119894)

119896rarr 119909119896as 119894 rarr infin

Using these infinitely many limits 1199090 1199091 1199092 we define thesequence 1199090 1199091 1199092 From the inclusion (52) for each119898 isin N and 119894 119895 ge 1198990(120598) we have

120573

119898

sum

119896=0

Δ [(119909119894

)119896

⊖ (119909119895

)119896

]

(119901119896)120573

le 119892⋆

(119909119894

⊖119909119895

)

119872120573

lt 120598119872120573

(54)

Take any 119894 ge 1198990(120598) First let 119895 rarr infin in (54) and then119898 rarr infin to obtain 119892⋆(119909119894 ⊖ 119909)

le 120598 We have by Minkowskirsquosinequality for each119898 isin N that

(120573sum

119896

(Δ119909)119896

(119901119896)120573

)

(1119872)120573

le 119892⋆

(119909119894

⊖119909)

+ 119892⋆

(119909119894

)

le 120598

+ 119892⋆

(119909119894

)

(55)

which implies that 119909 isin 119887V⋆(119901) Since 119892⋆(119909119894 ⊖ 119909)

le 120598 for all119894 ge 1198990(120598) it follows that 119909

119894

rarr 119909 as 119894 rarr infin Therefore wehave shown that 119887V⋆(119901) is ⋆-complete

It is trivial to show that the ⋆-paranormed spacesℓ⋆

infin(119901) 119888⋆

(119901) 119888⋆

0 (119901) ℓ⋆

(119901) 119887119904⋆

(119901) 119888119904⋆

(119901) 119888119904⋆

0 (119901) 119887V⋆

(119901) and119887V⋆infin(119901)may be reduced to some new sequence spaces in the

special cases of the sequences (119901119896) and generator functions

For instance the sequence space 119887V⋆(119901) corresponds in thecase 119901

119896= 119901 for all 119896 isin N to the sequence space 119887V⋆

119901of 119901-

bounded variation sequences in [20]Now as a consequence of Corollary 18 the following

corollary presents the relations between ⋆-paranormed andclassic paranormed spaces

Corollary 24 The following statements hold

(i) The space 120583⋆

(119901) is norm isomorphic to the usualparanormed space 120583(119901) where 120583 isin ℓ

infin 119888 1198880 ℓ119901 119887119904 119888119904

1198881199040 119887V 119887Vinfin(ii) 120583(119901) sube 120583

⋆

(119901) and ℓ⋆1 (119901) sube 119888119904⋆

(119901) sube 119888⋆

0 (119901) sube 119888⋆

(119901) sube

ℓ⋆

infin(119901) sube 120596

⋆

41 Duality Properties Following [24] we give the alpha-beta- and gamma-duals of a ⋆-paranormed sequence space120582⋆

(119901) sub 120596⋆ which are respectively denoted by 120582

⋆

(119901)120572

120582⋆

(119901)120573 and 120582⋆(119901)120574 as follows

120582⋆

(119901)

120572

= 119908= (119908119896) isin 120596⋆

119908 ⊙ 119911 = (119908119896⊙ 119911119896)

isin ℓ⋆

1 (119901) forall119911 = (119911119896) isin 120582⋆

(119901)

120582⋆

(119901)

120573

= 119908= (119908119896) isin 120596⋆

119908 ⊙ 119911 = (119908119896⊙ 119911119896)

isin 119888119904⋆

(119901) forall119911 = (119911119896) isin 120582⋆

(119901)

120582⋆

(119901)

120574

= 119908= (119908119896) isin 120596⋆

119908 ⊙ 119911 = (119908119896⊙ 119911119896)

isin 119887119904⋆

(119901) forall119911 = (119911119896) isin 120582⋆

(119901)

(56)


where (119908119896⊙ 119911119896) is the coordinatewise product of ⋆-complex

numbers 119908 and 119911 for all 119896 isin N Throughout the text wealso use the notation ldquoltrdquo for a ⋆-linear subspace which wascreated in [18]

Theorem 25 Let 0 = 120582⋆

(119901) sub 120596⋆ Then the following

statements are valid

(a) 120582⋆(119901)120573 is a sequence space if 120582⋆(119901)120573 sub 120596⋆

(b) If 120582⋆(119901) sub 120583⋆

(119901) sub 120596⋆ then 120583⋆(119901)120573 sub 120582

⋆

(119901)120573

(c) 120582⋆ sub 120582⋆

120573120573

= (120582⋆

120573

)120573

(d) 119888⋆0 (119901)120573

= 119888⋆

(119901)120573

= ℓ⋆

infin(119901)120573

= ℓ⋆

1 (119901)

(e) ℓ⋆1 (119901)120573

= ℓ⋆

infin(119901)

Proof Since the proofs are trivial for the conditions (b) and(c) we prove only (a) (d) and (e) Let 119908 = (119908

119896) 119898 = (119898

119896)

and 119899 = (119899119896) isin 120582

⋆

(119901)120573

(a) It is trivial that 120582⋆(119901)120573 sub 120596⋆ holds from the

hypothesis We show that119898oplus119899 isin 120582⋆

(119901)120573 for119898 119899 isin

120582⋆

(119901)120573 Suppose that 119897 isin 120582

⋆

(119901) Then (119898119896⊙ 119897119896) isin

119888119904⋆

(119901) and (119899119896⊙ 119897119896) isin 119888119904⋆

(119901) for all 119897 isin 120582⋆(119901) We candeduce that

((119898119896oplus 119899119896) ⊙ 119897119896) = (119898

119896⊙ 119897119896) oplus (119899119896⊙ 119897119896) isin 119888119904

⋆

(119901)

forall119897 isin 120582⋆

(119901)

(57)

Hence 119898 oplus 119899 isin 120582⋆

(119901)120573 Now we show that 119905 ⊙ 119908 isin

120582⋆

(119901)120573 for any 119905 isin C⋆ and 119908 = (119908

119896) isin 120582

⋆

(119901)120573

Since (119908119896⊙ 119897119896) isin 119888119904

⋆

(119901) for all 119897 isin 120582⋆

(119901) andcombining this with ((119905

119896⊙119908119896) ⊙ 119897119896) = 119905119896⊙ (119908119896⊙ 119897119896) isin

119888119904⋆

(119901) for all 119897 isin 120582⋆

(119901) we get 119905 ⊙ 119908 isin 120582⋆

(119901)120573

Therefore we have proved that 120582⋆(119901)120573 is a subspaceof the space 120596⋆

(d) Obviously ℓ⋆

infin(119901)120573

sub 119888⋆

(119901)120573

sub 119888⋆

0 (119901)120573 by

Theorem 25(b) Then we must show that ℓ⋆1 (119901) sub

ℓ⋆

infin(119901)120573 and 119888

⋆

0 (119901)120573

sub ℓ⋆

1 Now consider 119908 =

(119908119896) isin ℓ

⋆

1 (119901) and 119911 = (119911119896) isin ℓ

⋆

infin(119901) are given By

taking into account the cases ((a)-(b)) ofTheorem 12we have

⋆sum

119896

1003817100381710038171003817119908119896⊙ 119911119896

1003817100381710038171003817

(119901119896)120573

le sup119896isinN

119911119896

(1119872)120573

times (120573sum

119896

119908119896

(119901119896)120573

)

(1119872)120573

le 119892⋆

(119911)

times 119892⋆

(119908) lt infin

(58)

where sup119901119896le 1 which implies that 119908 ⊙ 119911 isin 119888119904

⋆

(119901)So the condition ℓ⋆1 (119901) sub ℓ

⋆

infin(119901)120573 holds

Conversely for a given 119910 = (119910119896) isin 120596⋆

ℓ⋆

1 (119901)we provethe existence of an 119909 isin 119888

⋆

0 (119901) with 119910 ⊙ 119909 notin 119888119904⋆

(119901)According to 119910 notin ℓ

⋆

1 (119901) we can choose an index

sequence (119899119903) which is strictly increasing with 1198990 = 0

and⋆sum

119899119903minus1119896=119899119903minus1

119910119896

(119901119896)120573

gt 119903 (119903 isin N) By taking intoaccount Remark 9(i) we define 119909 = (119909

119896) isin 119888

⋆

0 (119901)by 119909119896= (sgn⋆119910

119896⊘ 119903) where the ⋆-complex signum

function is defined by

sgn⋆ (119910) =

119910 ⊘

119910

119910 = 120579⋆

120579⋆

119910 = 120579⋆

(59)

for all 119910 = (119910119896) isin C⋆ Finally by using Remark 9(ii)

and taking the generators 120572 = 120573 we get

⋆

119899119903minus1

sum

119896=119899119903minus1

(119910119896⊙119909119896)

(119901119896)120573

=⋆

119899119903minus1

sum

119896=119899119903minus1

[119910119896⊙ (sgn⋆119910

119896⊘ 119903)]

(119901119896)120573

=

1119903(119901119896)120573

⊙⋆

119899119903minus1

sum

119896=119899119903minus1

119910119896

(119901119896)120573

ge

1 (sup119901119896le 1)

(60)

Therefore 119910 ⊙ 119909 notin 119888119904⋆

(119901) and thus 119910 notin 119888⋆

0 (119901)120573

Hence 119888⋆0 (119901)120573

sub ℓ⋆

1 (119901) The other part of this casecan be obtained similarly

(e) From the condition (c) we have ℓ⋆

infin(119901) sub

(ℓ⋆

infin(119901)120573

)120573

= ℓ⋆

1 (119901)120573 since ℓ

⋆

infin(119901)120573

= ℓ⋆

1 (119901)Now we assume the existence of a 119908 = (119908

119899) isin

ℓ⋆

1 (119901)120573

ℓ⋆

infin(119901) Since 119908 is an ⋆-unbounded there

exists a subsequence (119908119899119896

) of (119908119899) and we can find a

number (119896 + 1)2120573 such that

119908119899119896

(119901119896)120573

ge (119896 + 1)2120573for all 119896 isin N1 The sequence (119909

119899) is defined by

119909119899= (sgn⋆(119908

119899119896

) ⊘ (119896 + 1)2120573) if 119899 = 119899119896 and 120579

⋆

otherwise Then 119909 isin ℓ⋆

1 (119901) However

⋆sum

119899

(119908119899⊙119909119899)

(119901119896)120573

=⋆sum

119896

119908119899119896

(119901119896)120573

(119896 + 1)(2119901119896)120573= infin

(61)

Hence 119908 notin ℓ⋆

1 (119901)120573 which contradicts our assump-

tion and ℓ⋆

1 (119901)120573

sub ℓ⋆

infin(119901) This step completes the

proof

In addition toTheorem 25we give the following corollarywhich is immediate consequences of the 120577-duals (120577 isin 120572

120573 120574)

Corollary 26 For each 120577 isin 120572 120573 120574 the following statementshold

(a) 120582⋆(119901)120572 sub 120582⋆

(119901)120573

sub 120582⋆

(119901)120574

sub 120596⋆ in particular

120582⋆

(119901)120577 is a sequence space over C⋆

(b) ℓ⋆1 (119901)120577

= ℓ⋆

infin(119901) and ℓ⋆

infin(119901)120577

= ℓ⋆

1 (119901)

(c) If 120582⋆(119901) sub 120583⋆

(119901) sub 120596⋆ then 120583⋆(119901)120577 sub 120582

⋆

(119901)120577


Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors record their pleasure to the anonymous refereefor hisher constructive report and many helpful suggestionson the main results

References

[1] H Nakano ldquoModulared sequence spacesrdquo Proceedings of theJapan Academy vol 27 pp 508ndash512 1951

[2] S Simons ldquoThe sequence spaces l(119901V) and m(119901V)rdquo Proceedingsof the London Mathematical Society vol 15 pp 422ndash436 1965

[3] I J Maddox ldquoParanormed sequence spaces generated byinfinite matricesrdquo Mathematical Proceedings of the CambridgePhilosophical Society vol 64 no 2 pp 335ndash340 1968

[4] C G Lascarides ldquoA study of certain sequence spaces ofMaddoxand a generalization of a theorem of Iyerrdquo Pacific Journal ofMathematics vol 38 no 2 pp 487ndash500 1971

[5] C G Lascarides and I J Maddox ldquoMatrix transformationbetween some classes of sequencesrdquo Proceedings of the Cam-bridge Philosophical Society vol 68 pp 99ndash104 1970

[6] MMursaleen andAKNoman ldquoOn the spaces of120582-convergentand bounded sequencesrdquo Thai Journal of Mathematics vol 8no 2 pp 311ndash329 2010

[7] M Mursaleen and S A Mohiuddine ldquoSome matrix trans-formations of convex and paranormed sequence spaces intothe spaces of invariant meansrdquo Journal of Function Spaces andApplications vol 2012 Article ID 612671 6 pages 2012

[8] M Mursaleen and S A Mohiuddine ldquoAlmost bounded varia-tion of double sequences and some four dimensional summa-bility matricesrdquo Publicationes Mathematicae Debrecen vol 75no 3-4 pp 495ndash508 2009

[9] M Kirisci and F Basar ldquoSome new sequence spaces derivedby the domain of generalized difference matrixrdquo Computers ampMathematics with Applications vol 60 no 5 pp 1299ndash13092010

[10] F Basar and M Kirisci ldquoAlmost convergence and generalizeddifferencematrixrdquoComputers ampMathematics with Applicationsvol 61 no 3 pp 602ndash611 2011

[11] S Demiriz and C Cakan ldquoOn some new paranormed sequencespacesrdquoGeneralMathematicsNotes vol 1 no 2 pp 26ndash42 2010

[12] M Grossman and R Katz Non-Newtonian Calculus Lee Press1978

[13] M Grossman Bigeometric Calculus Archimedes FoundationBox 240 Rockport Mass USA 1983

[14] M Grossman The First Nonlinear System of Differential andIntegral Calculus Mathco 1979

[15] A E Bashirov E M Kurpınar and A Ozyapıcı ldquoMultiplicativecalculus and its applicationsrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 36ndash48 2008

[16] A E Bashirov and M Rıza ldquoOn complex multiplicativedifferentiationrdquo TWMS Journal of Applied and EngineeringMathematics vol 1 no 1 pp 75ndash85 2011

[17] E Misirli and Y Gurefe ldquoMultiplicative Adams Bashforth-Moulton methodsrdquo Numerical Algorithms vol 57 no 4 pp425ndash439 2011

[18] A F Cakmak and F Basar ldquoOn the classical sequence spacesand non-Newtonian calculusrdquo Journal of Inequalities and Appli-cations vol 2012 Article ID 932734 13 pages 2012

[19] S Tekin and F Basar ldquoCertain sequence spaces over the non-Newtonian complex fieldrdquo Abstract and Applied Analysis vol2013 Article ID 739319 11 pages 2013

[20] U Kadak ldquoDetermination of the Kothe-Toeplitz duals over thenon-Newtonian complex fieldrdquoTheScientificWorld Journal vol2014 Article ID 438924 10 pages 2014

[21] U Kadak and H Efe ldquoMatrix transformations between certainsequence spaces over the non-Newtonian complex fieldrdquo TheScientific World Journal vol 2014 Article ID 705818 12 pages2014

[22] U Kadak and H Efe ldquoThe construction of Hilbert spaces overthe non-Newtonian fieldrdquo International Journal of Analysis vol2014 Article ID 746059 10 pages 2014

[23] U Kadak and M Ozluk ldquoGeneralized Runge-Kutta methodwith respect to the non-Newtonian calculusrdquo Abstract andApplied Analysis vol 2015 Article ID 594685 10 pages 2015

[24] F Basar ldquoNormed and paranormed sequence spacesrdquo inSummability Theory and Its Applications pp 15ndash32 BenthamScience Publishers Istanbul Turkey 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


transformations between certain sequence spaces over thenon-Newtonian complex field and have generalized Runge-Kutta method with respect to the non-Newtonian calculus

2 Preliminaries Background and Notations

A generator is a one-to-one function whose domain is R

and whose range is a subset R120572of R where R

120572= 120572119909

119909 isin R Each generator generates exactly one arithmeticand conversely each arithmetic is generated by exactly onegenerator For example the identity function 119868 generatesclassical arithmetic and exponential function generates geo-metric (multiplicative) arithmetic As a generator we choosethe function 120572 such that those basic algebraic operations aredefined as follows

120572 - addition 119909

+ 119910 = 120572 120572minus1(119909) + 120572

minus1(119910)


minus 119910 = 120572 120572minus1(119909) minus 120572

minus1(119910)



minus1(119910)

120572 - division 119909

119910

= 120572 120572minus1(119909) divide 120572

minus1(119910)

120572 - order 119909

lt 119910 lArrrArr 120572minus1(119909) lt 120572

minus1(119910)

(1)

for all119909 119910 isin R120572sube R As an example if we choose the function

120572 = exp

120572 R 997888rarr Rexp sube R

119909 997891997888rarr 119910 = 120572 (119909) = 119890119909

(2)

120572-arithmetic can be interpreted as geometric arithmetic

120572 - addition 119909

+ 119910 = 119890ln119909+ln119910

= 119909 sdot 119910


minus 119910 = 119890ln119909minusln119910

= 119909divide119910


times 119910 = 119890ln119909 ln119910

= 119909ln119910

= 119910ln119909

120572 - division 119909

119910

= 119890ln119909ln119910

= 1199091 ln119910

(3)

By an arithmetic we mean a complete ordered field whoserealm is a subset of R There are infinitely many arithmeticsall of which are isomorphic that is structurally equivalentThe 120572-positive real numbers denoted byR+

120572 are the numbers

119909 in R120572such that

0

lt 119909 the 120572-negative real numbersdenoted by Rminus

120572 are those for which 119909

lt

0 The 120572-zero

0 and the 119886119897119901ℎ119886-one

1 turn out to be 120572(0) and 120572(1) Also120572(minus119901) = 120572minus120572

minus1(

119901 ) =

minus

119901 holds for all 119901 isin Z+ Thusthe set of all 120572-integers can be given by

Z120572= 120572 (minus2) 120572 (minus1) 120572 (0) 120572 (1) 120572 (2)

=

minus

2

minus

1

0

1

2 (4)

One can immediately conclude that the set of exp-integer canbe written as

Zexp =

11198902

1119890

1 119890 1198902 (5)

Besides the 120572-summation is defined by

120572

infin

sum

119896=0119909119896= 120572

infin

sum

119896=0120572minus1(119909119896)

= 120572 120572minus1(1199090) + sdot sdot sdot + 120572

minus1(119909119896) + sdot sdot sdot

(6)

for all 119909119896isin R120572sube R

Definition 1 (see [18]) Let119883 be a nonempty set and 119889120572 119883 times

119883 rarr R120572sube R be a function such that for all 119909 119910 119911 isin 119883 the

following axioms hold

(NM1) 119889120572(119909 119910) =

0 if and only if 119909 = 119910(NM2) 119889

120572(119909 119910) = 119889

120572(119910 119909)

(NM3) 119889120572(119909 119910)

le 119889120572(119909 119911)

+ 119889120572(119911 119910)

Then the pair (119883 119889120572) and 119889

120572are called an 120572-metric space and

an 120572-metric on119883 respectively

Throughout this paper we define the 119901th 120572-exponent 119909119901120572and 119902th 120572-root 119909(1119902)120572 of 119909 isin R

120572sube R as

1199092120572

= 119909


minus1(119909) = 120572 [120572

minus1(119909)]

2

1199093120572

= 1199092120572

times 119909

= 120572 120572minus1120572 [120572minus1(119909) times 120572

minus1(119909)] times 120572

minus1(119909)

= 120572 [120572minus1(119909)]

3

119909119901120572

= 119909(119901minus1)

120572

times 119909 = 120572 [120572minus1(119909)]

119901

(7)

Hence 120572radic119909 = 119909(12)120572= 119910 provided there exists an 119910 isin R

120572sube R

such that 1199102120572= 119909 For each 120572-nonnegative number 119909 the

symbol 120572radic119909 will be used to denote 120572[120572minus1(119909)]12 which is theunique 120572-nonnegative number 119910 whose 120572-square is equal to119909 For each number 119909 isin R

120572 120572radic1199092120572 = |119909|

120572= 120572(|120572

minus1(119909)|)where

the absolute value |119909|120572of 119909 isin R

120572is defined by

|119909|120572=

119909 119909

gt

0

0 119909 =

0

minus 119909 119909

lt

0

(8)

Definition 2 (see [18]) Let119883 = (119883 119889120572) be an 120572-metric space

Then the basic notions can be defined as follows

(a) A sequence 119909 = (119909119896) is a function from the set N into

the setR120572 The 120572-real number 119909

119896denotes the value of

the function at 119896 isin N and is called the 119896th term of thesequence

(b) A sequence (119909119899) in 119883 = (119883 119889

120572) is said to be 120572-

convergent if for every given 120576

gt

0 (120576 isin R120572) there



119899

minus 119909|120572


119899rarrinfin119909119899= 119909 or 119909

119899

120572


(c) A sequence (119909119899) in119883 = (119883 119889


if for every 120576

gt


lt 120576 for all119898 119899 gt 1198990




120573

+

minus

times

) and (R120572

+

minus

times

)




120573

such that

⋆ lim119909rarr119886

119891 (119909) = 119887


gt

0 exist120575

gt

0 ni

10038161003816100381610038161003816119891 (119909)

minus 119887

10038161003816100381610038161003816120573

lt 120576 forall119909 isin R120572 10038161003816100381610038161003816119909

minus 119886

10038161003816100381610038161003816120572

lt 120575

(9)





119899) 120573-converges

to 119887






(ii) 120580 is from R120572to R120573


120580 (119906

+ V) = 120580 (119906)

+ 120580 (V)

120580 (119906

minus V) = 120580 (119906)

minus 120580 (V)

120580 (119906

times V) = 120580 (119906)

times 120580 (V)

120580 (119906

V) = 120580 (119906)

120580 (V)

V =

0 119906 le V lArrrArr 120580 (119906)

le 120580 (V)

(10)


120572and that

120580(

119899 ) =


+ V =

120580minus1120580(119906)



119886 isin (R120572

+

minus

times

) and

119887 isin (R120573

+

minus

times


119886


C⋆

= 119911⋆

= (

119886

119887) |


119887 isinR120573subeR (11)


1= (

119886 1

119887 1) and 119911⋆

2= (

119886 2

119887 2)as


(119911⋆

1 119911⋆

2) 997891997888rarr 119911

⋆

1oplus 119911⋆

2= (120572 1198861 + 1198862 120573 1198871 + 1198872)

= (

119886 1

+

119886 2

119887 1

+

119887 2)


(119911⋆

1 119911⋆

2) 997891997888rarr 119911

⋆

1⊙ 119911⋆

2

= (120572 11988611198862 minus 11988711198872 120573 11988611198872 + 11988711198862)

(12)

where

119886 1

119886 2 isin R120572and

119887 1

119887 2 isin R120573





119886 1

119887 1) and 119911⋆

2= (

119886 2


119889⋆ C⋆ timesC⋆ 997888rarr [

0 infin) = 1198611015840

sub R120573

(119911⋆

1 119911⋆

2) 997891997888rarr 119889

⋆

(119911⋆

1 119911⋆

2)

= (120580 (

119886 1

minus

119886 2)2120572

+ (

119887 1

minus

119887 2)2120573

)

(12)120573


2

(13)



119909119896

119910119896) of


⋆

infin

sum

119896=0119911⋆

119896= 119911⋆

0 oplus 119911⋆

1 oplus 119911⋆



= (120572

infin

sum

119896=0

119909119896120573

infin

sum

119896=0

119910119896)

= (120572

infin

sum

119896=0120572minus1119909119896 120573

infin

sum

119896=0120573minus1119909119896)

isin C⋆

(14)


119899=⋆sum119899

119896=0 119911⋆



=⋆suminfin

119899=0 119911⋆





sdot



sdot


sdot


(NN1)

119909

=

0 hArr 119909 = 120579⋆ (120579⋆ = (

0

0 ))

(NN2)

120582 ⊙ 119909

=

| 120582

|

times

119909

(NN3)

119909 oplus 119910

le

119909

+

119910


sdot



119909 ⊖ 119910



119911⋆

= 119889⋆

(119911⋆

120579⋆

) = (120580 (

119886

minus

0 )2120572

+ (

119887

minus

0 )2120573

)

(12)120573

= 120573 radic1198862+ 119887

2

(15)

where 119911⋆ = (

119886

119887 ) and 120579⋆ = (

0



(N(MC)1)

119909

= 1 hArr 119909 = 120579⋆ (120579⋆ = (0 1))

(N(MC)2)

120582 ⊙ 119909

=

119909

|120582|

(N(MC)3)

119909 oplus 119910

le

119909

119910


sdot



119886


minus1(

119887 )) =

(

119886

minus


Re (119911⋆) = Re (119911⋆) = (119911⋆

oplus 119911⋆

)

2 =

119886

Im (119911⋆

) =

minus Im (119911⋆

) = (119911⋆

⊖ 119911⋆

)

2 =

119887

(16)


(i) Let 119911⋆1 = (

119886 1

119887 1) 119911⋆

2 = (

119886 2


119911⋆

1 ⊘ 119911⋆

2

= (120572

(11988611198862 + 11988711198872)

(1198862

2+ 1198872

2)

120573

(11988711198862 minus 11988611198872)

(1198862

2+ 1198872

2)

)

(17)


(

119886


⊙ 119911⋆=

119911⋆

2120573

holds Really

119911⋆

⊙ 119911⋆= (

119886

119887 ) ⊙ (

119886

minus

119887 ) = (120572 1198862+ 119887

2 120573 (0))

= 120573 1198862+ 119887

2 = 120573 (120573

minus1120573radic1198862+ 119887

2)

2

=

119911⋆

2120573

(18)



sdot

defined by

119911⋆

= (120580(

119886 )2120572

+

119887

2120573

)(12)120573 119911⋆ = (

119886

119887 ) isin

C⋆


119911⋆

= 119889⋆

(119911⋆

120579⋆

)



norm sdot ⋆

infindefined by

119911⋆


119911⋆

119896

119911 = (119911⋆

119896) isin 120582⋆

120582 isin ℓinfin 119888 1198880 (19)


⋆

119901

defined by

119911⋆

119901= (⋆sum

119896

119911⋆

119896

119901120573

)

(1119901)120573

119901 ge 1 119911 = (119911⋆

119896) isin ℓ⋆

119901 (20)




⋆


119909⋆

119887119904= 119909

⋆

119888119904= sup119899isinN

⋆

119899

sum

119896=0119909119896

119909 = (119909119896) isin 120583⋆

120583 isin 119887119904 119888119904 1198881199040

(21)


infinare Banach


119909⋆

119887V = ⋆sum119896

(Δ1015840

119909)119896

119909⋆

119887V119901

= (⋆sum

119896

(Δ119909)119896

119901120573

)

(1119901)120573

119909⋆

119887Vinfin

= sup119896isinN

(Δ119909)119896

(22)

where (Δ1015840119909)119896= (119909119896⊖119909119896+1) and (Δ119909)119896 = (119909

119896⊖119909119896minus1) 119909minus1 = 120579

⋆



119894 120583⋆

rarr C⋆ defined by119901119894(119909) = 119909







119899) such that



1003817100381710038171003817

⋆

= 120579⋆ (23)




⋆sum119896120585119896⊙119887119896The


Let 119890 = (119890119896) and 119890(119899) = (119890

(119899)



119896= 120575⋆

119899119896 where 120575⋆

119899119896denotes


120575⋆

119899119896=

1⋆ 119899 = 119896

120579⋆

119899 = 119896

(24)




119909 = 120585 ⊙ 119890 oplus⋆sum

119896

(119909119896⊖ 120585) ⊙ 119890

(119896)


119909119896= 120585

(25)



⋆

infin




119879119909 = (⋆sum119896



(119910119896) isin ℓ⋆


119896) by 119909

119896= 119910119896⊖119910119896minus1



sup119896isinN

⋆

119896

sum

119895=0119909119895

= sup119896isinN

⋆

119896

sum

119895=0(119910119895⊖119910119895minus1)

= sup119896isinN

119910119896

=

1003817100381710038171003817119910

1003817100381710038171003817

⋆

infinlt infin

(26)







infin 119888 1198880 ℓ119901 119887119904 119888119904 1198881199040

119887V 119887Vinfin

(ii) ℓ⋆1 sube 119888119904⋆

sube 119888⋆

0 sube 119888⋆

sube ℓ⋆


119888⋆ where 119887V⋆0 = 119887V⋆ cap 119888⋆0


⋆

0 sube 119888 sube 119888⋆








every 120576

gt


(119911119896 ℓ)



119911119896

le

119911119896⊖ ℓ

+

ℓ

le 120598

+

ℓ

(27)


minus1 119911119896

le 120573minus1(120598

+

ℓ



+

ℓ





similarly



⋆ 119888119904⋆ 119887V⋆119901 and ℓ⋆

119901are


respectively



119906

times V

le

119906119901120572

119901

+

V119902120572

119902

(28)



120572minus1(119906) 120572minus1(V) le

(120572minus1(119906))

119901

119901

+

(120572minus1(V))119902

119902

(29)


0 or V =



119891 (119905) = (120580 (119905))120582120573

minus

120582

times 120580 (119905) = 120573 (120572minus1(119905))

120582

minus120582120572minus1(119905) (30)



119891⋆

(119905) = 120573

(120573minus1∘ 119891)

1015840

(119905)

(120572minus1)

1015840

(119905)

= 120573

((120572minus1)

120582


1015840

(119905)

(120572minus1)

1015840

(119905)

= 120573120582 (120572minus1(119905))


(31)


⋆

(119905) =



119891⋆⋆

(119905) = 120573

(120573minus1119891⋆

(119905))

1015840

(120572minus1)

1015840

(119905)

= 120573120582 (120582 minus 1) (120572minus1 (119905))120582

(32)


1 ) = 120573120582(120582 minus 1)

lt


1 that is 119891(


[120572minus1(119905)]

120582


120572 (33)

Now taking 119905 = (119906)119901120572

(V)119902120572 = 120572(120572minus1(119906))119901

(120572minus1(V))119902 and

120582 = 1119901 in (33) we get

(

(120572minus1(119906))

119901

(120572minus1(V))119902

)

1119901

minus

1119901

(120572minus1(119906))

119901

(120572minus1(V))119902

le 1minus 1119901

120572minus1(119906)

120572minus1(V)119902119901

(120572minus1(V))119902

le (1minus 1119901

+

1119901

120572minus1(119906)119901

120572minus1(V)119902

)(120572minus1(V))119902

(34)



119896 V⋆119896isin C⋆ for 119896 isin 0 1 2 119899


120573

119899

sum

119896=0

119906⋆

119896⊙ V⋆119896

le (120573

119899

sum

119896=0

119906⋆

119896

119901120573

)

(1119901)120573

times (120573

119899

sum

119896=0

V⋆119896

119902120573

)

(1119902)120573

(35)


= (

0

0 ) orV = (

0


Let

120576 = (120573

119899

sum

119896=0

119906⋆

119896

119901120573

)

(1119901)120573

120575 = (120573

119899

sum

119896=0

V⋆119896

119902120573

)

(1119902)120573

(36)

and 119904⋆119896= 119906⋆

119896⊘ 120576 119905⋆119896= V⋆119896⊘ 120575 where 120576 = (

0

120576 ) isin R120573sube C⋆

and 120575 = (

0

120575 ) isin R120573


119904⋆

119896⊙ 119905⋆

119896

=

119904⋆

119896

times

119905⋆

119896

le

119904⋆

119896

119901120573

119901

+

119905⋆

119896

119902120573

119902

(37)

which implies that

120573

119899

sum

119896=0

119904⋆

119896⊙ 119905⋆

119896

le120573

119899

sum

119896=0

119904⋆

119896

119901120573

119901

+120573

119899

sum

119896=0

119905⋆

119896

119902120573

119902

(38)


120573

119899

sum

119896=0

(119906⋆

119896⊘ 120576) ⊙ (V⋆

119896⊘ 120575)

le120573

119899

sum

119896=0

119906⋆

119896

119901120573

120576

times

119901

+120573

119899

sum

119896=0

V⋆119896

119902120573

120575

times

119902

=

1

119901

+

1

119902

=

1

(39)




120573

119899

sum

119896=0

119906⋆

119896⊙ V⋆119896

le (120573

119899

sum

119896=0

119906⋆

119896

2120573

times120573

119899

sum

119896=0

V⋆119896

2120573

)

(12)120573

(40)




(120573

119899

sum

119896=0

119906⋆

119896oplus V⋆119896

119901120573

)

(1119901)120573

le (120573

119899

sum

119896=0

119906⋆

119896

119901120573

)

(1119901)120573

+ (120573

119899

sum

119896=0

V⋆119896

119901120573

)

(1119901)120573

(41)



120573

119899

sum

119896=0

119906⋆

119896oplus V⋆119896

119901120573

le120573

119899

sum

119896=0

119906⋆

119896

times

119906⋆

119896oplus V⋆119896

(119901minus1)120573

+120573

119899

sum

119896=0

V⋆119896

times

119906⋆

119896oplus V⋆119896

(119901minus1)120573

(42)


120573

119899

sum

119896=0

119906⋆

119896oplus V⋆119896

119901120573

le[

[

(120573

119899

sum

119896=0

119906⋆

119896

119901120573

)

(1119901)120573

+ (120573

119899

sum

119896=0

V⋆119896

119901120573

)

(1119901)120573

]

]

times (120573

119899

sum

119896=0

119906⋆

119896oplus V⋆119896

(119902119901minus119902)120573

)

(1119902)120573

(43)





120573sube



(N(PN)1) 119892⋆

(119909) =

0 if 119909 = 120579⋆ (120579⋆ = (0⋆ 0⋆ ))

(N(PN)2) 119892⋆

(⊖119909) = 119892⋆


⋆

(119909 oplus 119910)

le 119892⋆

(119909)

+ 119892⋆

(119910)


that is 120582⋆ = (

120582

120582 ) with 120582⋆

119899

119889⋆



119899

119892⋆

997888997888rarr 119909 then

120582⋆

119899⊙ 119909119899

119892⋆

997888997888rarr 120582⋆

⊙ 119909 as 119899 rarr infin



(BG(PN)3) 119892⋆

(119909 oplus 119910) le 119892⋆

(119909)119892⋆

(119910)



119896=


119896times 1199011015840

119896= 119901119896+ 1199011015840





119888⋆

0 and ℓ⋆



119901over


infin 119888⋆

119888⋆

0 and ℓ⋆



infin(119901) 119888⋆

(119901) 119888⋆

0 (119901) ℓ⋆

(119901) 119887119904⋆

(119901) 119888119904⋆

(119901) 119888119904⋆

0 (119901) 119887V⋆



ℓ⋆

infin(119901) = 119909= (119909

119896) isin 120596⋆

sup119896isinN

119909119896

(119901119896)120573

ltinfin

= 119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆

sup119896isinN

120573radic1205762

119896+ 120575

2119896

119901119896

ltinfin

119888⋆

(119901) = 119909= (119909119896) isin 120596⋆



119909119896⊖ 119897

(119901119896)120573

= 120579⋆

= 119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆

exist 119897 = (

ℓ

ℓ ) isinC⋆


120573(ℓradic2 (1205762119896+ 120575

2119896))

119901119896

=

0

119888⋆

0 (119901) = 119909= (119909119896) isin 120596⋆


119909119896

(119901119896)120573

= 120579⋆

= 119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆


120573radic1205762

119896+ 120575

2119896

119901119896

ltinfin


ℓ⋆

(119901) = 119909= (119909119896) isin 120596⋆

120573sum

119896

119909119896

(119901119896)120573

ltinfin

(0 lt 119901119896lt infin)

= 119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆

120573sum

119896

radic1205762

119896+ 120575

2119896

119901119896

ltinfin

(44)


(119901) 119888119904⋆

0 (119901) 119887V⋆



119887119904⋆

(119901) =

119909= (119909119896) isin 120596⋆

(⋆

119896

sum

119895=0119909119895)isinℓ⋆

infin(119901)

=

119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆

sup119896isinN

120573

radic(

119896

sum

119895=0120576119895)

2

+ (

119896

sum

119895=0120575119895)

2119901119896

ltinfin

119888119904⋆

(119901) =

119909= (119909119896) isin 120596⋆

(⋆

119896

sum

119895=0119909119895)isin 119888⋆

(119901)

=

119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆


120573

radic(

119896

sum

119895=0120576119895minus ℓ)

2

+ (

119896

sum

119895=0120575119895minus ℓ)

2119901119896

ltinfin

(ℓ = (

ℓ

ℓ ) isin C⋆

)

119888119904⋆

0 (119901) = 119909= (119909119896) isin 120596⋆

(⋆

119899

sum

119896=0119909119896)isin 119888⋆

0 (119901)

119887V⋆ (119901) = 119909= (119909119896) isin 120596⋆

120573

infin

sum

119896=0

Δ119909119896

(119901119896)120573

ltinfin

(Δ119909119896= 119909119896⊖ 119909119896minus1 119909minus1 = 120579

⋆

)

= 119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆

120573

infin

sum

119896=0


2+ (120575119896minus 120575119896minus1)

2119901119896

ltinfin

119887V⋆infin(119901) = 119909= (119909

119896) isin 120596⋆

sup119896isinN

Δ119909119896

(119901119896)120573

ltinfin

(45)


(119901)

119888⋆

0 (119901) ℓ⋆

(119901) 119887119904⋆

(119901) 119888119904⋆

(119901) 119888119904⋆

0 (119901) 119887V⋆

(119901) and 119887V⋆infin(119901) is a

⋆-linear space



119892⋆

(119909) = sup119896isinN

119909119896

(119901119896119872)120573

119892⋆

(119909) = (120573sum

119896

119909119896

(119901119896)120573

)

(1119872)120573

(46)

Then 119888⋆

(119901) and 119888⋆


119896isin ℓ⋆


infin(119901) and

ℓ⋆


119896gt 0


119892⋆

1 (119909) = sup119896isinN

⋆

119896

sum

119895=0119909119895

(119901119896119872)120573

iff inf 119901119896gt 0 (47)



119892⋆

2 (119909) = (120573sum

119896

Δ119909119896

(119901119896)120573

)

(1119872)120573

119892⋆

3 (119909) = sup119896isinN

Δ119909119896

(1119872)120573

iff inf 119901119896gt 0

(48)







(⋆sum

119896

Δ (119906119896oplus119909119896)

(119901119896)120573

)

(1119872)120573

le (⋆sum

119896

Δ119906119896

(119901119896)120573

)

(1119872)120573

+ (⋆sum

119896

Δ119909119896

(119901119896)120573

)

(1119872)120573

(49)

and the condition

120582⋆

(119901119896)120573

le max

1

120582⋆

119872120573

(50)


= (

120582


0 and 119892⋆

(⊖119909) = 119892⋆


lemax

1

120582⋆

times 119892⋆



119899) be any


rarr 120582⋆ with


(119909119899

)

le 119892⋆

(119909)

+ 119892⋆

(119909119899


119892⋆

(120582⋆

119899⊙119909119899

⊖120582⋆

⊙119909)

= (120573sum

119896

Δ (120582⋆

119899⊙119909(119899)

119896⊖120582⋆

⊙119909119896)

(119901119896)120573

)

(1119872)120573

le

120582⋆

119899⊖120582⋆

times 119892⋆

(119909119899

)

+

120582⋆

times 119892⋆

(119909119899

⊖119909)

(51)

which tends to



119894= 119909(119894)

0 119909(119894)

1 119909(119894)


gt


119892⋆

(119909119894

⊖119909119895

)

lt

120598 forall119894 119895 ge 1198990 (120598) (52)


Δ [(119909119894

)119896

⊖ (119909119895

)119896

]

le (120573sum

119896

Δ [(119909119894

)119896

⊖ (119909119895

)119896

]

(119901119896)120573

)

(1119872)120573

lt 120598

forall119894 119895 ge 1198990 (120598)

(53)


1)119896 (Δ119909

2)119896 is a




120573

119898

sum

119896=0

Δ [(119909119894

)119896

⊖ (119909119895

)119896

]

(119901119896)120573

le 119892⋆

(119909119894

⊖119909119895

)

119872120573

lt 120598119872120573

(54)



(120573sum

119896

(Δ119909)119896

(119901119896)120573

)

(1119872)120573

le 119892⋆

(119909119894

⊖119909)

+ 119892⋆

(119909119894

)

le 120598

+ 119892⋆

(119909119894

)

(55)



119894



infin(119901) 119888⋆

(119901) 119888⋆

0 (119901) ℓ⋆

(119901) 119887119904⋆

(119901) 119888119904⋆

(119901) 119888119904⋆

0 (119901) 119887V⋆





119901of 119901-






infin 119888 1198880 ℓ119901 119887119904 119888119904

1198881199040 119887V 119887Vinfin(ii) 120583(119901) sube 120583

⋆

(119901) and ℓ⋆1 (119901) sube 119888119904⋆

(119901) sube 119888⋆

0 (119901) sube 119888⋆

(119901) sube

ℓ⋆

infin(119901) sube 120596

⋆



⋆

(119901)120572

120582⋆

(119901)120573 and 120582⋆(119901)120574 as follows

120582⋆

(119901)

120572

= 119908= (119908119896) isin 120596⋆

119908 ⊙ 119911 = (119908119896⊙ 119911119896)

isin ℓ⋆

1 (119901) forall119911 = (119911119896) isin 120582⋆

(119901)

120582⋆

(119901)

120573

= 119908= (119908119896) isin 120596⋆

119908 ⊙ 119911 = (119908119896⊙ 119911119896)

isin 119888119904⋆

(119901) forall119911 = (119911119896) isin 120582⋆

(119901)

120582⋆

(119901)

120574

= 119908= (119908119896) isin 120596⋆

119908 ⊙ 119911 = (119908119896⊙ 119911119896)

isin 119887119904⋆

(119901) forall119911 = (119911119896) isin 120582⋆

(119901)

(56)




Theorem 25 Let 0 = 120582⋆




(b) If 120582⋆(119901) sub 120583⋆

(119901) sub 120596⋆ then 120583⋆(119901)120573 sub 120582

⋆

(119901)120573

(c) 120582⋆ sub 120582⋆

120573120573

= (120582⋆

120573

)120573

(d) 119888⋆0 (119901)120573

= 119888⋆

(119901)120573

= ℓ⋆

infin(119901)120573

= ℓ⋆

1 (119901)

(e) ℓ⋆1 (119901)120573

= ℓ⋆

infin(119901)


119896) 119898 = (119898

119896)

and 119899 = (119899119896) isin 120582

⋆

(119901)120573



(119901)120573 for119898 119899 isin

120582⋆


⋆

(119901) Then (119898119896⊙ 119897119896) isin

119888119904⋆

(119901) and (119899119896⊙ 119897119896) isin 119888119904⋆


((119898119896oplus 119899119896) ⊙ 119897119896) = (119898

119896⊙ 119897119896) oplus (119899119896⊙ 119897119896) isin 119888119904

⋆

(119901)


(119901)

(57)

Hence 119898 oplus 119899 isin 120582⋆


120582⋆

(119901)120573 for any 119905 isin C⋆ and 119908 = (119908

119896) isin 120582

⋆

(119901)120573

Since (119908119896⊙ 119897119896) isin 119888119904

⋆

(119901) for all 119897 isin 120582⋆


119896⊙119908119896) ⊙ 119897119896) = 119905119896⊙ (119908119896⊙ 119897119896) isin

119888119904⋆

(119901) for all 119897 isin 120582⋆

(119901) we get 119905 ⊙ 119908 isin 120582⋆

(119901)120573



infin(119901)120573

sub 119888⋆

(119901)120573

sub 119888⋆

0 (119901)120573 by


ℓ⋆

infin(119901)120573 and 119888

⋆

0 (119901)120573

sub ℓ⋆


(119908119896) isin ℓ

⋆

1 (119901) and 119911 = (119911119896) isin ℓ

⋆



⋆sum

119896

1003817100381710038171003817119908119896⊙ 119911119896

1003817100381710038171003817

(119901119896)120573

le sup119896isinN

119911119896

(1119872)120573

times (120573sum

119896

119908119896

(119901119896)120573

)

(1119872)120573

le 119892⋆

(119911)

times 119892⋆

(119908) lt infin

(58)


⋆


⋆



ℓ⋆


⋆

0 (119901) with 119910 ⊙ 119909 notin 119888119904⋆


⋆



and⋆sum

119899119903minus1119896=119899119903minus1

119910119896

(119901119896)120573


119896) isin 119888

⋆

0 (119901)by 119909119896= (sgn⋆119910



sgn⋆ (119910) =

119910 ⊘

119910

119910 = 120579⋆

120579⋆

119910 = 120579⋆

(59)



⋆

119899119903minus1

sum

119896=119899119903minus1

(119910119896⊙119909119896)

(119901119896)120573

=⋆

119899119903minus1

sum

119896=119899119903minus1

[119910119896⊙ (sgn⋆119910

119896⊘ 119903)]

(119901119896)120573

=

1119903(119901119896)120573

⊙⋆

119899119903minus1

sum

119896=119899119903minus1

119910119896

(119901119896)120573

ge

1 (sup119901119896le 1)

(60)

Therefore 119910 ⊙ 119909 notin 119888119904⋆

(119901) and thus 119910 notin 119888⋆

0 (119901)120573

Hence 119888⋆0 (119901)120573

sub ℓ⋆



infin(119901) sub

(ℓ⋆

infin(119901)120573

)120573

= ℓ⋆

1 (119901)120573 since ℓ

⋆

infin(119901)120573

= ℓ⋆


119899) isin

ℓ⋆

1 (119901)120573

ℓ⋆




number (119896 + 1)2120573 such that

119908119899119896

(119901119896)120573



119909119899= (sgn⋆(119908

119899119896

) ⊘ (119896 + 1)2120573) if 119899 = 119899119896 and 120579

⋆


1 (119901) However

⋆sum

119899

(119908119899⊙119909119899)

(119901119896)120573

=⋆sum

119896

119908119899119896

(119901119896)120573

(119896 + 1)(2119901119896)120573= infin

(61)



tion and ℓ⋆

1 (119901)120573

sub ℓ⋆


proof


120573 120574)


(a) 120582⋆(119901)120572 sub 120582⋆

(119901)120573

sub 120582⋆

(119901)120574


120582⋆


(b) ℓ⋆1 (119901)120577

= ℓ⋆


infin(119901)120577

= ℓ⋆

1 (119901)

(c) If 120582⋆(119901) sub 120583⋆

(119901) sub 120596⋆ then 120583⋆(119901)120577 sub 120582

⋆

(119901)120577




Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119899

minus 119909|120572


119899rarrinfin119909119899= 119909 or 119909

119899

120572


(c) A sequence (119909119899) in119883 = (119883 119889


if for every 120576

gt


lt 120576 for all119898 119899 gt 1198990




120573

+

minus

times

) and (R120572

+

minus

times

)




120573

such that

⋆ lim119909rarr119886

119891 (119909) = 119887


gt

0 exist120575

gt

0 ni

10038161003816100381610038161003816119891 (119909)

minus 119887

10038161003816100381610038161003816120573

lt 120576 forall119909 isin R120572 10038161003816100381610038161003816119909

minus 119886

10038161003816100381610038161003816120572

lt 120575

(9)





119899) 120573-converges

to 119887






(ii) 120580 is from R120572to R120573


120580 (119906

+ V) = 120580 (119906)

+ 120580 (V)

120580 (119906

minus V) = 120580 (119906)

minus 120580 (V)

120580 (119906

times V) = 120580 (119906)

times 120580 (V)

120580 (119906

V) = 120580 (119906)

120580 (V)

V =

0 119906 le V lArrrArr 120580 (119906)

le 120580 (V)

(10)


120572and that

120580(

119899 ) =


+ V =

120580minus1120580(119906)



119886 isin (R120572

+

minus

times

) and

119887 isin (R120573

+

minus

times


119886


C⋆

= 119911⋆

= (

119886

119887) |


119887 isinR120573subeR (11)


1= (

119886 1

119887 1) and 119911⋆

2= (

119886 2

119887 2)as


(119911⋆

1 119911⋆

2) 997891997888rarr 119911

⋆

1oplus 119911⋆

2= (120572 1198861 + 1198862 120573 1198871 + 1198872)

= (

119886 1

+

119886 2

119887 1

+

119887 2)


(119911⋆

1 119911⋆

2) 997891997888rarr 119911

⋆

1⊙ 119911⋆

2

= (120572 11988611198862 minus 11988711198872 120573 11988611198872 + 11988711198862)

(12)

where

119886 1

119886 2 isin R120572and

119887 1

119887 2 isin R120573





119886 1

119887 1) and 119911⋆

2= (

119886 2


119889⋆ C⋆ timesC⋆ 997888rarr [

0 infin) = 1198611015840

sub R120573

(119911⋆

1 119911⋆

2) 997891997888rarr 119889

⋆

(119911⋆

1 119911⋆

2)

= (120580 (

119886 1

minus

119886 2)2120572

+ (

119887 1

minus

119887 2)2120573

)

(12)120573


2

(13)



119909119896

119910119896) of


⋆

infin

sum

119896=0119911⋆

119896= 119911⋆

0 oplus 119911⋆

1 oplus 119911⋆



= (120572

infin

sum

119896=0

119909119896120573

infin

sum

119896=0

119910119896)

= (120572

infin

sum

119896=0120572minus1119909119896 120573

infin

sum

119896=0120573minus1119909119896)

isin C⋆

(14)


119899=⋆sum119899

119896=0 119911⋆



=⋆suminfin

119899=0 119911⋆





sdot



sdot


sdot


(NN1)

119909

=

0 hArr 119909 = 120579⋆ (120579⋆ = (

0

0 ))

(NN2)

120582 ⊙ 119909

=

| 120582

|

times

119909

(NN3)

119909 oplus 119910

le

119909

+

119910


sdot



119909 ⊖ 119910



119911⋆

= 119889⋆

(119911⋆

120579⋆

) = (120580 (

119886

minus

0 )2120572

+ (

119887

minus

0 )2120573

)

(12)120573

= 120573 radic1198862+ 119887

2

(15)

where 119911⋆ = (

119886

119887 ) and 120579⋆ = (

0



(N(MC)1)

119909

= 1 hArr 119909 = 120579⋆ (120579⋆ = (0 1))

(N(MC)2)

120582 ⊙ 119909

=

119909

|120582|

(N(MC)3)

119909 oplus 119910

le

119909

119910


sdot



119886


minus1(

119887 )) =

(

119886

minus


Re (119911⋆) = Re (119911⋆) = (119911⋆

oplus 119911⋆

)

2 =

119886

Im (119911⋆

) =

minus Im (119911⋆

) = (119911⋆

⊖ 119911⋆

)

2 =

119887

(16)


(i) Let 119911⋆1 = (

119886 1

119887 1) 119911⋆

2 = (

119886 2


119911⋆

1 ⊘ 119911⋆

2

= (120572

(11988611198862 + 11988711198872)

(1198862

2+ 1198872

2)

120573

(11988711198862 minus 11988611198872)

(1198862

2+ 1198872

2)

)

(17)


(

119886


⊙ 119911⋆=

119911⋆

2120573

holds Really

119911⋆

⊙ 119911⋆= (

119886

119887 ) ⊙ (

119886

minus

119887 ) = (120572 1198862+ 119887

2 120573 (0))

= 120573 1198862+ 119887

2 = 120573 (120573

minus1120573radic1198862+ 119887

2)

2

=

119911⋆

2120573

(18)



sdot

defined by

119911⋆

= (120580(

119886 )2120572

+

119887

2120573

)(12)120573 119911⋆ = (

119886

119887 ) isin

C⋆


119911⋆

= 119889⋆

(119911⋆

120579⋆

)



norm sdot ⋆

infindefined by

119911⋆


119911⋆

119896

119911 = (119911⋆

119896) isin 120582⋆

120582 isin ℓinfin 119888 1198880 (19)


⋆

119901

defined by

119911⋆

119901= (⋆sum

119896

119911⋆

119896

119901120573

)

(1119901)120573

119901 ge 1 119911 = (119911⋆

119896) isin ℓ⋆

119901 (20)




⋆


119909⋆

119887119904= 119909

⋆

119888119904= sup119899isinN

⋆

119899

sum

119896=0119909119896

119909 = (119909119896) isin 120583⋆

120583 isin 119887119904 119888119904 1198881199040

(21)


infinare Banach


119909⋆

119887V = ⋆sum119896

(Δ1015840

119909)119896

119909⋆

119887V119901

= (⋆sum

119896

(Δ119909)119896

119901120573

)

(1119901)120573

119909⋆

119887Vinfin

= sup119896isinN

(Δ119909)119896

(22)

where (Δ1015840119909)119896= (119909119896⊖119909119896+1) and (Δ119909)119896 = (119909

119896⊖119909119896minus1) 119909minus1 = 120579

⋆



119894 120583⋆

rarr C⋆ defined by119901119894(119909) = 119909







119899) such that



1003817100381710038171003817

⋆

= 120579⋆ (23)




⋆sum119896120585119896⊙119887119896The


Let 119890 = (119890119896) and 119890(119899) = (119890

(119899)



119896= 120575⋆

119899119896 where 120575⋆

119899119896denotes


120575⋆

119899119896=

1⋆ 119899 = 119896

120579⋆

119899 = 119896

(24)




119909 = 120585 ⊙ 119890 oplus⋆sum

119896

(119909119896⊖ 120585) ⊙ 119890

(119896)


119909119896= 120585

(25)



⋆

infin




119879119909 = (⋆sum119896



(119910119896) isin ℓ⋆


119896) by 119909

119896= 119910119896⊖119910119896minus1



sup119896isinN

⋆

119896

sum

119895=0119909119895

= sup119896isinN

⋆

119896

sum

119895=0(119910119895⊖119910119895minus1)

= sup119896isinN

119910119896

=

1003817100381710038171003817119910

1003817100381710038171003817

⋆

infinlt infin

(26)







infin 119888 1198880 ℓ119901 119887119904 119888119904 1198881199040

119887V 119887Vinfin

(ii) ℓ⋆1 sube 119888119904⋆

sube 119888⋆

0 sube 119888⋆

sube ℓ⋆


119888⋆ where 119887V⋆0 = 119887V⋆ cap 119888⋆0


⋆

0 sube 119888 sube 119888⋆








every 120576

gt


(119911119896 ℓ)



119911119896

le

119911119896⊖ ℓ

+

ℓ

le 120598

+

ℓ

(27)


minus1 119911119896

le 120573minus1(120598

+

ℓ



+

ℓ





similarly



⋆ 119888119904⋆ 119887V⋆119901 and ℓ⋆

119901are


respectively



119906

times V

le

119906119901120572

119901

+

V119902120572

119902

(28)



120572minus1(119906) 120572minus1(V) le

(120572minus1(119906))

119901

119901

+

(120572minus1(V))119902

119902

(29)


0 or V =



119891 (119905) = (120580 (119905))120582120573

minus

120582

times 120580 (119905) = 120573 (120572minus1(119905))

120582

minus120582120572minus1(119905) (30)



119891⋆

(119905) = 120573

(120573minus1∘ 119891)

1015840

(119905)

(120572minus1)

1015840

(119905)

= 120573

((120572minus1)

120582


1015840

(119905)

(120572minus1)

1015840

(119905)

= 120573120582 (120572minus1(119905))


(31)


⋆

(119905) =



119891⋆⋆

(119905) = 120573

(120573minus1119891⋆

(119905))

1015840

(120572minus1)

1015840

(119905)

= 120573120582 (120582 minus 1) (120572minus1 (119905))120582

(32)


1 ) = 120573120582(120582 minus 1)

lt


1 that is 119891(


[120572minus1(119905)]

120582


120572 (33)

Now taking 119905 = (119906)119901120572

(V)119902120572 = 120572(120572minus1(119906))119901

(120572minus1(V))119902 and

120582 = 1119901 in (33) we get

(

(120572minus1(119906))

119901

(120572minus1(V))119902

)

1119901

minus

1119901

(120572minus1(119906))

119901

(120572minus1(V))119902

le 1minus 1119901

120572minus1(119906)

120572minus1(V)119902119901

(120572minus1(V))119902

le (1minus 1119901

+

1119901

120572minus1(119906)119901

120572minus1(V)119902

)(120572minus1(V))119902

(34)



119896 V⋆119896isin C⋆ for 119896 isin 0 1 2 119899


120573

119899

sum

119896=0

119906⋆

119896⊙ V⋆119896

le (120573

119899

sum

119896=0

119906⋆

119896

119901120573

)

(1119901)120573

times (120573

119899

sum

119896=0

V⋆119896

119902120573

)

(1119902)120573

(35)


= (

0

0 ) orV = (

0


Let

120576 = (120573

119899

sum

119896=0

119906⋆

119896

119901120573

)

(1119901)120573

120575 = (120573

119899

sum

119896=0

V⋆119896

119902120573

)

(1119902)120573

(36)

and 119904⋆119896= 119906⋆

119896⊘ 120576 119905⋆119896= V⋆119896⊘ 120575 where 120576 = (

0

120576 ) isin R120573sube C⋆

and 120575 = (

0

120575 ) isin R120573


119904⋆

119896⊙ 119905⋆

119896

=

119904⋆

119896

times

119905⋆

119896

le

119904⋆

119896

119901120573

119901

+

119905⋆

119896

119902120573

119902

(37)

which implies that

120573

119899

sum

119896=0

119904⋆

119896⊙ 119905⋆

119896

le120573

119899

sum

119896=0

119904⋆

119896

119901120573

119901

+120573

119899

sum

119896=0

119905⋆

119896

119902120573

119902

(38)


120573

119899

sum

119896=0

(119906⋆

119896⊘ 120576) ⊙ (V⋆

119896⊘ 120575)

le120573

119899

sum

119896=0

119906⋆

119896

119901120573

120576

times

119901

+120573

119899

sum

119896=0

V⋆119896

119902120573

120575

times

119902

=

1

119901

+

1

119902

=

1

(39)




120573

119899

sum

119896=0

119906⋆

119896⊙ V⋆119896

le (120573

119899

sum

119896=0

119906⋆

119896

2120573

times120573

119899

sum

119896=0

V⋆119896

2120573

)

(12)120573

(40)




(120573

119899

sum

119896=0

119906⋆

119896oplus V⋆119896

119901120573

)

(1119901)120573

le (120573

119899

sum

119896=0

119906⋆

119896

119901120573

)

(1119901)120573

+ (120573

119899

sum

119896=0

V⋆119896

119901120573

)

(1119901)120573

(41)



120573

119899

sum

119896=0

119906⋆

119896oplus V⋆119896

119901120573

le120573

119899

sum

119896=0

119906⋆

119896

times

119906⋆

119896oplus V⋆119896

(119901minus1)120573

+120573

119899

sum

119896=0

V⋆119896

times

119906⋆

119896oplus V⋆119896

(119901minus1)120573

(42)


120573

119899

sum

119896=0

119906⋆

119896oplus V⋆119896

119901120573

le[

[

(120573

119899

sum

119896=0

119906⋆

119896

119901120573

)

(1119901)120573

+ (120573

119899

sum

119896=0

V⋆119896

119901120573

)

(1119901)120573

]

]

times (120573

119899

sum

119896=0

119906⋆

119896oplus V⋆119896

(119902119901minus119902)120573

)

(1119902)120573

(43)





120573sube



(N(PN)1) 119892⋆

(119909) =

0 if 119909 = 120579⋆ (120579⋆ = (0⋆ 0⋆ ))

(N(PN)2) 119892⋆

(⊖119909) = 119892⋆


⋆

(119909 oplus 119910)

le 119892⋆

(119909)

+ 119892⋆

(119910)


that is 120582⋆ = (

120582

120582 ) with 120582⋆

119899

119889⋆



119899

119892⋆

997888997888rarr 119909 then

120582⋆

119899⊙ 119909119899

119892⋆

997888997888rarr 120582⋆

⊙ 119909 as 119899 rarr infin



(BG(PN)3) 119892⋆

(119909 oplus 119910) le 119892⋆

(119909)119892⋆

(119910)



119896=


119896times 1199011015840

119896= 119901119896+ 1199011015840





119888⋆

0 and ℓ⋆



119901over


infin 119888⋆

119888⋆

0 and ℓ⋆



infin(119901) 119888⋆

(119901) 119888⋆

0 (119901) ℓ⋆

(119901) 119887119904⋆

(119901) 119888119904⋆

(119901) 119888119904⋆

0 (119901) 119887V⋆



ℓ⋆

infin(119901) = 119909= (119909

119896) isin 120596⋆

sup119896isinN

119909119896

(119901119896)120573

ltinfin

= 119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆

sup119896isinN

120573radic1205762

119896+ 120575

2119896

119901119896

ltinfin

119888⋆

(119901) = 119909= (119909119896) isin 120596⋆



119909119896⊖ 119897

(119901119896)120573

= 120579⋆

= 119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆

exist 119897 = (

ℓ

ℓ ) isinC⋆


120573(ℓradic2 (1205762119896+ 120575

2119896))

119901119896

=

0

119888⋆

0 (119901) = 119909= (119909119896) isin 120596⋆


119909119896

(119901119896)120573

= 120579⋆

= 119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆


120573radic1205762

119896+ 120575

2119896

119901119896

ltinfin


ℓ⋆

(119901) = 119909= (119909119896) isin 120596⋆

120573sum

119896

119909119896

(119901119896)120573

ltinfin

(0 lt 119901119896lt infin)

= 119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆

120573sum

119896

radic1205762

119896+ 120575

2119896

119901119896

ltinfin

(44)


(119901) 119888119904⋆

0 (119901) 119887V⋆



119887119904⋆

(119901) =

119909= (119909119896) isin 120596⋆

(⋆

119896

sum

119895=0119909119895)isinℓ⋆

infin(119901)

=

119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆

sup119896isinN

120573

radic(

119896

sum

119895=0120576119895)

2

+ (

119896

sum

119895=0120575119895)

2119901119896

ltinfin

119888119904⋆

(119901) =

119909= (119909119896) isin 120596⋆

(⋆

119896

sum

119895=0119909119895)isin 119888⋆

(119901)

=

119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆


120573

radic(

119896

sum

119895=0120576119895minus ℓ)

2

+ (

119896

sum

119895=0120575119895minus ℓ)

2119901119896

ltinfin

(ℓ = (

ℓ

ℓ ) isin C⋆

)

119888119904⋆

0 (119901) = 119909= (119909119896) isin 120596⋆

(⋆

119899

sum

119896=0119909119896)isin 119888⋆

0 (119901)

119887V⋆ (119901) = 119909= (119909119896) isin 120596⋆

120573

infin

sum

119896=0

Δ119909119896

(119901119896)120573

ltinfin

(Δ119909119896= 119909119896⊖ 119909119896minus1 119909minus1 = 120579

⋆

)

= 119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆

120573

infin

sum

119896=0


2+ (120575119896minus 120575119896minus1)

2119901119896

ltinfin

119887V⋆infin(119901) = 119909= (119909

119896) isin 120596⋆

sup119896isinN

Δ119909119896

(119901119896)120573

ltinfin

(45)


(119901)

119888⋆

0 (119901) ℓ⋆

(119901) 119887119904⋆

(119901) 119888119904⋆

(119901) 119888119904⋆

0 (119901) 119887V⋆

(119901) and 119887V⋆infin(119901) is a

⋆-linear space



119892⋆

(119909) = sup119896isinN

119909119896

(119901119896119872)120573

119892⋆

(119909) = (120573sum

119896

119909119896

(119901119896)120573

)

(1119872)120573

(46)

Then 119888⋆

(119901) and 119888⋆


119896isin ℓ⋆


infin(119901) and

ℓ⋆


119896gt 0


119892⋆

1 (119909) = sup119896isinN

⋆

119896

sum

119895=0119909119895

(119901119896119872)120573

iff inf 119901119896gt 0 (47)



119892⋆

2 (119909) = (120573sum

119896

Δ119909119896

(119901119896)120573

)

(1119872)120573

119892⋆

3 (119909) = sup119896isinN

Δ119909119896

(1119872)120573

iff inf 119901119896gt 0

(48)







(⋆sum

119896

Δ (119906119896oplus119909119896)

(119901119896)120573

)

(1119872)120573

le (⋆sum

119896

Δ119906119896

(119901119896)120573

)

(1119872)120573

+ (⋆sum

119896

Δ119909119896

(119901119896)120573

)

(1119872)120573

(49)

and the condition

120582⋆

(119901119896)120573

le max

1

120582⋆

119872120573

(50)


= (

120582


0 and 119892⋆

(⊖119909) = 119892⋆


lemax

1

120582⋆

times 119892⋆



119899) be any


rarr 120582⋆ with


(119909119899

)

le 119892⋆

(119909)

+ 119892⋆

(119909119899


119892⋆

(120582⋆

119899⊙119909119899

⊖120582⋆

⊙119909)

= (120573sum

119896

Δ (120582⋆

119899⊙119909(119899)

119896⊖120582⋆

⊙119909119896)

(119901119896)120573

)

(1119872)120573

le

120582⋆

119899⊖120582⋆

times 119892⋆

(119909119899

)

+

120582⋆

times 119892⋆

(119909119899

⊖119909)

(51)

which tends to



119894= 119909(119894)

0 119909(119894)

1 119909(119894)


gt


119892⋆

(119909119894

⊖119909119895

)

lt

120598 forall119894 119895 ge 1198990 (120598) (52)


Δ [(119909119894

)119896

⊖ (119909119895

)119896

]

le (120573sum

119896

Δ [(119909119894

)119896

⊖ (119909119895

)119896

]

(119901119896)120573

)

(1119872)120573

lt 120598

forall119894 119895 ge 1198990 (120598)

(53)


1)119896 (Δ119909

2)119896 is a




120573

119898

sum

119896=0

Δ [(119909119894

)119896

⊖ (119909119895

)119896

]

(119901119896)120573

le 119892⋆

(119909119894

⊖119909119895

)

119872120573

lt 120598119872120573

(54)



(120573sum

119896

(Δ119909)119896

(119901119896)120573

)

(1119872)120573

le 119892⋆

(119909119894

⊖119909)

+ 119892⋆

(119909119894

)

le 120598

+ 119892⋆

(119909119894

)

(55)



119894



infin(119901) 119888⋆

(119901) 119888⋆

0 (119901) ℓ⋆

(119901) 119887119904⋆

(119901) 119888119904⋆

(119901) 119888119904⋆

0 (119901) 119887V⋆





119901of 119901-






infin 119888 1198880 ℓ119901 119887119904 119888119904

1198881199040 119887V 119887Vinfin(ii) 120583(119901) sube 120583

⋆

(119901) and ℓ⋆1 (119901) sube 119888119904⋆

(119901) sube 119888⋆

0 (119901) sube 119888⋆

(119901) sube

ℓ⋆

infin(119901) sube 120596

⋆



⋆

(119901)120572

120582⋆

(119901)120573 and 120582⋆(119901)120574 as follows

120582⋆

(119901)

120572

= 119908= (119908119896) isin 120596⋆

119908 ⊙ 119911 = (119908119896⊙ 119911119896)

isin ℓ⋆

1 (119901) forall119911 = (119911119896) isin 120582⋆

(119901)

120582⋆

(119901)

120573

= 119908= (119908119896) isin 120596⋆

119908 ⊙ 119911 = (119908119896⊙ 119911119896)

isin 119888119904⋆

(119901) forall119911 = (119911119896) isin 120582⋆

(119901)

120582⋆

(119901)

120574

= 119908= (119908119896) isin 120596⋆

119908 ⊙ 119911 = (119908119896⊙ 119911119896)

isin 119887119904⋆

(119901) forall119911 = (119911119896) isin 120582⋆

(119901)

(56)




Theorem 25 Let 0 = 120582⋆




(b) If 120582⋆(119901) sub 120583⋆

(119901) sub 120596⋆ then 120583⋆(119901)120573 sub 120582

⋆

(119901)120573

(c) 120582⋆ sub 120582⋆

120573120573

= (120582⋆

120573

)120573

(d) 119888⋆0 (119901)120573

= 119888⋆

(119901)120573

= ℓ⋆

infin(119901)120573

= ℓ⋆

1 (119901)

(e) ℓ⋆1 (119901)120573

= ℓ⋆

infin(119901)


119896) 119898 = (119898

119896)

and 119899 = (119899119896) isin 120582

⋆

(119901)120573



(119901)120573 for119898 119899 isin

120582⋆


⋆

(119901) Then (119898119896⊙ 119897119896) isin

119888119904⋆

(119901) and (119899119896⊙ 119897119896) isin 119888119904⋆


((119898119896oplus 119899119896) ⊙ 119897119896) = (119898

119896⊙ 119897119896) oplus (119899119896⊙ 119897119896) isin 119888119904

⋆

(119901)


(119901)

(57)

Hence 119898 oplus 119899 isin 120582⋆


120582⋆

(119901)120573 for any 119905 isin C⋆ and 119908 = (119908

119896) isin 120582

⋆

(119901)120573

Since (119908119896⊙ 119897119896) isin 119888119904

⋆

(119901) for all 119897 isin 120582⋆


119896⊙119908119896) ⊙ 119897119896) = 119905119896⊙ (119908119896⊙ 119897119896) isin

119888119904⋆

(119901) for all 119897 isin 120582⋆

(119901) we get 119905 ⊙ 119908 isin 120582⋆

(119901)120573



infin(119901)120573

sub 119888⋆

(119901)120573

sub 119888⋆

0 (119901)120573 by


ℓ⋆

infin(119901)120573 and 119888

⋆

0 (119901)120573

sub ℓ⋆


(119908119896) isin ℓ

⋆

1 (119901) and 119911 = (119911119896) isin ℓ

⋆



⋆sum

119896

1003817100381710038171003817119908119896⊙ 119911119896

1003817100381710038171003817

(119901119896)120573

le sup119896isinN

119911119896

(1119872)120573

times (120573sum

119896

119908119896

(119901119896)120573

)

(1119872)120573

le 119892⋆

(119911)

times 119892⋆

(119908) lt infin

(58)


⋆


⋆



ℓ⋆


⋆

0 (119901) with 119910 ⊙ 119909 notin 119888119904⋆


⋆



and⋆sum

119899119903minus1119896=119899119903minus1

119910119896

(119901119896)120573


119896) isin 119888

⋆

0 (119901)by 119909119896= (sgn⋆119910



sgn⋆ (119910) =

119910 ⊘

119910

119910 = 120579⋆

120579⋆

119910 = 120579⋆

(59)



⋆

119899119903minus1

sum

119896=119899119903minus1

(119910119896⊙119909119896)

(119901119896)120573

=⋆

119899119903minus1

sum

119896=119899119903minus1

[119910119896⊙ (sgn⋆119910

119896⊘ 119903)]

(119901119896)120573

=

1119903(119901119896)120573

⊙⋆

119899119903minus1

sum

119896=119899119903minus1

119910119896

(119901119896)120573

ge

1 (sup119901119896le 1)

(60)

Therefore 119910 ⊙ 119909 notin 119888119904⋆

(119901) and thus 119910 notin 119888⋆

0 (119901)120573

Hence 119888⋆0 (119901)120573

sub ℓ⋆



infin(119901) sub

(ℓ⋆

infin(119901)120573

)120573

= ℓ⋆

1 (119901)120573 since ℓ

⋆

infin(119901)120573

= ℓ⋆


119899) isin

ℓ⋆

1 (119901)120573

ℓ⋆




number (119896 + 1)2120573 such that

119908119899119896

(119901119896)120573



119909119899= (sgn⋆(119908

119899119896

) ⊘ (119896 + 1)2120573) if 119899 = 119899119896 and 120579

⋆


1 (119901) However

⋆sum

119899

(119908119899⊙119909119899)

(119901119896)120573

=⋆sum

119896

119908119899119896

(119901119896)120573

(119896 + 1)(2119901119896)120573= infin

(61)



tion and ℓ⋆

1 (119901)120573

sub ℓ⋆


proof


120573 120574)


(a) 120582⋆(119901)120572 sub 120582⋆

(119901)120573

sub 120582⋆

(119901)120574


120582⋆


(b) ℓ⋆1 (119901)120577

= ℓ⋆


infin(119901)120577

= ℓ⋆

1 (119901)

(c) If 120582⋆(119901) sub 120583⋆

(119901) sub 120596⋆ then 120583⋆(119901)120577 sub 120582

⋆

(119901)120577




Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119909119896

119910119896) of


⋆

infin

sum

119896=0119911⋆

119896= 119911⋆

0 oplus 119911⋆

1 oplus 119911⋆



= (120572

infin

sum

119896=0

119909119896120573

infin

sum

119896=0

119910119896)

= (120572

infin

sum

119896=0120572minus1119909119896 120573

infin

sum

119896=0120573minus1119909119896)

isin C⋆

(14)


119899=⋆sum119899

119896=0 119911⋆



=⋆suminfin

119899=0 119911⋆





sdot



sdot


sdot


(NN1)

119909

=

0 hArr 119909 = 120579⋆ (120579⋆ = (

0

0 ))

(NN2)

120582 ⊙ 119909

=

| 120582

|

times

119909

(NN3)

119909 oplus 119910

le

119909

+

119910


sdot



119909 ⊖ 119910



119911⋆

= 119889⋆

(119911⋆

120579⋆

) = (120580 (

119886

minus

0 )2120572

+ (

119887

minus

0 )2120573

)

(12)120573

= 120573 radic1198862+ 119887

2

(15)

where 119911⋆ = (

119886

119887 ) and 120579⋆ = (

0



(N(MC)1)

119909

= 1 hArr 119909 = 120579⋆ (120579⋆ = (0 1))

(N(MC)2)

120582 ⊙ 119909

=

119909

|120582|

(N(MC)3)

119909 oplus 119910

le

119909

119910


sdot



119886


minus1(

119887 )) =

(

119886

minus


Re (119911⋆) = Re (119911⋆) = (119911⋆

oplus 119911⋆

)

2 =

119886

Im (119911⋆

) =

minus Im (119911⋆

) = (119911⋆

⊖ 119911⋆

)

2 =

119887

(16)


(i) Let 119911⋆1 = (

119886 1

119887 1) 119911⋆

2 = (

119886 2


119911⋆

1 ⊘ 119911⋆

2

= (120572

(11988611198862 + 11988711198872)

(1198862

2+ 1198872

2)

120573

(11988711198862 minus 11988611198872)

(1198862

2+ 1198872

2)

)

(17)


(

119886


⊙ 119911⋆=

119911⋆

2120573

holds Really

119911⋆

⊙ 119911⋆= (

119886

119887 ) ⊙ (

119886

minus

119887 ) = (120572 1198862+ 119887

2 120573 (0))

= 120573 1198862+ 119887

2 = 120573 (120573

minus1120573radic1198862+ 119887

2)

2

=

119911⋆

2120573

(18)



sdot

defined by

119911⋆

= (120580(

119886 )2120572

+

119887

2120573

)(12)120573 119911⋆ = (

119886

119887 ) isin

C⋆


119911⋆

= 119889⋆

(119911⋆

120579⋆

)



norm sdot ⋆

infindefined by

119911⋆


119911⋆

119896

119911 = (119911⋆

119896) isin 120582⋆

120582 isin ℓinfin 119888 1198880 (19)


⋆

119901

defined by

119911⋆

119901= (⋆sum

119896

119911⋆

119896

119901120573

)

(1119901)120573

119901 ge 1 119911 = (119911⋆

119896) isin ℓ⋆

119901 (20)




⋆


119909⋆

119887119904= 119909

⋆

119888119904= sup119899isinN

⋆

119899

sum

119896=0119909119896

119909 = (119909119896) isin 120583⋆

120583 isin 119887119904 119888119904 1198881199040

(21)


infinare Banach


119909⋆

119887V = ⋆sum119896

(Δ1015840

119909)119896

119909⋆

119887V119901

= (⋆sum

119896

(Δ119909)119896

119901120573

)

(1119901)120573

119909⋆

119887Vinfin

= sup119896isinN

(Δ119909)119896

(22)

where (Δ1015840119909)119896= (119909119896⊖119909119896+1) and (Δ119909)119896 = (119909

119896⊖119909119896minus1) 119909minus1 = 120579

⋆



119894 120583⋆

rarr C⋆ defined by119901119894(119909) = 119909







119899) such that



1003817100381710038171003817

⋆

= 120579⋆ (23)




⋆sum119896120585119896⊙119887119896The


Let 119890 = (119890119896) and 119890(119899) = (119890

(119899)



119896= 120575⋆

119899119896 where 120575⋆

119899119896denotes


120575⋆

119899119896=

1⋆ 119899 = 119896

120579⋆

119899 = 119896

(24)




119909 = 120585 ⊙ 119890 oplus⋆sum

119896

(119909119896⊖ 120585) ⊙ 119890

(119896)


119909119896= 120585

(25)



⋆

infin




119879119909 = (⋆sum119896



(119910119896) isin ℓ⋆


119896) by 119909

119896= 119910119896⊖119910119896minus1



sup119896isinN

⋆

119896

sum

119895=0119909119895

= sup119896isinN

⋆

119896

sum

119895=0(119910119895⊖119910119895minus1)

= sup119896isinN

119910119896

=

1003817100381710038171003817119910

1003817100381710038171003817

⋆

infinlt infin

(26)







infin 119888 1198880 ℓ119901 119887119904 119888119904 1198881199040

119887V 119887Vinfin

(ii) ℓ⋆1 sube 119888119904⋆

sube 119888⋆

0 sube 119888⋆

sube ℓ⋆


119888⋆ where 119887V⋆0 = 119887V⋆ cap 119888⋆0


⋆

0 sube 119888 sube 119888⋆








every 120576

gt


(119911119896 ℓ)



119911119896

le

119911119896⊖ ℓ

+

ℓ

le 120598

+

ℓ

(27)


minus1 119911119896

le 120573minus1(120598

+

ℓ



+

ℓ





similarly



⋆ 119888119904⋆ 119887V⋆119901 and ℓ⋆

119901are


respectively



119906

times V

le

119906119901120572

119901

+

V119902120572

119902

(28)



120572minus1(119906) 120572minus1(V) le

(120572minus1(119906))

119901

119901

+

(120572minus1(V))119902

119902

(29)


0 or V =



119891 (119905) = (120580 (119905))120582120573

minus

120582

times 120580 (119905) = 120573 (120572minus1(119905))

120582

minus120582120572minus1(119905) (30)



119891⋆

(119905) = 120573

(120573minus1∘ 119891)

1015840

(119905)

(120572minus1)

1015840

(119905)

= 120573

((120572minus1)

120582


1015840

(119905)

(120572minus1)

1015840

(119905)

= 120573120582 (120572minus1(119905))


(31)


⋆

(119905) =



119891⋆⋆

(119905) = 120573

(120573minus1119891⋆

(119905))

1015840

(120572minus1)

1015840

(119905)

= 120573120582 (120582 minus 1) (120572minus1 (119905))120582

(32)


1 ) = 120573120582(120582 minus 1)

lt


1 that is 119891(


[120572minus1(119905)]

120582


120572 (33)

Now taking 119905 = (119906)119901120572

(V)119902120572 = 120572(120572minus1(119906))119901

(120572minus1(V))119902 and

120582 = 1119901 in (33) we get

(

(120572minus1(119906))

119901

(120572minus1(V))119902

)

1119901

minus

1119901

(120572minus1(119906))

119901

(120572minus1(V))119902

le 1minus 1119901

120572minus1(119906)

120572minus1(V)119902119901

(120572minus1(V))119902

le (1minus 1119901

+

1119901

120572minus1(119906)119901

120572minus1(V)119902

)(120572minus1(V))119902

(34)



119896 V⋆119896isin C⋆ for 119896 isin 0 1 2 119899


120573

119899

sum

119896=0

119906⋆

119896⊙ V⋆119896

le (120573

119899

sum

119896=0

119906⋆

119896

119901120573

)

(1119901)120573

times (120573

119899

sum

119896=0

V⋆119896

119902120573

)

(1119902)120573

(35)


= (

0

0 ) orV = (

0


Let

120576 = (120573

119899

sum

119896=0

119906⋆

119896

119901120573

)

(1119901)120573

120575 = (120573

119899

sum

119896=0

V⋆119896

119902120573

)

(1119902)120573

(36)

and 119904⋆119896= 119906⋆

119896⊘ 120576 119905⋆119896= V⋆119896⊘ 120575 where 120576 = (

0

120576 ) isin R120573sube C⋆

and 120575 = (

0

120575 ) isin R120573


119904⋆

119896⊙ 119905⋆

119896

=

119904⋆

119896

times

119905⋆

119896

le

119904⋆

119896

119901120573

119901

+

119905⋆

119896

119902120573

119902

(37)

which implies that

120573

119899

sum

119896=0

119904⋆

119896⊙ 119905⋆

119896

le120573

119899

sum

119896=0

119904⋆

119896

119901120573

119901

+120573

119899

sum

119896=0

119905⋆

119896

119902120573

119902

(38)


120573

119899

sum

119896=0

(119906⋆

119896⊘ 120576) ⊙ (V⋆

119896⊘ 120575)

le120573

119899

sum

119896=0

119906⋆

119896

119901120573

120576

times

119901

+120573

119899

sum

119896=0

V⋆119896

119902120573

120575

times

119902

=

1

119901

+

1

119902

=

1

(39)




120573

119899

sum

119896=0

119906⋆

119896⊙ V⋆119896

le (120573

119899

sum

119896=0

119906⋆

119896

2120573

times120573

119899

sum

119896=0

V⋆119896

2120573

)

(12)120573

(40)




(120573

119899

sum

119896=0

119906⋆

119896oplus V⋆119896

119901120573

)

(1119901)120573

le (120573

119899

sum

119896=0

119906⋆

119896

119901120573

)

(1119901)120573

+ (120573

119899

sum

119896=0

V⋆119896

119901120573

)

(1119901)120573

(41)



120573

119899

sum

119896=0

119906⋆

119896oplus V⋆119896

119901120573

le120573

119899

sum

119896=0

119906⋆

119896

times

119906⋆

119896oplus V⋆119896

(119901minus1)120573

+120573

119899

sum

119896=0

V⋆119896

times

119906⋆

119896oplus V⋆119896

(119901minus1)120573

(42)


120573

119899

sum

119896=0

119906⋆

119896oplus V⋆119896

119901120573

le[

[

(120573

119899

sum

119896=0

119906⋆

119896

119901120573

)

(1119901)120573

+ (120573

119899

sum

119896=0

V⋆119896

119901120573

)

(1119901)120573

]

]

times (120573

119899

sum

119896=0

119906⋆

119896oplus V⋆119896

(119902119901minus119902)120573

)

(1119902)120573

(43)





120573sube



(N(PN)1) 119892⋆

(119909) =

0 if 119909 = 120579⋆ (120579⋆ = (0⋆ 0⋆ ))

(N(PN)2) 119892⋆

(⊖119909) = 119892⋆


⋆

(119909 oplus 119910)

le 119892⋆

(119909)

+ 119892⋆

(119910)


that is 120582⋆ = (

120582

120582 ) with 120582⋆

119899

119889⋆



119899

119892⋆

997888997888rarr 119909 then

120582⋆

119899⊙ 119909119899

119892⋆

997888997888rarr 120582⋆

⊙ 119909 as 119899 rarr infin



(BG(PN)3) 119892⋆

(119909 oplus 119910) le 119892⋆

(119909)119892⋆

(119910)



119896=


119896times 1199011015840

119896= 119901119896+ 1199011015840





119888⋆

0 and ℓ⋆



119901over


infin 119888⋆

119888⋆

0 and ℓ⋆



infin(119901) 119888⋆

(119901) 119888⋆

0 (119901) ℓ⋆

(119901) 119887119904⋆

(119901) 119888119904⋆

(119901) 119888119904⋆

0 (119901) 119887V⋆



ℓ⋆

infin(119901) = 119909= (119909

119896) isin 120596⋆

sup119896isinN

119909119896

(119901119896)120573

ltinfin

= 119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆

sup119896isinN

120573radic1205762

119896+ 120575

2119896

119901119896

ltinfin

119888⋆

(119901) = 119909= (119909119896) isin 120596⋆



119909119896⊖ 119897

(119901119896)120573

= 120579⋆

= 119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆

exist 119897 = (

ℓ

ℓ ) isinC⋆


120573(ℓradic2 (1205762119896+ 120575

2119896))

119901119896

=

0

119888⋆

0 (119901) = 119909= (119909119896) isin 120596⋆


119909119896

(119901119896)120573

= 120579⋆

= 119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆


120573radic1205762

119896+ 120575

2119896

119901119896

ltinfin


ℓ⋆

(119901) = 119909= (119909119896) isin 120596⋆

120573sum

119896

119909119896

(119901119896)120573

ltinfin

(0 lt 119901119896lt infin)

= 119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆

120573sum

119896

radic1205762

119896+ 120575

2119896

119901119896

ltinfin

(44)


(119901) 119888119904⋆

0 (119901) 119887V⋆



119887119904⋆

(119901) =

119909= (119909119896) isin 120596⋆

(⋆

119896

sum

119895=0119909119895)isinℓ⋆

infin(119901)

=

119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆

sup119896isinN

120573

radic(

119896

sum

119895=0120576119895)

2

+ (

119896

sum

119895=0120575119895)

2119901119896

ltinfin

119888119904⋆

(119901) =

119909= (119909119896) isin 120596⋆

(⋆

119896

sum

119895=0119909119895)isin 119888⋆

(119901)

=

119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆


120573

radic(

119896

sum

119895=0120576119895minus ℓ)

2

+ (

119896

sum

119895=0120575119895minus ℓ)

2119901119896

ltinfin

(ℓ = (

ℓ

ℓ ) isin C⋆

)

119888119904⋆

0 (119901) = 119909= (119909119896) isin 120596⋆

(⋆

119899

sum

119896=0119909119896)isin 119888⋆

0 (119901)

119887V⋆ (119901) = 119909= (119909119896) isin 120596⋆

120573

infin

sum

119896=0

Δ119909119896

(119901119896)120573

ltinfin

(Δ119909119896= 119909119896⊖ 119909119896minus1 119909minus1 = 120579

⋆

)

= 119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆

120573

infin

sum

119896=0


2+ (120575119896minus 120575119896minus1)

2119901119896

ltinfin

119887V⋆infin(119901) = 119909= (119909

119896) isin 120596⋆

sup119896isinN

Δ119909119896

(119901119896)120573

ltinfin

(45)


(119901)

119888⋆

0 (119901) ℓ⋆

(119901) 119887119904⋆

(119901) 119888119904⋆

(119901) 119888119904⋆

0 (119901) 119887V⋆

(119901) and 119887V⋆infin(119901) is a

⋆-linear space



119892⋆

(119909) = sup119896isinN

119909119896

(119901119896119872)120573

119892⋆

(119909) = (120573sum

119896

119909119896

(119901119896)120573

)

(1119872)120573

(46)

Then 119888⋆

(119901) and 119888⋆


119896isin ℓ⋆


infin(119901) and

ℓ⋆


119896gt 0


119892⋆

1 (119909) = sup119896isinN

⋆

119896

sum

119895=0119909119895

(119901119896119872)120573

iff inf 119901119896gt 0 (47)



119892⋆

2 (119909) = (120573sum

119896

Δ119909119896

(119901119896)120573

)

(1119872)120573

119892⋆

3 (119909) = sup119896isinN

Δ119909119896

(1119872)120573

iff inf 119901119896gt 0

(48)







(⋆sum

119896

Δ (119906119896oplus119909119896)

(119901119896)120573

)

(1119872)120573

le (⋆sum

119896

Δ119906119896

(119901119896)120573

)

(1119872)120573

+ (⋆sum

119896

Δ119909119896

(119901119896)120573

)

(1119872)120573

(49)

and the condition

120582⋆

(119901119896)120573

le max

1

120582⋆

119872120573

(50)


= (

120582


0 and 119892⋆

(⊖119909) = 119892⋆


lemax

1

120582⋆

times 119892⋆



119899) be any


rarr 120582⋆ with


(119909119899

)

le 119892⋆

(119909)

+ 119892⋆

(119909119899


119892⋆

(120582⋆

119899⊙119909119899

⊖120582⋆

⊙119909)

= (120573sum

119896

Δ (120582⋆

119899⊙119909(119899)

119896⊖120582⋆

⊙119909119896)

(119901119896)120573

)

(1119872)120573

le

120582⋆

119899⊖120582⋆

times 119892⋆

(119909119899

)

+

120582⋆

times 119892⋆

(119909119899

⊖119909)

(51)

which tends to



119894= 119909(119894)

0 119909(119894)

1 119909(119894)


gt


119892⋆

(119909119894

⊖119909119895

)

lt

120598 forall119894 119895 ge 1198990 (120598) (52)


Δ [(119909119894

)119896

⊖ (119909119895

)119896

]

le (120573sum

119896

Δ [(119909119894

)119896

⊖ (119909119895

)119896

]

(119901119896)120573

)

(1119872)120573

lt 120598

forall119894 119895 ge 1198990 (120598)

(53)


1)119896 (Δ119909

2)119896 is a




120573

119898

sum

119896=0

Δ [(119909119894

)119896

⊖ (119909119895

)119896

]

(119901119896)120573

le 119892⋆

(119909119894

⊖119909119895

)

119872120573

lt 120598119872120573

(54)



(120573sum

119896

(Δ119909)119896

(119901119896)120573

)

(1119872)120573

le 119892⋆

(119909119894

⊖119909)

+ 119892⋆

(119909119894

)

le 120598

+ 119892⋆

(119909119894

)

(55)



119894



infin(119901) 119888⋆

(119901) 119888⋆

0 (119901) ℓ⋆

(119901) 119887119904⋆

(119901) 119888119904⋆

(119901) 119888119904⋆

0 (119901) 119887V⋆





119901of 119901-






infin 119888 1198880 ℓ119901 119887119904 119888119904

1198881199040 119887V 119887Vinfin(ii) 120583(119901) sube 120583

⋆

(119901) and ℓ⋆1 (119901) sube 119888119904⋆

(119901) sube 119888⋆

0 (119901) sube 119888⋆

(119901) sube

ℓ⋆

infin(119901) sube 120596

⋆



⋆

(119901)120572

120582⋆

(119901)120573 and 120582⋆(119901)120574 as follows

120582⋆

(119901)

120572

= 119908= (119908119896) isin 120596⋆

119908 ⊙ 119911 = (119908119896⊙ 119911119896)

isin ℓ⋆

1 (119901) forall119911 = (119911119896) isin 120582⋆

(119901)

120582⋆

(119901)

120573

= 119908= (119908119896) isin 120596⋆

119908 ⊙ 119911 = (119908119896⊙ 119911119896)

isin 119888119904⋆

(119901) forall119911 = (119911119896) isin 120582⋆

(119901)

120582⋆

(119901)

120574

= 119908= (119908119896) isin 120596⋆

119908 ⊙ 119911 = (119908119896⊙ 119911119896)

isin 119887119904⋆

(119901) forall119911 = (119911119896) isin 120582⋆

(119901)

(56)




Theorem 25 Let 0 = 120582⋆




(b) If 120582⋆(119901) sub 120583⋆

(119901) sub 120596⋆ then 120583⋆(119901)120573 sub 120582

⋆

(119901)120573

(c) 120582⋆ sub 120582⋆

120573120573

= (120582⋆

120573

)120573

(d) 119888⋆0 (119901)120573

= 119888⋆

(119901)120573

= ℓ⋆

infin(119901)120573

= ℓ⋆

1 (119901)

(e) ℓ⋆1 (119901)120573

= ℓ⋆

infin(119901)


119896) 119898 = (119898

119896)

and 119899 = (119899119896) isin 120582

⋆

(119901)120573



(119901)120573 for119898 119899 isin

120582⋆


⋆

(119901) Then (119898119896⊙ 119897119896) isin

119888119904⋆

(119901) and (119899119896⊙ 119897119896) isin 119888119904⋆


((119898119896oplus 119899119896) ⊙ 119897119896) = (119898

119896⊙ 119897119896) oplus (119899119896⊙ 119897119896) isin 119888119904

⋆

(119901)


(119901)

(57)

Hence 119898 oplus 119899 isin 120582⋆


120582⋆

(119901)120573 for any 119905 isin C⋆ and 119908 = (119908

119896) isin 120582

⋆

(119901)120573

Since (119908119896⊙ 119897119896) isin 119888119904

⋆

(119901) for all 119897 isin 120582⋆


119896⊙119908119896) ⊙ 119897119896) = 119905119896⊙ (119908119896⊙ 119897119896) isin

119888119904⋆

(119901) for all 119897 isin 120582⋆

(119901) we get 119905 ⊙ 119908 isin 120582⋆

(119901)120573



infin(119901)120573

sub 119888⋆

(119901)120573

sub 119888⋆

0 (119901)120573 by


ℓ⋆

infin(119901)120573 and 119888

⋆

0 (119901)120573

sub ℓ⋆


(119908119896) isin ℓ

⋆

1 (119901) and 119911 = (119911119896) isin ℓ

⋆



⋆sum

119896

1003817100381710038171003817119908119896⊙ 119911119896

1003817100381710038171003817

(119901119896)120573

le sup119896isinN

119911119896

(1119872)120573

times (120573sum

119896

119908119896

(119901119896)120573

)

(1119872)120573

le 119892⋆

(119911)

times 119892⋆

(119908) lt infin

(58)


⋆


⋆



ℓ⋆


⋆

0 (119901) with 119910 ⊙ 119909 notin 119888119904⋆


⋆



and⋆sum

119899119903minus1119896=119899119903minus1

119910119896

(119901119896)120573


119896) isin 119888

⋆

0 (119901)by 119909119896= (sgn⋆119910



sgn⋆ (119910) =

119910 ⊘

119910

119910 = 120579⋆

120579⋆

119910 = 120579⋆

(59)



⋆

119899119903minus1

sum

119896=119899119903minus1

(119910119896⊙119909119896)

(119901119896)120573

=⋆

119899119903minus1

sum

119896=119899119903minus1

[119910119896⊙ (sgn⋆119910

119896⊘ 119903)]

(119901119896)120573

=

1119903(119901119896)120573

⊙⋆

119899119903minus1

sum

119896=119899119903minus1

119910119896

(119901119896)120573

ge

1 (sup119901119896le 1)

(60)

Therefore 119910 ⊙ 119909 notin 119888119904⋆

(119901) and thus 119910 notin 119888⋆

0 (119901)120573

Hence 119888⋆0 (119901)120573

sub ℓ⋆



infin(119901) sub

(ℓ⋆

infin(119901)120573

)120573

= ℓ⋆

1 (119901)120573 since ℓ

⋆

infin(119901)120573

= ℓ⋆


119899) isin

ℓ⋆

1 (119901)120573

ℓ⋆




number (119896 + 1)2120573 such that

119908119899119896

(119901119896)120573



119909119899= (sgn⋆(119908

119899119896

) ⊘ (119896 + 1)2120573) if 119899 = 119899119896 and 120579

⋆


1 (119901) However

⋆sum

119899

(119908119899⊙119909119899)

(119901119896)120573

=⋆sum

119896

119908119899119896

(119901119896)120573

(119896 + 1)(2119901119896)120573= infin

(61)



tion and ℓ⋆

1 (119901)120573

sub ℓ⋆


proof


120573 120574)


(a) 120582⋆(119901)120572 sub 120582⋆

(119901)120573

sub 120582⋆

(119901)120574


120582⋆


(b) ℓ⋆1 (119901)120577

= ℓ⋆


infin(119901)120577

= ℓ⋆

1 (119901)

(c) If 120582⋆(119901) sub 120583⋆

(119901) sub 120596⋆ then 120583⋆(119901)120577 sub 120582

⋆

(119901)120577




Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












⋆


119909⋆

119887119904= 119909

⋆

119888119904= sup119899isinN

⋆

119899

sum

119896=0119909119896

119909 = (119909119896) isin 120583⋆

120583 isin 119887119904 119888119904 1198881199040

(21)


infinare Banach


119909⋆

119887V = ⋆sum119896

(Δ1015840

119909)119896

119909⋆

119887V119901

= (⋆sum

119896

(Δ119909)119896

119901120573

)

(1119901)120573

119909⋆

119887Vinfin

= sup119896isinN

(Δ119909)119896

(22)

where (Δ1015840119909)119896= (119909119896⊖119909119896+1) and (Δ119909)119896 = (119909

119896⊖119909119896minus1) 119909minus1 = 120579

⋆



119894 120583⋆

rarr C⋆ defined by119901119894(119909) = 119909







119899) such that



1003817100381710038171003817

⋆

= 120579⋆ (23)




⋆sum119896120585119896⊙119887119896The


Let 119890 = (119890119896) and 119890(119899) = (119890

(119899)



119896= 120575⋆

119899119896 where 120575⋆

119899119896denotes


120575⋆

119899119896=

1⋆ 119899 = 119896

120579⋆

119899 = 119896

(24)




119909 = 120585 ⊙ 119890 oplus⋆sum

119896

(119909119896⊖ 120585) ⊙ 119890

(119896)


119909119896= 120585

(25)



⋆

infin




119879119909 = (⋆sum119896



(119910119896) isin ℓ⋆


119896) by 119909

119896= 119910119896⊖119910119896minus1



sup119896isinN

⋆

119896

sum

119895=0119909119895

= sup119896isinN

⋆

119896

sum

119895=0(119910119895⊖119910119895minus1)

= sup119896isinN

119910119896

=

1003817100381710038171003817119910

1003817100381710038171003817

⋆

infinlt infin

(26)







infin 119888 1198880 ℓ119901 119887119904 119888119904 1198881199040

119887V 119887Vinfin

(ii) ℓ⋆1 sube 119888119904⋆

sube 119888⋆

0 sube 119888⋆

sube ℓ⋆


119888⋆ where 119887V⋆0 = 119887V⋆ cap 119888⋆0


⋆

0 sube 119888 sube 119888⋆








every 120576

gt


(119911119896 ℓ)



119911119896

le

119911119896⊖ ℓ

+

ℓ

le 120598

+

ℓ

(27)


minus1 119911119896

le 120573minus1(120598

+

ℓ



+

ℓ





similarly



⋆ 119888119904⋆ 119887V⋆119901 and ℓ⋆

119901are


respectively



119906

times V

le

119906119901120572

119901

+

V119902120572

119902

(28)



120572minus1(119906) 120572minus1(V) le

(120572minus1(119906))

119901

119901

+

(120572minus1(V))119902

119902

(29)


0 or V =



119891 (119905) = (120580 (119905))120582120573

minus

120582

times 120580 (119905) = 120573 (120572minus1(119905))

120582

minus120582120572minus1(119905) (30)



119891⋆

(119905) = 120573

(120573minus1∘ 119891)

1015840

(119905)

(120572minus1)

1015840

(119905)

= 120573

((120572minus1)

120582


1015840

(119905)

(120572minus1)

1015840

(119905)

= 120573120582 (120572minus1(119905))


(31)


⋆

(119905) =



119891⋆⋆

(119905) = 120573

(120573minus1119891⋆

(119905))

1015840

(120572minus1)

1015840

(119905)

= 120573120582 (120582 minus 1) (120572minus1 (119905))120582

(32)


1 ) = 120573120582(120582 minus 1)

lt


1 that is 119891(


[120572minus1(119905)]

120582


120572 (33)

Now taking 119905 = (119906)119901120572

(V)119902120572 = 120572(120572minus1(119906))119901

(120572minus1(V))119902 and

120582 = 1119901 in (33) we get

(

(120572minus1(119906))

119901

(120572minus1(V))119902

)

1119901

minus

1119901

(120572minus1(119906))

119901

(120572minus1(V))119902

le 1minus 1119901

120572minus1(119906)

120572minus1(V)119902119901

(120572minus1(V))119902

le (1minus 1119901

+

1119901

120572minus1(119906)119901

120572minus1(V)119902

)(120572minus1(V))119902

(34)



119896 V⋆119896isin C⋆ for 119896 isin 0 1 2 119899


120573

119899

sum

119896=0

119906⋆

119896⊙ V⋆119896

le (120573

119899

sum

119896=0

119906⋆

119896

119901120573

)

(1119901)120573

times (120573

119899

sum

119896=0

V⋆119896

119902120573

)

(1119902)120573

(35)


= (

0

0 ) orV = (

0


Let

120576 = (120573

119899

sum

119896=0

119906⋆

119896

119901120573

)

(1119901)120573

120575 = (120573

119899

sum

119896=0

V⋆119896

119902120573

)

(1119902)120573

(36)

and 119904⋆119896= 119906⋆

119896⊘ 120576 119905⋆119896= V⋆119896⊘ 120575 where 120576 = (

0

120576 ) isin R120573sube C⋆

and 120575 = (

0

120575 ) isin R120573


119904⋆

119896⊙ 119905⋆

119896

=

119904⋆

119896

times

119905⋆

119896

le

119904⋆

119896

119901120573

119901

+

119905⋆

119896

119902120573

119902

(37)

which implies that

120573

119899

sum

119896=0

119904⋆

119896⊙ 119905⋆

119896

le120573

119899

sum

119896=0

119904⋆

119896

119901120573

119901

+120573

119899

sum

119896=0

119905⋆

119896

119902120573

119902

(38)


120573

119899

sum

119896=0

(119906⋆

119896⊘ 120576) ⊙ (V⋆

119896⊘ 120575)

le120573

119899

sum

119896=0

119906⋆

119896

119901120573

120576

times

119901

+120573

119899

sum

119896=0

V⋆119896

119902120573

120575

times

119902

=

1

119901

+

1

119902

=

1

(39)




120573

119899

sum

119896=0

119906⋆

119896⊙ V⋆119896

le (120573

119899

sum

119896=0

119906⋆

119896

2120573

times120573

119899

sum

119896=0

V⋆119896

2120573

)

(12)120573

(40)




(120573

119899

sum

119896=0

119906⋆

119896oplus V⋆119896

119901120573

)

(1119901)120573

le (120573

119899

sum

119896=0

119906⋆

119896

119901120573

)

(1119901)120573

+ (120573

119899

sum

119896=0

V⋆119896

119901120573

)

(1119901)120573

(41)



120573

119899

sum

119896=0

119906⋆

119896oplus V⋆119896

119901120573

le120573

119899

sum

119896=0

119906⋆

119896

times

119906⋆

119896oplus V⋆119896

(119901minus1)120573

+120573

119899

sum

119896=0

V⋆119896

times

119906⋆

119896oplus V⋆119896

(119901minus1)120573

(42)


120573

119899

sum

119896=0

119906⋆

119896oplus V⋆119896

119901120573

le[

[

(120573

119899

sum

119896=0

119906⋆

119896

119901120573

)

(1119901)120573

+ (120573

119899

sum

119896=0

V⋆119896

119901120573

)

(1119901)120573

]

]

times (120573

119899

sum

119896=0

119906⋆

119896oplus V⋆119896

(119902119901minus119902)120573

)

(1119902)120573

(43)





120573sube



(N(PN)1) 119892⋆

(119909) =

0 if 119909 = 120579⋆ (120579⋆ = (0⋆ 0⋆ ))

(N(PN)2) 119892⋆

(⊖119909) = 119892⋆


⋆

(119909 oplus 119910)

le 119892⋆

(119909)

+ 119892⋆

(119910)


that is 120582⋆ = (

120582

120582 ) with 120582⋆

119899

119889⋆



119899

119892⋆

997888997888rarr 119909 then

120582⋆

119899⊙ 119909119899

119892⋆

997888997888rarr 120582⋆

⊙ 119909 as 119899 rarr infin



(BG(PN)3) 119892⋆

(119909 oplus 119910) le 119892⋆

(119909)119892⋆

(119910)



119896=


119896times 1199011015840

119896= 119901119896+ 1199011015840





119888⋆

0 and ℓ⋆



119901over


infin 119888⋆

119888⋆

0 and ℓ⋆



infin(119901) 119888⋆

(119901) 119888⋆

0 (119901) ℓ⋆

(119901) 119887119904⋆

(119901) 119888119904⋆

(119901) 119888119904⋆

0 (119901) 119887V⋆



ℓ⋆

infin(119901) = 119909= (119909

119896) isin 120596⋆

sup119896isinN

119909119896

(119901119896)120573

ltinfin

= 119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆

sup119896isinN

120573radic1205762

119896+ 120575

2119896

119901119896

ltinfin

119888⋆

(119901) = 119909= (119909119896) isin 120596⋆



119909119896⊖ 119897

(119901119896)120573

= 120579⋆

= 119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆

exist 119897 = (

ℓ

ℓ ) isinC⋆


120573(ℓradic2 (1205762119896+ 120575

2119896))

119901119896

=

0

119888⋆

0 (119901) = 119909= (119909119896) isin 120596⋆


119909119896

(119901119896)120573

= 120579⋆

= 119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆


120573radic1205762

119896+ 120575

2119896

119901119896

ltinfin


ℓ⋆

(119901) = 119909= (119909119896) isin 120596⋆

120573sum

119896

119909119896

(119901119896)120573

ltinfin

(0 lt 119901119896lt infin)

= 119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆

120573sum

119896

radic1205762

119896+ 120575

2119896

119901119896

ltinfin

(44)


(119901) 119888119904⋆

0 (119901) 119887V⋆



119887119904⋆

(119901) =

119909= (119909119896) isin 120596⋆

(⋆

119896

sum

119895=0119909119895)isinℓ⋆

infin(119901)

=

119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆

sup119896isinN

120573

radic(

119896

sum

119895=0120576119895)

2

+ (

119896

sum

119895=0120575119895)

2119901119896

ltinfin

119888119904⋆

(119901) =

119909= (119909119896) isin 120596⋆

(⋆

119896

sum

119895=0119909119895)isin 119888⋆

(119901)

=

119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆


120573

radic(

119896

sum

119895=0120576119895minus ℓ)

2

+ (

119896

sum

119895=0120575119895minus ℓ)

2119901119896

ltinfin

(ℓ = (

ℓ

ℓ ) isin C⋆

)

119888119904⋆

0 (119901) = 119909= (119909119896) isin 120596⋆

(⋆

119899

sum

119896=0119909119896)isin 119888⋆

0 (119901)

119887V⋆ (119901) = 119909= (119909119896) isin 120596⋆

120573

infin

sum

119896=0

Δ119909119896

(119901119896)120573

ltinfin

(Δ119909119896= 119909119896⊖ 119909119896minus1 119909minus1 = 120579

⋆

)

= 119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆

120573

infin

sum

119896=0


2+ (120575119896minus 120575119896minus1)

2119901119896

ltinfin

119887V⋆infin(119901) = 119909= (119909

119896) isin 120596⋆

sup119896isinN

Δ119909119896

(119901119896)120573

ltinfin

(45)


(119901)

119888⋆

0 (119901) ℓ⋆

(119901) 119887119904⋆

(119901) 119888119904⋆

(119901) 119888119904⋆

0 (119901) 119887V⋆

(119901) and 119887V⋆infin(119901) is a

⋆-linear space



119892⋆

(119909) = sup119896isinN

119909119896

(119901119896119872)120573

119892⋆

(119909) = (120573sum

119896

119909119896

(119901119896)120573

)

(1119872)120573

(46)

Then 119888⋆

(119901) and 119888⋆


119896isin ℓ⋆


infin(119901) and

ℓ⋆


119896gt 0


119892⋆

1 (119909) = sup119896isinN

⋆

119896

sum

119895=0119909119895

(119901119896119872)120573

iff inf 119901119896gt 0 (47)



119892⋆

2 (119909) = (120573sum

119896

Δ119909119896

(119901119896)120573

)

(1119872)120573

119892⋆

3 (119909) = sup119896isinN

Δ119909119896

(1119872)120573

iff inf 119901119896gt 0

(48)







(⋆sum

119896

Δ (119906119896oplus119909119896)

(119901119896)120573

)

(1119872)120573

le (⋆sum

119896

Δ119906119896

(119901119896)120573

)

(1119872)120573

+ (⋆sum

119896

Δ119909119896

(119901119896)120573

)

(1119872)120573

(49)

and the condition

120582⋆

(119901119896)120573

le max

1

120582⋆

119872120573

(50)


= (

120582


0 and 119892⋆

(⊖119909) = 119892⋆


lemax

1

120582⋆

times 119892⋆



119899) be any


rarr 120582⋆ with


(119909119899

)

le 119892⋆

(119909)

+ 119892⋆

(119909119899


119892⋆

(120582⋆

119899⊙119909119899

⊖120582⋆

⊙119909)

= (120573sum

119896

Δ (120582⋆

119899⊙119909(119899)

119896⊖120582⋆

⊙119909119896)

(119901119896)120573

)

(1119872)120573

le

120582⋆

119899⊖120582⋆

times 119892⋆

(119909119899

)

+

120582⋆

times 119892⋆

(119909119899

⊖119909)

(51)

which tends to



119894= 119909(119894)

0 119909(119894)

1 119909(119894)


gt


119892⋆

(119909119894

⊖119909119895

)

lt

120598 forall119894 119895 ge 1198990 (120598) (52)


Δ [(119909119894

)119896

⊖ (119909119895

)119896

]

le (120573sum

119896

Δ [(119909119894

)119896

⊖ (119909119895

)119896

]

(119901119896)120573

)

(1119872)120573

lt 120598

forall119894 119895 ge 1198990 (120598)

(53)


1)119896 (Δ119909

2)119896 is a




120573

119898

sum

119896=0

Δ [(119909119894

)119896

⊖ (119909119895

)119896

]

(119901119896)120573

le 119892⋆

(119909119894

⊖119909119895

)

119872120573

lt 120598119872120573

(54)



(120573sum

119896

(Δ119909)119896

(119901119896)120573

)

(1119872)120573

le 119892⋆

(119909119894

⊖119909)

+ 119892⋆

(119909119894

)

le 120598

+ 119892⋆

(119909119894

)

(55)



119894



infin(119901) 119888⋆

(119901) 119888⋆

0 (119901) ℓ⋆

(119901) 119887119904⋆

(119901) 119888119904⋆

(119901) 119888119904⋆

0 (119901) 119887V⋆





119901of 119901-






infin 119888 1198880 ℓ119901 119887119904 119888119904

1198881199040 119887V 119887Vinfin(ii) 120583(119901) sube 120583

⋆

(119901) and ℓ⋆1 (119901) sube 119888119904⋆

(119901) sube 119888⋆

0 (119901) sube 119888⋆

(119901) sube

ℓ⋆

infin(119901) sube 120596

⋆



⋆

(119901)120572

120582⋆

(119901)120573 and 120582⋆(119901)120574 as follows

120582⋆

(119901)

120572

= 119908= (119908119896) isin 120596⋆

119908 ⊙ 119911 = (119908119896⊙ 119911119896)

isin ℓ⋆

1 (119901) forall119911 = (119911119896) isin 120582⋆

(119901)

120582⋆

(119901)

120573

= 119908= (119908119896) isin 120596⋆

119908 ⊙ 119911 = (119908119896⊙ 119911119896)

isin 119888119904⋆

(119901) forall119911 = (119911119896) isin 120582⋆

(119901)

120582⋆

(119901)

120574

= 119908= (119908119896) isin 120596⋆

119908 ⊙ 119911 = (119908119896⊙ 119911119896)

isin 119887119904⋆

(119901) forall119911 = (119911119896) isin 120582⋆

(119901)

(56)




Theorem 25 Let 0 = 120582⋆




(b) If 120582⋆(119901) sub 120583⋆

(119901) sub 120596⋆ then 120583⋆(119901)120573 sub 120582

⋆

(119901)120573

(c) 120582⋆ sub 120582⋆

120573120573

= (120582⋆

120573

)120573

(d) 119888⋆0 (119901)120573

= 119888⋆

(119901)120573

= ℓ⋆

infin(119901)120573

= ℓ⋆

1 (119901)

(e) ℓ⋆1 (119901)120573

= ℓ⋆

infin(119901)


119896) 119898 = (119898

119896)

and 119899 = (119899119896) isin 120582

⋆

(119901)120573



(119901)120573 for119898 119899 isin

120582⋆


⋆

(119901) Then (119898119896⊙ 119897119896) isin

119888119904⋆

(119901) and (119899119896⊙ 119897119896) isin 119888119904⋆


((119898119896oplus 119899119896) ⊙ 119897119896) = (119898

119896⊙ 119897119896) oplus (119899119896⊙ 119897119896) isin 119888119904

⋆

(119901)


(119901)

(57)

Hence 119898 oplus 119899 isin 120582⋆


120582⋆

(119901)120573 for any 119905 isin C⋆ and 119908 = (119908

119896) isin 120582

⋆

(119901)120573

Since (119908119896⊙ 119897119896) isin 119888119904

⋆

(119901) for all 119897 isin 120582⋆


119896⊙119908119896) ⊙ 119897119896) = 119905119896⊙ (119908119896⊙ 119897119896) isin

119888119904⋆

(119901) for all 119897 isin 120582⋆

(119901) we get 119905 ⊙ 119908 isin 120582⋆

(119901)120573



infin(119901)120573

sub 119888⋆

(119901)120573

sub 119888⋆

0 (119901)120573 by


ℓ⋆

infin(119901)120573 and 119888

⋆

0 (119901)120573

sub ℓ⋆


(119908119896) isin ℓ

⋆

1 (119901) and 119911 = (119911119896) isin ℓ

⋆



⋆sum

119896

1003817100381710038171003817119908119896⊙ 119911119896

1003817100381710038171003817

(119901119896)120573

le sup119896isinN

119911119896

(1119872)120573

times (120573sum

119896

119908119896

(119901119896)120573

)

(1119872)120573

le 119892⋆

(119911)

times 119892⋆

(119908) lt infin

(58)


⋆


⋆



ℓ⋆


⋆

0 (119901) with 119910 ⊙ 119909 notin 119888119904⋆


⋆



and⋆sum

119899119903minus1119896=119899119903minus1

119910119896

(119901119896)120573


119896) isin 119888

⋆

0 (119901)by 119909119896= (sgn⋆119910



sgn⋆ (119910) =

119910 ⊘

119910

119910 = 120579⋆

120579⋆

119910 = 120579⋆

(59)



⋆

119899119903minus1

sum

119896=119899119903minus1

(119910119896⊙119909119896)

(119901119896)120573

=⋆

119899119903minus1

sum

119896=119899119903minus1

[119910119896⊙ (sgn⋆119910

119896⊘ 119903)]

(119901119896)120573

=

1119903(119901119896)120573

⊙⋆

119899119903minus1

sum

119896=119899119903minus1

119910119896

(119901119896)120573

ge

1 (sup119901119896le 1)

(60)

Therefore 119910 ⊙ 119909 notin 119888119904⋆

(119901) and thus 119910 notin 119888⋆

0 (119901)120573

Hence 119888⋆0 (119901)120573

sub ℓ⋆



infin(119901) sub

(ℓ⋆

infin(119901)120573

)120573

= ℓ⋆

1 (119901)120573 since ℓ

⋆

infin(119901)120573

= ℓ⋆


119899) isin

ℓ⋆

1 (119901)120573

ℓ⋆




number (119896 + 1)2120573 such that

119908119899119896

(119901119896)120573



119909119899= (sgn⋆(119908

119899119896

) ⊘ (119896 + 1)2120573) if 119899 = 119899119896 and 120579

⋆


1 (119901) However

⋆sum

119899

(119908119899⊙119909119899)

(119901119896)120573

=⋆sum

119896

119908119899119896

(119901119896)120573

(119896 + 1)(2119901119896)120573= infin

(61)



tion and ℓ⋆

1 (119901)120573

sub ℓ⋆


proof


120573 120574)


(a) 120582⋆(119901)120572 sub 120582⋆

(119901)120573

sub 120582⋆

(119901)120574


120582⋆


(b) ℓ⋆1 (119901)120577

= ℓ⋆


infin(119901)120577

= ℓ⋆

1 (119901)

(c) If 120582⋆(119901) sub 120583⋆

(119901) sub 120596⋆ then 120583⋆(119901)120577 sub 120582

⋆

(119901)120577




Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











⋆ 119888119904⋆ 119887V⋆119901 and ℓ⋆

119901are


respectively



119906

times V

le

119906119901120572

119901

+

V119902120572

119902

(28)



120572minus1(119906) 120572minus1(V) le

(120572minus1(119906))

119901

119901

+

(120572minus1(V))119902

119902

(29)


0 or V =



119891 (119905) = (120580 (119905))120582120573

minus

120582

times 120580 (119905) = 120573 (120572minus1(119905))

120582

minus120582120572minus1(119905) (30)



119891⋆

(119905) = 120573

(120573minus1∘ 119891)

1015840

(119905)

(120572minus1)

1015840

(119905)

= 120573

((120572minus1)

120582


1015840

(119905)

(120572minus1)

1015840

(119905)

= 120573120582 (120572minus1(119905))


(31)


⋆

(119905) =



119891⋆⋆

(119905) = 120573

(120573minus1119891⋆

(119905))

1015840

(120572minus1)

1015840

(119905)

= 120573120582 (120582 minus 1) (120572minus1 (119905))120582

(32)


1 ) = 120573120582(120582 minus 1)

lt


1 that is 119891(


[120572minus1(119905)]

120582


120572 (33)

Now taking 119905 = (119906)119901120572

(V)119902120572 = 120572(120572minus1(119906))119901

(120572minus1(V))119902 and

120582 = 1119901 in (33) we get

(

(120572minus1(119906))

119901

(120572minus1(V))119902

)

1119901

minus

1119901

(120572minus1(119906))

119901

(120572minus1(V))119902

le 1minus 1119901

120572minus1(119906)

120572minus1(V)119902119901

(120572minus1(V))119902

le (1minus 1119901

+

1119901

120572minus1(119906)119901

120572minus1(V)119902

)(120572minus1(V))119902

(34)



119896 V⋆119896isin C⋆ for 119896 isin 0 1 2 119899


120573

119899

sum

119896=0

119906⋆

119896⊙ V⋆119896

le (120573

119899

sum

119896=0

119906⋆

119896

119901120573

)

(1119901)120573

times (120573

119899

sum

119896=0

V⋆119896

119902120573

)

(1119902)120573

(35)


= (

0

0 ) orV = (

0


Let

120576 = (120573

119899

sum

119896=0

119906⋆

119896

119901120573

)

(1119901)120573

120575 = (120573

119899

sum

119896=0

V⋆119896

119902120573

)

(1119902)120573

(36)

and 119904⋆119896= 119906⋆

119896⊘ 120576 119905⋆119896= V⋆119896⊘ 120575 where 120576 = (

0

120576 ) isin R120573sube C⋆

and 120575 = (

0

120575 ) isin R120573


119904⋆

119896⊙ 119905⋆

119896

=

119904⋆

119896

times

119905⋆

119896

le

119904⋆

119896

119901120573

119901

+

119905⋆

119896

119902120573

119902

(37)

which implies that

120573

119899

sum

119896=0

119904⋆

119896⊙ 119905⋆

119896

le120573

119899

sum

119896=0

119904⋆

119896

119901120573

119901

+120573

119899

sum

119896=0

119905⋆

119896

119902120573

119902

(38)


120573

119899

sum

119896=0

(119906⋆

119896⊘ 120576) ⊙ (V⋆

119896⊘ 120575)

le120573

119899

sum

119896=0

119906⋆

119896

119901120573

120576

times

119901

+120573

119899

sum

119896=0

V⋆119896

119902120573

120575

times

119902

=

1

119901

+

1

119902

=

1

(39)




120573

119899

sum

119896=0

119906⋆

119896⊙ V⋆119896

le (120573

119899

sum

119896=0

119906⋆

119896

2120573

times120573

119899

sum

119896=0

V⋆119896

2120573

)

(12)120573

(40)




(120573

119899

sum

119896=0

119906⋆

119896oplus V⋆119896

119901120573

)

(1119901)120573

le (120573

119899

sum

119896=0

119906⋆

119896

119901120573

)

(1119901)120573

+ (120573

119899

sum

119896=0

V⋆119896

119901120573

)

(1119901)120573

(41)



120573

119899

sum

119896=0

119906⋆

119896oplus V⋆119896

119901120573

le120573

119899

sum

119896=0

119906⋆

119896

times

119906⋆

119896oplus V⋆119896

(119901minus1)120573

+120573

119899

sum

119896=0

V⋆119896

times

119906⋆

119896oplus V⋆119896

(119901minus1)120573

(42)


120573

119899

sum

119896=0

119906⋆

119896oplus V⋆119896

119901120573

le[

[

(120573

119899

sum

119896=0

119906⋆

119896

119901120573

)

(1119901)120573

+ (120573

119899

sum

119896=0

V⋆119896

119901120573

)

(1119901)120573

]

]

times (120573

119899

sum

119896=0

119906⋆

119896oplus V⋆119896

(119902119901minus119902)120573

)

(1119902)120573

(43)





120573sube



(N(PN)1) 119892⋆

(119909) =

0 if 119909 = 120579⋆ (120579⋆ = (0⋆ 0⋆ ))

(N(PN)2) 119892⋆

(⊖119909) = 119892⋆


⋆

(119909 oplus 119910)

le 119892⋆

(119909)

+ 119892⋆

(119910)


that is 120582⋆ = (

120582

120582 ) with 120582⋆

119899

119889⋆



119899

119892⋆

997888997888rarr 119909 then

120582⋆

119899⊙ 119909119899

119892⋆

997888997888rarr 120582⋆

⊙ 119909 as 119899 rarr infin



(BG(PN)3) 119892⋆

(119909 oplus 119910) le 119892⋆

(119909)119892⋆

(119910)



119896=


119896times 1199011015840

119896= 119901119896+ 1199011015840





119888⋆

0 and ℓ⋆



119901over


infin 119888⋆

119888⋆

0 and ℓ⋆



infin(119901) 119888⋆

(119901) 119888⋆

0 (119901) ℓ⋆

(119901) 119887119904⋆

(119901) 119888119904⋆

(119901) 119888119904⋆

0 (119901) 119887V⋆



ℓ⋆

infin(119901) = 119909= (119909

119896) isin 120596⋆

sup119896isinN

119909119896

(119901119896)120573

ltinfin

= 119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆

sup119896isinN

120573radic1205762

119896+ 120575

2119896

119901119896

ltinfin

119888⋆

(119901) = 119909= (119909119896) isin 120596⋆



119909119896⊖ 119897

(119901119896)120573

= 120579⋆

= 119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆

exist 119897 = (

ℓ

ℓ ) isinC⋆


120573(ℓradic2 (1205762119896+ 120575

2119896))

119901119896

=

0

119888⋆

0 (119901) = 119909= (119909119896) isin 120596⋆


119909119896

(119901119896)120573

= 120579⋆

= 119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆


120573radic1205762

119896+ 120575

2119896

119901119896

ltinfin


ℓ⋆

(119901) = 119909= (119909119896) isin 120596⋆

120573sum

119896

119909119896

(119901119896)120573

ltinfin

(0 lt 119901119896lt infin)

= 119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆

120573sum

119896

radic1205762

119896+ 120575

2119896

119901119896

ltinfin

(44)


(119901) 119888119904⋆

0 (119901) 119887V⋆



119887119904⋆

(119901) =

119909= (119909119896) isin 120596⋆

(⋆

119896

sum

119895=0119909119895)isinℓ⋆

infin(119901)

=

119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆

sup119896isinN

120573

radic(

119896

sum

119895=0120576119895)

2

+ (

119896

sum

119895=0120575119895)

2119901119896

ltinfin

119888119904⋆

(119901) =

119909= (119909119896) isin 120596⋆

(⋆

119896

sum

119895=0119909119895)isin 119888⋆

(119901)

=

119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆


120573

radic(

119896

sum

119895=0120576119895minus ℓ)

2

+ (

119896

sum

119895=0120575119895minus ℓ)

2119901119896

ltinfin

(ℓ = (

ℓ

ℓ ) isin C⋆

)

119888119904⋆

0 (119901) = 119909= (119909119896) isin 120596⋆

(⋆

119899

sum

119896=0119909119896)isin 119888⋆

0 (119901)

119887V⋆ (119901) = 119909= (119909119896) isin 120596⋆

120573

infin

sum

119896=0

Δ119909119896

(119901119896)120573

ltinfin

(Δ119909119896= 119909119896⊖ 119909119896minus1 119909minus1 = 120579

⋆

)

= 119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆

120573

infin

sum

119896=0


2+ (120575119896minus 120575119896minus1)

2119901119896

ltinfin

119887V⋆infin(119901) = 119909= (119909

119896) isin 120596⋆

sup119896isinN

Δ119909119896

(119901119896)120573

ltinfin

(45)


(119901)

119888⋆

0 (119901) ℓ⋆

(119901) 119887119904⋆

(119901) 119888119904⋆

(119901) 119888119904⋆

0 (119901) 119887V⋆

(119901) and 119887V⋆infin(119901) is a

⋆-linear space



119892⋆

(119909) = sup119896isinN

119909119896

(119901119896119872)120573

119892⋆

(119909) = (120573sum

119896

119909119896

(119901119896)120573

)

(1119872)120573

(46)

Then 119888⋆

(119901) and 119888⋆


119896isin ℓ⋆


infin(119901) and

ℓ⋆


119896gt 0


119892⋆

1 (119909) = sup119896isinN

⋆

119896

sum

119895=0119909119895

(119901119896119872)120573

iff inf 119901119896gt 0 (47)



119892⋆

2 (119909) = (120573sum

119896

Δ119909119896

(119901119896)120573

)

(1119872)120573

119892⋆

3 (119909) = sup119896isinN

Δ119909119896

(1119872)120573

iff inf 119901119896gt 0

(48)







(⋆sum

119896

Δ (119906119896oplus119909119896)

(119901119896)120573

)

(1119872)120573

le (⋆sum

119896

Δ119906119896

(119901119896)120573

)

(1119872)120573

+ (⋆sum

119896

Δ119909119896

(119901119896)120573

)

(1119872)120573

(49)

and the condition

120582⋆

(119901119896)120573

le max

1

120582⋆

119872120573

(50)


= (

120582


0 and 119892⋆

(⊖119909) = 119892⋆


lemax

1

120582⋆

times 119892⋆



119899) be any


rarr 120582⋆ with


(119909119899

)

le 119892⋆

(119909)

+ 119892⋆

(119909119899


119892⋆

(120582⋆

119899⊙119909119899

⊖120582⋆

⊙119909)

= (120573sum

119896

Δ (120582⋆

119899⊙119909(119899)

119896⊖120582⋆

⊙119909119896)

(119901119896)120573

)

(1119872)120573

le

120582⋆

119899⊖120582⋆

times 119892⋆

(119909119899

)

+

120582⋆

times 119892⋆

(119909119899

⊖119909)

(51)

which tends to



119894= 119909(119894)

0 119909(119894)

1 119909(119894)


gt


119892⋆

(119909119894

⊖119909119895

)

lt

120598 forall119894 119895 ge 1198990 (120598) (52)


Δ [(119909119894

)119896

⊖ (119909119895

)119896

]

le (120573sum

119896

Δ [(119909119894

)119896

⊖ (119909119895

)119896

]

(119901119896)120573

)

(1119872)120573

lt 120598

forall119894 119895 ge 1198990 (120598)

(53)


1)119896 (Δ119909

2)119896 is a




120573

119898

sum

119896=0

Δ [(119909119894

)119896

⊖ (119909119895

)119896

]

(119901119896)120573

le 119892⋆

(119909119894

⊖119909119895

)

119872120573

lt 120598119872120573

(54)



(120573sum

119896

(Δ119909)119896

(119901119896)120573

)

(1119872)120573

le 119892⋆

(119909119894

⊖119909)

+ 119892⋆

(119909119894

)

le 120598

+ 119892⋆

(119909119894

)

(55)



119894



infin(119901) 119888⋆

(119901) 119888⋆

0 (119901) ℓ⋆

(119901) 119887119904⋆

(119901) 119888119904⋆

(119901) 119888119904⋆

0 (119901) 119887V⋆





119901of 119901-






infin 119888 1198880 ℓ119901 119887119904 119888119904

1198881199040 119887V 119887Vinfin(ii) 120583(119901) sube 120583

⋆

(119901) and ℓ⋆1 (119901) sube 119888119904⋆

(119901) sube 119888⋆

0 (119901) sube 119888⋆

(119901) sube

ℓ⋆

infin(119901) sube 120596

⋆



⋆

(119901)120572

120582⋆

(119901)120573 and 120582⋆(119901)120574 as follows

120582⋆

(119901)

120572

= 119908= (119908119896) isin 120596⋆

119908 ⊙ 119911 = (119908119896⊙ 119911119896)

isin ℓ⋆

1 (119901) forall119911 = (119911119896) isin 120582⋆

(119901)

120582⋆

(119901)

120573

= 119908= (119908119896) isin 120596⋆

119908 ⊙ 119911 = (119908119896⊙ 119911119896)

isin 119888119904⋆

(119901) forall119911 = (119911119896) isin 120582⋆

(119901)

120582⋆

(119901)

120574

= 119908= (119908119896) isin 120596⋆

119908 ⊙ 119911 = (119908119896⊙ 119911119896)

isin 119887119904⋆

(119901) forall119911 = (119911119896) isin 120582⋆

(119901)

(56)




Theorem 25 Let 0 = 120582⋆




(b) If 120582⋆(119901) sub 120583⋆

(119901) sub 120596⋆ then 120583⋆(119901)120573 sub 120582

⋆

(119901)120573

(c) 120582⋆ sub 120582⋆

120573120573

= (120582⋆

120573

)120573

(d) 119888⋆0 (119901)120573

= 119888⋆

(119901)120573

= ℓ⋆

infin(119901)120573

= ℓ⋆

1 (119901)

(e) ℓ⋆1 (119901)120573

= ℓ⋆

infin(119901)


119896) 119898 = (119898

119896)

and 119899 = (119899119896) isin 120582

⋆

(119901)120573



(119901)120573 for119898 119899 isin

120582⋆


⋆

(119901) Then (119898119896⊙ 119897119896) isin

119888119904⋆

(119901) and (119899119896⊙ 119897119896) isin 119888119904⋆


((119898119896oplus 119899119896) ⊙ 119897119896) = (119898

119896⊙ 119897119896) oplus (119899119896⊙ 119897119896) isin 119888119904

⋆

(119901)


(119901)

(57)

Hence 119898 oplus 119899 isin 120582⋆


120582⋆

(119901)120573 for any 119905 isin C⋆ and 119908 = (119908

119896) isin 120582

⋆

(119901)120573

Since (119908119896⊙ 119897119896) isin 119888119904

⋆

(119901) for all 119897 isin 120582⋆


119896⊙119908119896) ⊙ 119897119896) = 119905119896⊙ (119908119896⊙ 119897119896) isin

119888119904⋆

(119901) for all 119897 isin 120582⋆

(119901) we get 119905 ⊙ 119908 isin 120582⋆

(119901)120573



infin(119901)120573

sub 119888⋆

(119901)120573

sub 119888⋆

0 (119901)120573 by


ℓ⋆

infin(119901)120573 and 119888

⋆

0 (119901)120573

sub ℓ⋆


(119908119896) isin ℓ

⋆

1 (119901) and 119911 = (119911119896) isin ℓ

⋆



⋆sum

119896

1003817100381710038171003817119908119896⊙ 119911119896

1003817100381710038171003817

(119901119896)120573

le sup119896isinN

119911119896

(1119872)120573

times (120573sum

119896

119908119896

(119901119896)120573

)

(1119872)120573

le 119892⋆

(119911)

times 119892⋆

(119908) lt infin

(58)


⋆


⋆



ℓ⋆


⋆

0 (119901) with 119910 ⊙ 119909 notin 119888119904⋆


⋆



and⋆sum

119899119903minus1119896=119899119903minus1

119910119896

(119901119896)120573


119896) isin 119888

⋆

0 (119901)by 119909119896= (sgn⋆119910



sgn⋆ (119910) =

119910 ⊘

119910

119910 = 120579⋆

120579⋆

119910 = 120579⋆

(59)



⋆

119899119903minus1

sum

119896=119899119903minus1

(119910119896⊙119909119896)

(119901119896)120573

=⋆

119899119903minus1

sum

119896=119899119903minus1

[119910119896⊙ (sgn⋆119910

119896⊘ 119903)]

(119901119896)120573

=

1119903(119901119896)120573

⊙⋆

119899119903minus1

sum

119896=119899119903minus1

119910119896

(119901119896)120573

ge

1 (sup119901119896le 1)

(60)

Therefore 119910 ⊙ 119909 notin 119888119904⋆

(119901) and thus 119910 notin 119888⋆

0 (119901)120573

Hence 119888⋆0 (119901)120573

sub ℓ⋆



infin(119901) sub

(ℓ⋆

infin(119901)120573

)120573

= ℓ⋆

1 (119901)120573 since ℓ

⋆

infin(119901)120573

= ℓ⋆


119899) isin

ℓ⋆

1 (119901)120573

ℓ⋆




number (119896 + 1)2120573 such that

119908119899119896

(119901119896)120573



119909119899= (sgn⋆(119908

119899119896

) ⊘ (119896 + 1)2120573) if 119899 = 119899119896 and 120579

⋆


1 (119901) However

⋆sum

119899

(119908119899⊙119909119899)

(119901119896)120573

=⋆sum

119896

119908119899119896

(119901119896)120573

(119896 + 1)(2119901119896)120573= infin

(61)



tion and ℓ⋆

1 (119901)120573

sub ℓ⋆


proof


120573 120574)


(a) 120582⋆(119901)120572 sub 120582⋆

(119901)120573

sub 120582⋆

(119901)120574


120582⋆


(b) ℓ⋆1 (119901)120577

= ℓ⋆


infin(119901)120577

= ℓ⋆

1 (119901)

(c) If 120582⋆(119901) sub 120583⋆

(119901) sub 120596⋆ then 120583⋆(119901)120577 sub 120582

⋆

(119901)120577




Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120573

119899

sum

119896=0

119906⋆

119896⊙ V⋆119896

le (120573

119899

sum

119896=0

119906⋆

119896

2120573

times120573

119899

sum

119896=0

V⋆119896

2120573

)

(12)120573

(40)




(120573

119899

sum

119896=0

119906⋆

119896oplus V⋆119896

119901120573

)

(1119901)120573

le (120573

119899

sum

119896=0

119906⋆

119896

119901120573

)

(1119901)120573

+ (120573

119899

sum

119896=0

V⋆119896

119901120573

)

(1119901)120573

(41)



120573

119899

sum

119896=0

119906⋆

119896oplus V⋆119896

119901120573

le120573

119899

sum

119896=0

119906⋆

119896

times

119906⋆

119896oplus V⋆119896

(119901minus1)120573

+120573

119899

sum

119896=0

V⋆119896

times

119906⋆

119896oplus V⋆119896

(119901minus1)120573

(42)


120573

119899

sum

119896=0

119906⋆

119896oplus V⋆119896

119901120573

le[

[

(120573

119899

sum

119896=0

119906⋆

119896

119901120573

)

(1119901)120573

+ (120573

119899

sum

119896=0

V⋆119896

119901120573

)

(1119901)120573

]

]

times (120573

119899

sum

119896=0

119906⋆

119896oplus V⋆119896

(119902119901minus119902)120573

)

(1119902)120573

(43)





120573sube



(N(PN)1) 119892⋆

(119909) =

0 if 119909 = 120579⋆ (120579⋆ = (0⋆ 0⋆ ))

(N(PN)2) 119892⋆

(⊖119909) = 119892⋆


⋆

(119909 oplus 119910)

le 119892⋆

(119909)

+ 119892⋆

(119910)


that is 120582⋆ = (

120582

120582 ) with 120582⋆

119899

119889⋆



119899

119892⋆

997888997888rarr 119909 then

120582⋆

119899⊙ 119909119899

119892⋆

997888997888rarr 120582⋆

⊙ 119909 as 119899 rarr infin



(BG(PN)3) 119892⋆

(119909 oplus 119910) le 119892⋆

(119909)119892⋆

(119910)



119896=


119896times 1199011015840

119896= 119901119896+ 1199011015840





119888⋆

0 and ℓ⋆



119901over


infin 119888⋆

119888⋆

0 and ℓ⋆



infin(119901) 119888⋆

(119901) 119888⋆

0 (119901) ℓ⋆

(119901) 119887119904⋆

(119901) 119888119904⋆

(119901) 119888119904⋆

0 (119901) 119887V⋆



ℓ⋆

infin(119901) = 119909= (119909

119896) isin 120596⋆

sup119896isinN

119909119896

(119901119896)120573

ltinfin

= 119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆

sup119896isinN

120573radic1205762

119896+ 120575

2119896

119901119896

ltinfin

119888⋆

(119901) = 119909= (119909119896) isin 120596⋆



119909119896⊖ 119897

(119901119896)120573

= 120579⋆

= 119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆

exist 119897 = (

ℓ

ℓ ) isinC⋆


120573(ℓradic2 (1205762119896+ 120575

2119896))

119901119896

=

0

119888⋆

0 (119901) = 119909= (119909119896) isin 120596⋆


119909119896

(119901119896)120573

= 120579⋆

= 119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆


120573radic1205762

119896+ 120575

2119896

119901119896

ltinfin


ℓ⋆

(119901) = 119909= (119909119896) isin 120596⋆

120573sum

119896

119909119896

(119901119896)120573

ltinfin

(0 lt 119901119896lt infin)

= 119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆

120573sum

119896

radic1205762

119896+ 120575

2119896

119901119896

ltinfin

(44)


(119901) 119888119904⋆

0 (119901) 119887V⋆



119887119904⋆

(119901) =

119909= (119909119896) isin 120596⋆

(⋆

119896

sum

119895=0119909119895)isinℓ⋆

infin(119901)

=

119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆

sup119896isinN

120573

radic(

119896

sum

119895=0120576119895)

2

+ (

119896

sum

119895=0120575119895)

2119901119896

ltinfin

119888119904⋆

(119901) =

119909= (119909119896) isin 120596⋆

(⋆

119896

sum

119895=0119909119895)isin 119888⋆

(119901)

=

119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆


120573

radic(

119896

sum

119895=0120576119895minus ℓ)

2

+ (

119896

sum

119895=0120575119895minus ℓ)

2119901119896

ltinfin

(ℓ = (

ℓ

ℓ ) isin C⋆

)

119888119904⋆

0 (119901) = 119909= (119909119896) isin 120596⋆

(⋆

119899

sum

119896=0119909119896)isin 119888⋆

0 (119901)

119887V⋆ (119901) = 119909= (119909119896) isin 120596⋆

120573

infin

sum

119896=0

Δ119909119896

(119901119896)120573

ltinfin

(Δ119909119896= 119909119896⊖ 119909119896minus1 119909minus1 = 120579

⋆

)

= 119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆

120573

infin

sum

119896=0


2+ (120575119896minus 120575119896minus1)

2119901119896

ltinfin

119887V⋆infin(119901) = 119909= (119909

119896) isin 120596⋆

sup119896isinN

Δ119909119896

(119901119896)120573

ltinfin

(45)


(119901)

119888⋆

0 (119901) ℓ⋆

(119901) 119887119904⋆

(119901) 119888119904⋆

(119901) 119888119904⋆

0 (119901) 119887V⋆

(119901) and 119887V⋆infin(119901) is a

⋆-linear space



119892⋆

(119909) = sup119896isinN

119909119896

(119901119896119872)120573

119892⋆

(119909) = (120573sum

119896

119909119896

(119901119896)120573

)

(1119872)120573

(46)

Then 119888⋆

(119901) and 119888⋆


119896isin ℓ⋆


infin(119901) and

ℓ⋆


119896gt 0


119892⋆

1 (119909) = sup119896isinN

⋆

119896

sum

119895=0119909119895

(119901119896119872)120573

iff inf 119901119896gt 0 (47)



119892⋆

2 (119909) = (120573sum

119896

Δ119909119896

(119901119896)120573

)

(1119872)120573

119892⋆

3 (119909) = sup119896isinN

Δ119909119896

(1119872)120573

iff inf 119901119896gt 0

(48)







(⋆sum

119896

Δ (119906119896oplus119909119896)

(119901119896)120573

)

(1119872)120573

le (⋆sum

119896

Δ119906119896

(119901119896)120573

)

(1119872)120573

+ (⋆sum

119896

Δ119909119896

(119901119896)120573

)

(1119872)120573

(49)

and the condition

120582⋆

(119901119896)120573

le max

1

120582⋆

119872120573

(50)


= (

120582


0 and 119892⋆

(⊖119909) = 119892⋆


lemax

1

120582⋆

times 119892⋆



119899) be any


rarr 120582⋆ with


(119909119899

)

le 119892⋆

(119909)

+ 119892⋆

(119909119899


119892⋆

(120582⋆

119899⊙119909119899

⊖120582⋆

⊙119909)

= (120573sum

119896

Δ (120582⋆

119899⊙119909(119899)

119896⊖120582⋆

⊙119909119896)

(119901119896)120573

)

(1119872)120573

le

120582⋆

119899⊖120582⋆

times 119892⋆

(119909119899

)

+

120582⋆

times 119892⋆

(119909119899

⊖119909)

(51)

which tends to



119894= 119909(119894)

0 119909(119894)

1 119909(119894)


gt


119892⋆

(119909119894

⊖119909119895

)

lt

120598 forall119894 119895 ge 1198990 (120598) (52)


Δ [(119909119894

)119896

⊖ (119909119895

)119896

]

le (120573sum

119896

Δ [(119909119894

)119896

⊖ (119909119895

)119896

]

(119901119896)120573

)

(1119872)120573

lt 120598

forall119894 119895 ge 1198990 (120598)

(53)


1)119896 (Δ119909

2)119896 is a




120573

119898

sum

119896=0

Δ [(119909119894

)119896

⊖ (119909119895

)119896

]

(119901119896)120573

le 119892⋆

(119909119894

⊖119909119895

)

119872120573

lt 120598119872120573

(54)



(120573sum

119896

(Δ119909)119896

(119901119896)120573

)

(1119872)120573

le 119892⋆

(119909119894

⊖119909)

+ 119892⋆

(119909119894

)

le 120598

+ 119892⋆

(119909119894

)

(55)



119894



infin(119901) 119888⋆

(119901) 119888⋆

0 (119901) ℓ⋆

(119901) 119887119904⋆

(119901) 119888119904⋆

(119901) 119888119904⋆

0 (119901) 119887V⋆





119901of 119901-






infin 119888 1198880 ℓ119901 119887119904 119888119904

1198881199040 119887V 119887Vinfin(ii) 120583(119901) sube 120583

⋆

(119901) and ℓ⋆1 (119901) sube 119888119904⋆

(119901) sube 119888⋆

0 (119901) sube 119888⋆

(119901) sube

ℓ⋆

infin(119901) sube 120596

⋆



⋆

(119901)120572

120582⋆

(119901)120573 and 120582⋆(119901)120574 as follows

120582⋆

(119901)

120572

= 119908= (119908119896) isin 120596⋆

119908 ⊙ 119911 = (119908119896⊙ 119911119896)

isin ℓ⋆

1 (119901) forall119911 = (119911119896) isin 120582⋆

(119901)

120582⋆

(119901)

120573

= 119908= (119908119896) isin 120596⋆

119908 ⊙ 119911 = (119908119896⊙ 119911119896)

isin 119888119904⋆

(119901) forall119911 = (119911119896) isin 120582⋆

(119901)

120582⋆

(119901)

120574

= 119908= (119908119896) isin 120596⋆

119908 ⊙ 119911 = (119908119896⊙ 119911119896)

isin 119887119904⋆

(119901) forall119911 = (119911119896) isin 120582⋆

(119901)

(56)




Theorem 25 Let 0 = 120582⋆




(b) If 120582⋆(119901) sub 120583⋆

(119901) sub 120596⋆ then 120583⋆(119901)120573 sub 120582

⋆

(119901)120573

(c) 120582⋆ sub 120582⋆

120573120573

= (120582⋆

120573

)120573

(d) 119888⋆0 (119901)120573

= 119888⋆

(119901)120573

= ℓ⋆

infin(119901)120573

= ℓ⋆

1 (119901)

(e) ℓ⋆1 (119901)120573

= ℓ⋆

infin(119901)


119896) 119898 = (119898

119896)

and 119899 = (119899119896) isin 120582

⋆

(119901)120573



(119901)120573 for119898 119899 isin

120582⋆


⋆

(119901) Then (119898119896⊙ 119897119896) isin

119888119904⋆

(119901) and (119899119896⊙ 119897119896) isin 119888119904⋆


((119898119896oplus 119899119896) ⊙ 119897119896) = (119898

119896⊙ 119897119896) oplus (119899119896⊙ 119897119896) isin 119888119904

⋆

(119901)


(119901)

(57)

Hence 119898 oplus 119899 isin 120582⋆


120582⋆

(119901)120573 for any 119905 isin C⋆ and 119908 = (119908

119896) isin 120582

⋆

(119901)120573

Since (119908119896⊙ 119897119896) isin 119888119904

⋆

(119901) for all 119897 isin 120582⋆


119896⊙119908119896) ⊙ 119897119896) = 119905119896⊙ (119908119896⊙ 119897119896) isin

119888119904⋆

(119901) for all 119897 isin 120582⋆

(119901) we get 119905 ⊙ 119908 isin 120582⋆

(119901)120573



infin(119901)120573

sub 119888⋆

(119901)120573

sub 119888⋆

0 (119901)120573 by


ℓ⋆

infin(119901)120573 and 119888

⋆

0 (119901)120573

sub ℓ⋆


(119908119896) isin ℓ

⋆

1 (119901) and 119911 = (119911119896) isin ℓ

⋆



⋆sum

119896

1003817100381710038171003817119908119896⊙ 119911119896

1003817100381710038171003817

(119901119896)120573

le sup119896isinN

119911119896

(1119872)120573

times (120573sum

119896

119908119896

(119901119896)120573

)

(1119872)120573

le 119892⋆

(119911)

times 119892⋆

(119908) lt infin

(58)


⋆


⋆



ℓ⋆


⋆

0 (119901) with 119910 ⊙ 119909 notin 119888119904⋆


⋆



and⋆sum

119899119903minus1119896=119899119903minus1

119910119896

(119901119896)120573


119896) isin 119888

⋆

0 (119901)by 119909119896= (sgn⋆119910



sgn⋆ (119910) =

119910 ⊘

119910

119910 = 120579⋆

120579⋆

119910 = 120579⋆

(59)



⋆

119899119903minus1

sum

119896=119899119903minus1

(119910119896⊙119909119896)

(119901119896)120573

=⋆

119899119903minus1

sum

119896=119899119903minus1

[119910119896⊙ (sgn⋆119910

119896⊘ 119903)]

(119901119896)120573

=

1119903(119901119896)120573

⊙⋆

119899119903minus1

sum

119896=119899119903minus1

119910119896

(119901119896)120573

ge

1 (sup119901119896le 1)

(60)

Therefore 119910 ⊙ 119909 notin 119888119904⋆

(119901) and thus 119910 notin 119888⋆

0 (119901)120573

Hence 119888⋆0 (119901)120573

sub ℓ⋆



infin(119901) sub

(ℓ⋆

infin(119901)120573

)120573

= ℓ⋆

1 (119901)120573 since ℓ

⋆

infin(119901)120573

= ℓ⋆


119899) isin

ℓ⋆

1 (119901)120573

ℓ⋆




number (119896 + 1)2120573 such that

119908119899119896

(119901119896)120573



119909119899= (sgn⋆(119908

119899119896

) ⊘ (119896 + 1)2120573) if 119899 = 119899119896 and 120579

⋆


1 (119901) However

⋆sum

119899

(119908119899⊙119909119899)

(119901119896)120573

=⋆sum

119896

119908119899119896

(119901119896)120573

(119896 + 1)(2119901119896)120573= infin

(61)



tion and ℓ⋆

1 (119901)120573

sub ℓ⋆


proof


120573 120574)


(a) 120582⋆(119901)120572 sub 120582⋆

(119901)120573

sub 120582⋆

(119901)120574


120582⋆


(b) ℓ⋆1 (119901)120577

= ℓ⋆


infin(119901)120577

= ℓ⋆

1 (119901)

(c) If 120582⋆(119901) sub 120583⋆

(119901) sub 120596⋆ then 120583⋆(119901)120577 sub 120582

⋆

(119901)120577




Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










ℓ⋆

(119901) = 119909= (119909119896) isin 120596⋆

120573sum

119896

119909119896

(119901119896)120573

ltinfin

(0 lt 119901119896lt infin)

= 119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆

120573sum

119896

radic1205762

119896+ 120575

2119896

119901119896

ltinfin

(44)


(119901) 119888119904⋆

0 (119901) 119887V⋆



119887119904⋆

(119901) =

119909= (119909119896) isin 120596⋆

(⋆

119896

sum

119895=0119909119895)isinℓ⋆

infin(119901)

=

119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆

sup119896isinN

120573

radic(

119896

sum

119895=0120576119895)

2

+ (

119896

sum

119895=0120575119895)

2119901119896

ltinfin

119888119904⋆

(119901) =

119909= (119909119896) isin 120596⋆

(⋆

119896

sum

119895=0119909119895)isin 119888⋆

(119901)

=

119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆


120573

radic(

119896

sum

119895=0120576119895minus ℓ)

2

+ (

119896

sum

119895=0120575119895minus ℓ)

2119901119896

ltinfin

(ℓ = (

ℓ

ℓ ) isin C⋆

)

119888119904⋆

0 (119901) = 119909= (119909119896) isin 120596⋆

(⋆

119899

sum

119896=0119909119896)isin 119888⋆

0 (119901)

119887V⋆ (119901) = 119909= (119909119896) isin 120596⋆

120573

infin

sum

119896=0

Δ119909119896

(119901119896)120573

ltinfin

(Δ119909119896= 119909119896⊖ 119909119896minus1 119909minus1 = 120579

⋆

)

= 119909= (119909119896) = (

120576119896

120575 119896) isin120596⋆

120573

infin

sum

119896=0


2+ (120575119896minus 120575119896minus1)

2119901119896

ltinfin

119887V⋆infin(119901) = 119909= (119909

119896) isin 120596⋆

sup119896isinN

Δ119909119896

(119901119896)120573

ltinfin

(45)


(119901)

119888⋆

0 (119901) ℓ⋆

(119901) 119887119904⋆

(119901) 119888119904⋆

(119901) 119888119904⋆

0 (119901) 119887V⋆

(119901) and 119887V⋆infin(119901) is a

⋆-linear space



119892⋆

(119909) = sup119896isinN

119909119896

(119901119896119872)120573

119892⋆

(119909) = (120573sum

119896

119909119896

(119901119896)120573

)

(1119872)120573

(46)

Then 119888⋆

(119901) and 119888⋆


119896isin ℓ⋆


infin(119901) and

ℓ⋆


119896gt 0


119892⋆

1 (119909) = sup119896isinN

⋆

119896

sum

119895=0119909119895

(119901119896119872)120573

iff inf 119901119896gt 0 (47)



119892⋆

2 (119909) = (120573sum

119896

Δ119909119896

(119901119896)120573

)

(1119872)120573

119892⋆

3 (119909) = sup119896isinN

Δ119909119896

(1119872)120573

iff inf 119901119896gt 0

(48)







(⋆sum

119896

Δ (119906119896oplus119909119896)

(119901119896)120573

)

(1119872)120573

le (⋆sum

119896

Δ119906119896

(119901119896)120573

)

(1119872)120573

+ (⋆sum

119896

Δ119909119896

(119901119896)120573

)

(1119872)120573

(49)

and the condition

120582⋆

(119901119896)120573

le max

1

120582⋆

119872120573

(50)


= (

120582


0 and 119892⋆

(⊖119909) = 119892⋆


lemax

1

120582⋆

times 119892⋆



119899) be any


rarr 120582⋆ with


(119909119899

)

le 119892⋆

(119909)

+ 119892⋆

(119909119899


119892⋆

(120582⋆

119899⊙119909119899

⊖120582⋆

⊙119909)

= (120573sum

119896

Δ (120582⋆

119899⊙119909(119899)

119896⊖120582⋆

⊙119909119896)

(119901119896)120573

)

(1119872)120573

le

120582⋆

119899⊖120582⋆

times 119892⋆

(119909119899

)

+

120582⋆

times 119892⋆

(119909119899

⊖119909)

(51)

which tends to



119894= 119909(119894)

0 119909(119894)

1 119909(119894)


gt


119892⋆

(119909119894

⊖119909119895

)

lt

120598 forall119894 119895 ge 1198990 (120598) (52)


Δ [(119909119894

)119896

⊖ (119909119895

)119896

]

le (120573sum

119896

Δ [(119909119894

)119896

⊖ (119909119895

)119896

]

(119901119896)120573

)

(1119872)120573

lt 120598

forall119894 119895 ge 1198990 (120598)

(53)


1)119896 (Δ119909

2)119896 is a




120573

119898

sum

119896=0

Δ [(119909119894

)119896

⊖ (119909119895

)119896

]

(119901119896)120573

le 119892⋆

(119909119894

⊖119909119895

)

119872120573

lt 120598119872120573

(54)



(120573sum

119896

(Δ119909)119896

(119901119896)120573

)

(1119872)120573

le 119892⋆

(119909119894

⊖119909)

+ 119892⋆

(119909119894

)

le 120598

+ 119892⋆

(119909119894

)

(55)



119894



infin(119901) 119888⋆

(119901) 119888⋆

0 (119901) ℓ⋆

(119901) 119887119904⋆

(119901) 119888119904⋆

(119901) 119888119904⋆

0 (119901) 119887V⋆





119901of 119901-






infin 119888 1198880 ℓ119901 119887119904 119888119904

1198881199040 119887V 119887Vinfin(ii) 120583(119901) sube 120583

⋆

(119901) and ℓ⋆1 (119901) sube 119888119904⋆

(119901) sube 119888⋆

0 (119901) sube 119888⋆

(119901) sube

ℓ⋆

infin(119901) sube 120596

⋆



⋆

(119901)120572

120582⋆

(119901)120573 and 120582⋆(119901)120574 as follows

120582⋆

(119901)

120572

= 119908= (119908119896) isin 120596⋆

119908 ⊙ 119911 = (119908119896⊙ 119911119896)

isin ℓ⋆

1 (119901) forall119911 = (119911119896) isin 120582⋆

(119901)

120582⋆

(119901)

120573

= 119908= (119908119896) isin 120596⋆

119908 ⊙ 119911 = (119908119896⊙ 119911119896)

isin 119888119904⋆

(119901) forall119911 = (119911119896) isin 120582⋆

(119901)

120582⋆

(119901)

120574

= 119908= (119908119896) isin 120596⋆

119908 ⊙ 119911 = (119908119896⊙ 119911119896)

isin 119887119904⋆

(119901) forall119911 = (119911119896) isin 120582⋆

(119901)

(56)




Theorem 25 Let 0 = 120582⋆




(b) If 120582⋆(119901) sub 120583⋆

(119901) sub 120596⋆ then 120583⋆(119901)120573 sub 120582

⋆

(119901)120573

(c) 120582⋆ sub 120582⋆

120573120573

= (120582⋆

120573

)120573

(d) 119888⋆0 (119901)120573

= 119888⋆

(119901)120573

= ℓ⋆

infin(119901)120573

= ℓ⋆

1 (119901)

(e) ℓ⋆1 (119901)120573

= ℓ⋆

infin(119901)


119896) 119898 = (119898

119896)

and 119899 = (119899119896) isin 120582

⋆

(119901)120573



(119901)120573 for119898 119899 isin

120582⋆


⋆

(119901) Then (119898119896⊙ 119897119896) isin

119888119904⋆

(119901) and (119899119896⊙ 119897119896) isin 119888119904⋆


((119898119896oplus 119899119896) ⊙ 119897119896) = (119898

119896⊙ 119897119896) oplus (119899119896⊙ 119897119896) isin 119888119904

⋆

(119901)


(119901)

(57)

Hence 119898 oplus 119899 isin 120582⋆


120582⋆

(119901)120573 for any 119905 isin C⋆ and 119908 = (119908

119896) isin 120582

⋆

(119901)120573

Since (119908119896⊙ 119897119896) isin 119888119904

⋆

(119901) for all 119897 isin 120582⋆


119896⊙119908119896) ⊙ 119897119896) = 119905119896⊙ (119908119896⊙ 119897119896) isin

119888119904⋆

(119901) for all 119897 isin 120582⋆

(119901) we get 119905 ⊙ 119908 isin 120582⋆

(119901)120573



infin(119901)120573

sub 119888⋆

(119901)120573

sub 119888⋆

0 (119901)120573 by


ℓ⋆

infin(119901)120573 and 119888

⋆

0 (119901)120573

sub ℓ⋆


(119908119896) isin ℓ

⋆

1 (119901) and 119911 = (119911119896) isin ℓ

⋆



⋆sum

119896

1003817100381710038171003817119908119896⊙ 119911119896

1003817100381710038171003817

(119901119896)120573

le sup119896isinN

119911119896

(1119872)120573

times (120573sum

119896

119908119896

(119901119896)120573

)

(1119872)120573

le 119892⋆

(119911)

times 119892⋆

(119908) lt infin

(58)


⋆


⋆



ℓ⋆


⋆

0 (119901) with 119910 ⊙ 119909 notin 119888119904⋆


⋆



and⋆sum

119899119903minus1119896=119899119903minus1

119910119896

(119901119896)120573


119896) isin 119888

⋆

0 (119901)by 119909119896= (sgn⋆119910



sgn⋆ (119910) =

119910 ⊘

119910

119910 = 120579⋆

120579⋆

119910 = 120579⋆

(59)



⋆

119899119903minus1

sum

119896=119899119903minus1

(119910119896⊙119909119896)

(119901119896)120573

=⋆

119899119903minus1

sum

119896=119899119903minus1

[119910119896⊙ (sgn⋆119910

119896⊘ 119903)]

(119901119896)120573

=

1119903(119901119896)120573

⊙⋆

119899119903minus1

sum

119896=119899119903minus1

119910119896

(119901119896)120573

ge

1 (sup119901119896le 1)

(60)

Therefore 119910 ⊙ 119909 notin 119888119904⋆

(119901) and thus 119910 notin 119888⋆

0 (119901)120573

Hence 119888⋆0 (119901)120573

sub ℓ⋆



infin(119901) sub

(ℓ⋆

infin(119901)120573

)120573

= ℓ⋆

1 (119901)120573 since ℓ

⋆

infin(119901)120573

= ℓ⋆


119899) isin

ℓ⋆

1 (119901)120573

ℓ⋆




number (119896 + 1)2120573 such that

119908119899119896

(119901119896)120573



119909119899= (sgn⋆(119908

119899119896

) ⊘ (119896 + 1)2120573) if 119899 = 119899119896 and 120579

⋆


1 (119901) However

⋆sum

119899

(119908119899⊙119909119899)

(119901119896)120573

=⋆sum

119896

119908119899119896

(119901119896)120573

(119896 + 1)(2119901119896)120573= infin

(61)



tion and ℓ⋆

1 (119901)120573

sub ℓ⋆


proof


120573 120574)


(a) 120582⋆(119901)120572 sub 120582⋆

(119901)120573

sub 120582⋆

(119901)120574


120582⋆


(b) ℓ⋆1 (119901)120577

= ℓ⋆


infin(119901)120577

= ℓ⋆

1 (119901)

(c) If 120582⋆(119901) sub 120583⋆

(119901) sub 120596⋆ then 120583⋆(119901)120577 sub 120582

⋆

(119901)120577




Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(⋆sum

119896

Δ (119906119896oplus119909119896)

(119901119896)120573

)

(1119872)120573

le (⋆sum

119896

Δ119906119896

(119901119896)120573

)

(1119872)120573

+ (⋆sum

119896

Δ119909119896

(119901119896)120573

)

(1119872)120573

(49)

and the condition

120582⋆

(119901119896)120573

le max

1

120582⋆

119872120573

(50)


= (

120582


0 and 119892⋆

(⊖119909) = 119892⋆


lemax

1

120582⋆

times 119892⋆



119899) be any


rarr 120582⋆ with


(119909119899

)

le 119892⋆

(119909)

+ 119892⋆

(119909119899


119892⋆

(120582⋆

119899⊙119909119899

⊖120582⋆

⊙119909)

= (120573sum

119896

Δ (120582⋆

119899⊙119909(119899)

119896⊖120582⋆

⊙119909119896)

(119901119896)120573

)

(1119872)120573

le

120582⋆

119899⊖120582⋆

times 119892⋆

(119909119899

)

+

120582⋆

times 119892⋆

(119909119899

⊖119909)

(51)

which tends to



119894= 119909(119894)

0 119909(119894)

1 119909(119894)


gt


119892⋆

(119909119894

⊖119909119895

)

lt

120598 forall119894 119895 ge 1198990 (120598) (52)


Δ [(119909119894

)119896

⊖ (119909119895

)119896

]

le (120573sum

119896

Δ [(119909119894

)119896

⊖ (119909119895

)119896

]

(119901119896)120573

)

(1119872)120573

lt 120598

forall119894 119895 ge 1198990 (120598)

(53)


1)119896 (Δ119909

2)119896 is a




120573

119898

sum

119896=0

Δ [(119909119894

)119896

⊖ (119909119895

)119896

]

(119901119896)120573

le 119892⋆

(119909119894

⊖119909119895

)

119872120573

lt 120598119872120573

(54)



(120573sum

119896

(Δ119909)119896

(119901119896)120573

)

(1119872)120573

le 119892⋆

(119909119894

⊖119909)

+ 119892⋆

(119909119894

)

le 120598

+ 119892⋆

(119909119894

)

(55)



119894



infin(119901) 119888⋆

(119901) 119888⋆

0 (119901) ℓ⋆

(119901) 119887119904⋆

(119901) 119888119904⋆

(119901) 119888119904⋆

0 (119901) 119887V⋆





119901of 119901-






infin 119888 1198880 ℓ119901 119887119904 119888119904

1198881199040 119887V 119887Vinfin(ii) 120583(119901) sube 120583

⋆

(119901) and ℓ⋆1 (119901) sube 119888119904⋆

(119901) sube 119888⋆

0 (119901) sube 119888⋆

(119901) sube

ℓ⋆

infin(119901) sube 120596

⋆



⋆

(119901)120572

120582⋆

(119901)120573 and 120582⋆(119901)120574 as follows

120582⋆

(119901)

120572

= 119908= (119908119896) isin 120596⋆

119908 ⊙ 119911 = (119908119896⊙ 119911119896)

isin ℓ⋆

1 (119901) forall119911 = (119911119896) isin 120582⋆

(119901)

120582⋆

(119901)

120573

= 119908= (119908119896) isin 120596⋆

119908 ⊙ 119911 = (119908119896⊙ 119911119896)

isin 119888119904⋆

(119901) forall119911 = (119911119896) isin 120582⋆

(119901)

120582⋆

(119901)

120574

= 119908= (119908119896) isin 120596⋆

119908 ⊙ 119911 = (119908119896⊙ 119911119896)

isin 119887119904⋆

(119901) forall119911 = (119911119896) isin 120582⋆

(119901)

(56)




Theorem 25 Let 0 = 120582⋆




(b) If 120582⋆(119901) sub 120583⋆

(119901) sub 120596⋆ then 120583⋆(119901)120573 sub 120582

⋆

(119901)120573

(c) 120582⋆ sub 120582⋆

120573120573

= (120582⋆

120573

)120573

(d) 119888⋆0 (119901)120573

= 119888⋆

(119901)120573

= ℓ⋆

infin(119901)120573

= ℓ⋆

1 (119901)

(e) ℓ⋆1 (119901)120573

= ℓ⋆

infin(119901)


119896) 119898 = (119898

119896)

and 119899 = (119899119896) isin 120582

⋆

(119901)120573



(119901)120573 for119898 119899 isin

120582⋆


⋆

(119901) Then (119898119896⊙ 119897119896) isin

119888119904⋆

(119901) and (119899119896⊙ 119897119896) isin 119888119904⋆


((119898119896oplus 119899119896) ⊙ 119897119896) = (119898

119896⊙ 119897119896) oplus (119899119896⊙ 119897119896) isin 119888119904

⋆

(119901)


(119901)

(57)

Hence 119898 oplus 119899 isin 120582⋆


120582⋆

(119901)120573 for any 119905 isin C⋆ and 119908 = (119908

119896) isin 120582

⋆

(119901)120573

Since (119908119896⊙ 119897119896) isin 119888119904

⋆

(119901) for all 119897 isin 120582⋆


119896⊙119908119896) ⊙ 119897119896) = 119905119896⊙ (119908119896⊙ 119897119896) isin

119888119904⋆

(119901) for all 119897 isin 120582⋆

(119901) we get 119905 ⊙ 119908 isin 120582⋆

(119901)120573



infin(119901)120573

sub 119888⋆

(119901)120573

sub 119888⋆

0 (119901)120573 by


ℓ⋆

infin(119901)120573 and 119888

⋆

0 (119901)120573

sub ℓ⋆


(119908119896) isin ℓ

⋆

1 (119901) and 119911 = (119911119896) isin ℓ

⋆



⋆sum

119896

1003817100381710038171003817119908119896⊙ 119911119896

1003817100381710038171003817

(119901119896)120573

le sup119896isinN

119911119896

(1119872)120573

times (120573sum

119896

119908119896

(119901119896)120573

)

(1119872)120573

le 119892⋆

(119911)

times 119892⋆

(119908) lt infin

(58)


⋆


⋆



ℓ⋆


⋆

0 (119901) with 119910 ⊙ 119909 notin 119888119904⋆


⋆



and⋆sum

119899119903minus1119896=119899119903minus1

119910119896

(119901119896)120573


119896) isin 119888

⋆

0 (119901)by 119909119896= (sgn⋆119910



sgn⋆ (119910) =

119910 ⊘

119910

119910 = 120579⋆

120579⋆

119910 = 120579⋆

(59)



⋆

119899119903minus1

sum

119896=119899119903minus1

(119910119896⊙119909119896)

(119901119896)120573

=⋆

119899119903minus1

sum

119896=119899119903minus1

[119910119896⊙ (sgn⋆119910

119896⊘ 119903)]

(119901119896)120573

=

1119903(119901119896)120573

⊙⋆

119899119903minus1

sum

119896=119899119903minus1

119910119896

(119901119896)120573

ge

1 (sup119901119896le 1)

(60)

Therefore 119910 ⊙ 119909 notin 119888119904⋆

(119901) and thus 119910 notin 119888⋆

0 (119901)120573

Hence 119888⋆0 (119901)120573

sub ℓ⋆



infin(119901) sub

(ℓ⋆

infin(119901)120573

)120573

= ℓ⋆

1 (119901)120573 since ℓ

⋆

infin(119901)120573

= ℓ⋆


119899) isin

ℓ⋆

1 (119901)120573

ℓ⋆




number (119896 + 1)2120573 such that

119908119899119896

(119901119896)120573



119909119899= (sgn⋆(119908

119899119896

) ⊘ (119896 + 1)2120573) if 119899 = 119899119896 and 120579

⋆


1 (119901) However

⋆sum

119899

(119908119899⊙119909119899)

(119901119896)120573

=⋆sum

119896

119908119899119896

(119901119896)120573

(119896 + 1)(2119901119896)120573= infin

(61)



tion and ℓ⋆

1 (119901)120573

sub ℓ⋆


proof


120573 120574)


(a) 120582⋆(119901)120572 sub 120582⋆

(119901)120573

sub 120582⋆

(119901)120574


120582⋆


(b) ℓ⋆1 (119901)120577

= ℓ⋆


infin(119901)120577

= ℓ⋆

1 (119901)

(c) If 120582⋆(119901) sub 120583⋆

(119901) sub 120596⋆ then 120583⋆(119901)120577 sub 120582

⋆

(119901)120577




Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Theorem 25 Let 0 = 120582⋆




(b) If 120582⋆(119901) sub 120583⋆

(119901) sub 120596⋆ then 120583⋆(119901)120573 sub 120582

⋆

(119901)120573

(c) 120582⋆ sub 120582⋆

120573120573

= (120582⋆

120573

)120573

(d) 119888⋆0 (119901)120573

= 119888⋆

(119901)120573

= ℓ⋆

infin(119901)120573

= ℓ⋆

1 (119901)

(e) ℓ⋆1 (119901)120573

= ℓ⋆

infin(119901)


119896) 119898 = (119898

119896)

and 119899 = (119899119896) isin 120582

⋆

(119901)120573



(119901)120573 for119898 119899 isin

120582⋆


⋆

(119901) Then (119898119896⊙ 119897119896) isin

119888119904⋆

(119901) and (119899119896⊙ 119897119896) isin 119888119904⋆


((119898119896oplus 119899119896) ⊙ 119897119896) = (119898

119896⊙ 119897119896) oplus (119899119896⊙ 119897119896) isin 119888119904

⋆

(119901)


(119901)

(57)

Hence 119898 oplus 119899 isin 120582⋆


120582⋆

(119901)120573 for any 119905 isin C⋆ and 119908 = (119908

119896) isin 120582

⋆

(119901)120573

Since (119908119896⊙ 119897119896) isin 119888119904

⋆

(119901) for all 119897 isin 120582⋆


119896⊙119908119896) ⊙ 119897119896) = 119905119896⊙ (119908119896⊙ 119897119896) isin

119888119904⋆

(119901) for all 119897 isin 120582⋆

(119901) we get 119905 ⊙ 119908 isin 120582⋆

(119901)120573



infin(119901)120573

sub 119888⋆

(119901)120573

sub 119888⋆

0 (119901)120573 by


ℓ⋆

infin(119901)120573 and 119888

⋆

0 (119901)120573

sub ℓ⋆


(119908119896) isin ℓ

⋆

1 (119901) and 119911 = (119911119896) isin ℓ

⋆



⋆sum

119896

1003817100381710038171003817119908119896⊙ 119911119896

1003817100381710038171003817

(119901119896)120573

le sup119896isinN

119911119896

(1119872)120573

times (120573sum

119896

119908119896

(119901119896)120573

)

(1119872)120573

le 119892⋆

(119911)

times 119892⋆

(119908) lt infin

(58)


⋆


⋆



ℓ⋆


⋆

0 (119901) with 119910 ⊙ 119909 notin 119888119904⋆


⋆



and⋆sum

119899119903minus1119896=119899119903minus1

119910119896

(119901119896)120573


119896) isin 119888

⋆

0 (119901)by 119909119896= (sgn⋆119910



sgn⋆ (119910) =

119910 ⊘

119910

119910 = 120579⋆

120579⋆

119910 = 120579⋆

(59)



⋆

119899119903minus1

sum

119896=119899119903minus1

(119910119896⊙119909119896)

(119901119896)120573

=⋆

119899119903minus1

sum

119896=119899119903minus1

[119910119896⊙ (sgn⋆119910

119896⊘ 119903)]

(119901119896)120573

=

1119903(119901119896)120573

⊙⋆

119899119903minus1

sum

119896=119899119903minus1

119910119896

(119901119896)120573

ge

1 (sup119901119896le 1)

(60)

Therefore 119910 ⊙ 119909 notin 119888119904⋆

(119901) and thus 119910 notin 119888⋆

0 (119901)120573

Hence 119888⋆0 (119901)120573

sub ℓ⋆



infin(119901) sub

(ℓ⋆

infin(119901)120573

)120573

= ℓ⋆

1 (119901)120573 since ℓ

⋆

infin(119901)120573

= ℓ⋆


119899) isin

ℓ⋆

1 (119901)120573

ℓ⋆




number (119896 + 1)2120573 such that

119908119899119896

(119901119896)120573



119909119899= (sgn⋆(119908

119899119896

) ⊘ (119896 + 1)2120573) if 119899 = 119899119896 and 120579

⋆


1 (119901) However

⋆sum

119899

(119908119899⊙119909119899)

(119901119896)120573

=⋆sum

119896

119908119899119896

(119901119896)120573

(119896 + 1)(2119901119896)120573= infin

(61)



tion and ℓ⋆

1 (119901)120573

sub ℓ⋆


proof


120573 120574)


(a) 120582⋆(119901)120572 sub 120582⋆

(119901)120573

sub 120582⋆

(119901)120574


120582⋆


(b) ℓ⋆1 (119901)120577

= ℓ⋆


infin(119901)120577

= ℓ⋆

1 (119901)

(c) If 120582⋆(119901) sub 120583⋆

(119901) sub 120596⋆ then 120583⋆(119901)120577 sub 120582

⋆

(119901)120577




Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article On the Classical Paranormed Sequence Spaces...

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