Open access peer-reviewed chapter

A Study of Bounded Variation Sequence Spaces

By Vakeel Ahmad Khan, Hira Fatima and Mobeen Ahmad

Submitted: September 19th 2018Reviewed: October 9th 2018Published: April 23rd 2019

DOI: 10.5772/intechopen.81907

Abstract

In the theory of classes of sequence, a wonderful application of Hahn-Banach extension theorem gave rise to the concept of Banach limit, i.e., the limit functional defined on c can be extended to the whole space l ∞ and this extended functional is called as the Banach limit. After that, in 1948 Lorentz used this concept of a week limit to introduce a new type of convergence, named as the almost convergence. Later on, Raimi generalized the concept of almost convergence known as σ − convergence and the sequence space BV σ was introduced and studied by Mursaleen. The main aim of this chapter is to study some new double sequence spaces of invariant means defined by ideal, modulus function and Orlicz function. Furthermore, we also study several properties relevant to topological structures and inclusion relations between these spaces.

Keywords

• invariant mean
• bounded variation
• ideal
• filter
• I-convergence
• Orlicz function
• modulus function
• paranorm

1. Introduction

The concept of convergence of a sequence of real numbers has been extended to statistical convergence independently by Fast  and Schoenberg . There has been an effort to introduce several generalizations and variants of statistical convergence in different spaces. One such very important generalization of this notion was introduced by Kostyrko et al.  by using an ideal I of subsets of the set of natural numbers, which they called I-convergence. After that the idea of I-convergence for double sequence was introduced by Das et al.  in 2008.

Throughout a double sequence is defined by x=xijand we denote 2ωshowing the space of all real or complex double sequences.

Let Xbe a nonempty set then a family I2Xis said to be an ideal in Xif ØI, Iis additive, i.e., for all A,BIABIand Iis hereditary, i.e., for all AI,BABI. A nonempty family of sets F2Xis said to be a filter on Xif for all A,BFimplies ABFand for all AFwith ABimplies BF. An ideal I2Xis said to be nontrivial if I2X, this non trivial ideal is said to be admissible if Ix:xXand is said to be maximal if there cannot exist any nontrivial ideal JIcontaining Ias a subset. For each ideal Ithere is a filter FIcalled as filter associate with ideal I, that is

FI=KX:KcI,whereKc=X\K.E1

A double sequence x=xij2ωis said to be I-convergent [5, 6, 7, 8] to a number Lif for every ϵ>0, we have ijN×N:xijLϵI.In this case, we write Ilimxij=L.A double sequence x=xij2ωis said to be I-Cauchy if for every ϵ>0there exists numbers m=mϵ,n=nϵsuch that ijN×N:xijxmnϵI.

A continuous linear functional ϕon lis said to be an invariant mean [9, 10] or σ-mean if and only if:

1. ϕx0where the sequence x=xkhas xk0for all k,

2. ϕe=1where e=1,1,1,1,

3. ϕxσn=ϕxfor all xl,

where σbe an injective mapping of the set of the positive integers into itself having no finite orbits.

If x=xk, write Tx=Txk=xσk, so we have

Vσ=x=xk:limmtm,kx=LuniformlyinkL=σlimxE2

where m0,k>0.

tm,kx=xk+xσk++xσmkm+1andt1,k=0,E3

where σmkdenote the mth-iterate of σkat k. In this case σis the translation mapping, that is, σk=k+1,σmean is called a Banach limit  and Vσ, the set of bounded sequences of all whose invariant means are equal, is the set of almost convergent sequences. The special case of (3) in which σk=k+1was given by Lorentz  and the general result can be proved in a similar way. It is familiar that a Banach limit extends the limit functional on cin the sense that

ϕx=limx,forallxc.E4

Definition 1.1 A sequence xlis of σ-bounded variation if and only if:

(i) ϕm,kxconverges uniformly in k,

(ii) limmtm,kx,which must exist, should take the same value for all k.

We denote by BVσ, the space of all sequences of σ-bounded variation:

BVσ=xl:mϕm,kx<uniformlyink.

is a Banach space normed by

x=supkm=0ϕm,kx.E5

A function M:00is said to be an Orlicz function [13, 14] if it satisfies the following conditions:

(i) M is continuous, convex and non-decreasing,

(ii) M0=0,Mx>0and Mxas x.

Remark 1.1 If the convexity of an Orlicz function is replaced by Mx+yMx+My, then this function is called Modulus function [15, 16, 17]. If Mis an Orlicz function, then MλXλMxfor all λwith 0<λ<1.An Orlicz function Mis said to satisfy Δ2-condition for all values of uif there exists a constant K>0such that MLuKLMufor all values of L>1.

Definition 1.2 A double sequence space Xis said to be:

[i] solid or normal if xijXimplies that αijxijXfor all sequence of scalars αijwith αij<1for all ijN×N.

[ii] symmetric if xπijXwhenever xijX, where πijis a permutation on N×N.

[iii] sequence algebra if xijyijEwhenever xij,yijX.

[iv] convergence free if yijXwhenever xijXand xij=0implies yij=0, for all ijN×N.

Definition 1.3 Let K=nikj:ij:n1<n2<n3<.andk1<k2<k3<.N×Nand Xbe a double sequence space. A K-step space of Xis a sequence space

λkE=αijxij:xijX.

A canonical preimage of a sequence xnikjXis a sequence bnkXdefined as follows:

bnk=ank,forn,kK0,otherwise.

A sequence space Xis said to be monotone if it contains the canonical preimages of all its step spaces.

The following subspaces lp,lp,cpandc0pwhere p=pkis a sequence of positive real numbers. These subspaces were first introduced and discussed by Maddox . The following inequalities will be used throughout the section. Let p=pijbe a double sequence of positive real numbers . For any complex λwith 0<pijsupijpij=G<, we have

λpijmax1λG.

Let D=max12G1andH=max1supijpij, then for the sequences aijand bijin the complex plane, we have

aij+bijpijCaijpij+bijpij.

2. Bounded variation sequence spaces defined by Orlicz function

In this section, we define and study the concepts of I-convergence for double sequences defined by Orlicz function and present some basic results on the above definitions [8, 20].

2BVσIM=xij2w:IlimMϕmnijxLρ=0forsomeLCρ>0E6
20BVσIM=xij2w:IlimMϕmnijxρ=0ρ>0,E7
2BVσIM=xij2w:ij:k>0s.tMϕmnijxρkIρ>0E8
2BVσM=xij2w:supMϕmnijxρ<ρ>0.E9

Now, we read some theorems based on these sequence spaces. These theorems are of general importance as indispensable tools in various theoretical and practical problems.

Theorem 2.1 Let M1,M2be two Orlicz functions with Δ2condition, then

(a)χM2χM1M2

(b) χM1χM2χM1+M2for χ=2BVσI,20BVσI.

Proof. (a) Let x=xij20BVσIM2be an arbitrary element, so there exists ρ>0such that

IlimM2ϕmnijxρ=0.E10

Let ϵ>0and choose δwith 0<δ<1such that M1t<εfor 0<tδ.

Write yij=M2ϕmnijxρ. Consider,

limijM1yij=limyijδ,i,jNM1yij+limyij>δ,i,jNM1yij.E11

Now, since M1is an Orlicz function so we have M1λxλM1x,0<λ<1. Therefore, we have

limyijδ,i,jNM1yijM12limyijδ,i,jNyij.E12

For yij>δ, we have yij<yijδ<1+yijδ. Now, since M1is non-decreasing and convex, it follows that,

M1yij<M11+yijδ<12M12+12M12yijδ.E13

Since M1satisfies the Δ2-condition, so we have

M1yij<12KyijδM12+12KM12yijδ<12KyijδM12+12KyijδM12=KyijδM12.E14

This implies that,

M1yij<KyijδM12.E15

Hence, we have

limyij>δ,i,jNM1yijmax1Kδ1M12limyij>δ,i,jNyij.E16

Therefore from (12) and (16), we have

IlimijM1yij=0.IlimijM1M2ϕmnijxρ=0.

This implies that x=xij20BVσIM1M2. Hence χM2χM1M2for χ=20BVσI.The other cases can be proved in similar way.

(b) Let x=xij20BVσIM120BVσIM2. Let ϵ>0be given. Then there exist ρ>0,such that

IlimijM1ϕmnijxρ=0,E17

and

IlimijM2ϕmnijxρ=0.E18

Therefore

IlimijM1+M2ϕmnijxρ=IlimijM1ϕmnijxρ+IlimijM2ϕmnijxρ,

from Eqs. (17) and (18), we get

IlimijM1+M2ϕmnijxρ=0.

so we have x=xij20BVσIM1+M2.

Hence, 20BVσIM120BVσIM220BVσIM1+M2.For χ=2BVσIthe inclusion are similar.

Corollary χχMfor χ=2BVσIand 2BVσI.

Proof. For this let Mx=x, for all x=xijX. Let us suppose that x=xij20BVσI. Then for any given ϵ>0, we have

ij:ϕmnijxϵI.

Now let A1=ij:ϕmnijx<ϵI,be such that A1cI. Consider for ρ>0,

Mϕmnijxρ=ϕmnijxρ<ϵρ<ϵ.

This implies that IlimMϕmnijxρ=0, which shows that x=xij20BVσIM.

Hence, we have

20BVσI20BVσIM.χχM.

Using the definition of convergence free sequence space, let us give another theorem which will be of particular importance in our future work:

Theorem 2.2 The spaces 20BVσIMand 2BVσIMare not convergence free.

Example 2.1 To show this let I=Ifand Mx=x, for all x=0. Now consider the double sequence xij,yijwhich defined as follows:

xij=1i+jandyij=i+j,i,jN.

Then we have xijbelong to both 20BVσIMand 2BVσIM,but yijdoes not belong to 20BVσIMand 2BVσIM. Hence, the spaces 20BVσIMand 2BVσIMare not convergence free.

To gain a good understanding of these double sequence spaces and related concepts, let us finally look at this theorem on inclusions:

Theorem 2.3 Let M be an Orlicz function. Then

20BVσIM2BVσIM2BVσIM.

Proof. For this let us consider x=xij20BVσIM.It is obvious that it must belong to 2BVσIM.Now consider

MϕmnijxLρMϕmnijxρ+MLρ.

Now taking the limit on both sides we get

IlimijMϕmnijxLρ=0.

Hence x=xij2BVσIM.Now it remains to show that

2BVσIM2BVσIM.

For this let us consider x=xij2BVσIMthis implies that there exist ρ>0s.t

IlimijMϕmnijxLρ=0.

Now consider,

MϕmnijxρMϕmnijxLρ+MLρ.

Now taking the supremum on both sides, we get

supijMϕmnijxρ<.

Hence, x=xij2BVσIM.

3. Paranorm bounded variation sequence spaces

In this section we study double sequence spaces by using the double sequences of strictly positive real numbers p=pijwith the help of BVσspace and an Orlicz function M. We study some of its properties and prove some inclusion relations related to these new spaces. For m, n 0, we have

2BVσIMp={xij2ω:ij:MϕmnijxLρpijϵI;forsomeLC,ρ>0}E19
20BVσIMp=xij2ω:ij:MϕmnijxρpijϵIρ>0,E20
2BVσIMp=xij2ω:ij:K>0:MϕmnijxρpijKIρ>0E21
2lMp=xij2ω:supMϕmnijxρpij<ρ>0.E22

We also denote

2MBVσIMp=2BVσIMp2lMp

and

20MBVσIMp=20BVσIMp2lMp.

We can now state and proof the theorems based on these double sequence spaces which are as follows:

Theorem 3.1 Let p=pij2lthen the classes of double sequence 2MBVσIMpand 20MBVσIMpare paranormed spaces, paranormed by

gxij=infi,j1ρpijH:supijMϕmnijxρpij1,forsomeρ>0}

where H=max1supijpij.

Proof. P1It is clear that gx=0if and only if x=0.

P2gx=gxis obvious.

P3Let x=xij,y=yij2MBVσIMp. Now for ρ1,ρ2>0, we denote

A1=ρ1:supijMϕmnijxρpij1E23
A2=ρ2:supijMϕmnijxρpij1E24

Let us take ρ3=ρ1+ρ2. Then by using the convexity of M, we have

Mϕmnijx+yρρ1ρ1+ρ2Mϕmnijxρ1+ρ2ρ1+ρ2Mϕmnijyρ2

which in terms give us

supijMϕmnijx+yρpij1

and

gxij+yij=infρ1+ρ2pijH:ρ1A1ρ2A2infρ1pijH:ρ1A1+infρ2pijH:ρ2A2=gxij+gyij.

Therefore gx+ygx+gy.

P4Let λijbe a double sequence of scalars with λijλijand xij,L2MBVσIMpsuch that

xijLij,

in the sense that

gxijL0ij.

Then, since the inequality

gxijgxijL+gL

holds by subadditivity of g, the sequence gxijis bounded.

Therefore,

gλijxijλL=gλijxijλxij+λxijλL=gλijλxij+λxijLgλijλxij+gλxijLλijλpijMgxij+λpijMgxijL0

as ij. That implies that the scalar multiplication is continuous. Hence 2MBVσIMpis a paranormed space. For another space 20MBVσIMp, the result is similar.

We shall see about the separability of these new defined double sequence spaces in the next theorem.

Theorem 3.2 The spaces 2MBVσIMpand 20MBVσIMpare not separable.

Example 3.1 By counter example, we prove the above result for the space 2MBVσIMp.

Let A be an infinite subset of increasing natural numbers, i.e., AN×Nsuch that AI.

Let

pij=1,ifijA2,otherwise.
Let P0=xij:xij=0or1forijMandxij=0otherwise.

Since A is infinite, so P0is uncountable. Consider the class of open balls

B1=Bz12:zP0.

Let C1be an open cover of 2MBVσIMpcontaining B1.

Since B1is uncountable, so C1cannot be reduced to a countable subcover for 2MBVσIMp. Thus 2MBVσIMpis not separable.

We shall now introduce a theorem which improves our work.

Theorem 3.3 Let pijand qijbe two double sequences of positive real numbers. Then 20MBVσIMp20MBVσIMqif and only if limi,jKinfpijqij>0, where KcN×Nsuch that KI.

Proof. Let limi,jKinfpijqij>0and xij20MBVσIMq.Then, there exists β>0such that pij>βqijfor sufficiently large ijK.

Since xij20MBVσIMq.For a given ϵ>0,there exist ρ>0such that

B0=ijN×N:MϕmnijxρqijϵI.

Let G0=KcB0.Then for all sufficiently large ijG0.

ij:Mϕmnijxρpijϵ ij:MϕmnijxρβqijϵI.

Therefore, xij20MBVσIMp. The converse part of the result follows obviously.

Remark 3.1 Let pijand qijbe two double sequences of positive real numbers. Then 20MBVσIMq20MBVσIMpif and only if limi,jKinfqijpij>0and 20MBVσIMq=20MBVσIMpif and only if limi,jKinfpijqij>0and limi,jKinfqijpij>0, where KcN×Nsuch that KI.

Theorem 3.4 The set 2MBVσIMpis closed subspace of 2lMp.

Proof. Let xijpqbe a Cauchy double sequence in 2MBVσIMpsuch that xpqx. We show that x2MBVσIMp.Since, xijpq2MBVσIMp, then there exists apq, and ρ>0such that

ij:MϕmnijxpqapqρpijϵI.

We need to show that

(1) apqconverges to a.

(2) If U=ij:Mϕmnijxpqaρpij<ϵ, then UcI.

Since xijpqbe a Cauchy double sequence in 2MBVσIMpthen for a given ϵ>0there exists k0Nsuch that

supijMϕmnijxpqϕmnijxrsρpij<ϵ3,forallp,q,r,sk0.

For a given ϵ>0, we have

Bpqrs=ij:Mϕmnijxpqϕmnijxrsρpij<ϵ3M,Bpq=ij:Mϕmnijxpqapqρpij<ϵ3M,Brs=ij:Mϕmnijxrsarsρpij<ϵ3M.

Then Bpqrsc,Bpqc,BrscI. Let Bc=BpqrscBpqcBrsc,

where B=ij:Mapqarsρpij<ϵ,then BcI. We choose k0Bc, then for each p,q,r,sk0, we have

ij:Mapqarsρpij<ϵ}[{ijN:Mϕmnijxpqϕmnijxrsρpij<ϵ3Mij:Mϕmnijxpqapqρpij<ϵ3Mij:Mϕmnijxrsarsρpij<ϵ3M].

Then apqis a Cauchy double sequence in C. So, there exists a scalar aCsuch that apqa,asp,q.

(2) For the next step, let 0<δ<1be given. Then, we show that if

U=ij:Mϕmnijxpqaρpijδ

then UcI.Since xpqx, then there exists p0,q0Nsuch that,

P=ij:Mϕmnijxp0q0ϕmnijxρpij<δ3DHE25

where D=max12G1,G=supijpij0andH=max1supijpijimplies PcI. The number p0q0can be so chosen that together with (25), we have

Q=ij:Map0q0aρpij<δ3DH

such that QcI.Since xijpq2MBVσIMp.

We have

ij:Mϕmnijxp0q0ap0q0ρpijδI.

Then we have a subset SN×Nsuch that ScI, where

S=ij:Mϕmnijxp0q0ap0q0ρpij<δ3DH.

Let Uc=PcQcSc,where

U=ij:Mϕmnijxaρpij<δ

Therefore, for ijUc, we have

ij:Mϕmnijxaρpij<δij:Mϕmnijxp0q0ϕmnijxρpij<δ3DHij:Map0q0aρpij<δ3DM{ij:Mϕmnijxp0q0ap0q0ρpij<δ3DH}.

Hence the result 2MBVσIMp2lMpfollows.

Since the inclusions 2MBVσIMp2lMpand 20MBVσIMp2lMpare strict so in view of Theorem (3.3), we have the following result.

The above theorem is interesting and itself will have various applications in our future work.

4. Bounded variation sequence spaces defined by modulus function

In this section, we study some new double sequence spaces of invariant means defined by ideal and modulus function. Furthermore, we also study several properties relevant to topological structures and inclusion relations between these spaces. The following classes of double sequence spaces are as follows:

2BVσIf=xij2ω:ij:m,n=0fϕmnijxLϵIforsomeLC;E26
20BVσIf=xij2ω:ij:m,n=0fϕmnijxϵI;E27
2BVσIf=xij2ω:ij:K>0:m,n=0fϕmnijxKI;E28
2BVσf=xij2ω:supi,jm,n=0fϕmnijx<.E29

We also denote

2MBVσIf=2BVσIf2BVσf

and

20MBVσIf=20BVσIf2BVσf.

We shall now consider important theorems of these double sequence spaces by using modulus function.

Theorem 4.1 For any modulus function f, the classes of double sequence 20BVσIf, 2BVσIf,20MBVσIfand 2MBVσIfare linear spaces.

Proof. Suppose x=xijand y=yij2BVσIfbe any two arbitrary elements. Let α,βare scalars. Now, since xij,yij2BVσIf. Then this implies that there exists some positive numbers L1,L2Cand such that the sets

A1=ij:m,n=0fϕmnijxL1ϵ2I,E30
A2=ij:m,n=0fϕmnijyL2ϵ2I.E31

Now, assume

B1=ij:m,n=0fϕmnijxL1<ϵ2FI,E32
B2=ij:m,n=0fϕmnijyL2<ϵ2FIE33

be such that B1c,B2cI.Since fis a modulus function, we have

m,n=0fϕmnijαx+βyαL1+βL2=m,n=0fαϕmnijx+βϕmnijyαL1+βL2=m,n=0fαϕmnijxL1+βϕmnijyL2m,n=0fαϕmnijxL1+m,n=0fβϕmnijyL2m,n=0fϕmnijxL1+m,n=0fϕmnijyL2ϵ2+ϵ2=ϵ.

This implies that ij:m,n=0fϕmnijαx+βyαL1+βL2ϵI.Thus αxij+βyij2BVσIf.As xijand yijare two arbitrary element then αxij+βyij2BVσIffor all xij,yij2BVσIfand for all scalars α,β. Hence 2BVσIfis linear space. The proof for other spaces will follow similarly.▪

We may go a step further and define another theorem on ideal convergence which basically depends upon the set in the filter associated with the same ideal.

Theorem 4.2 A sequence x=xij2MBVσIfI-convergent if and only if for every ϵ>0, there exists Mε,NεNsuch that

ij:m,n=0fϕmnijxijϕmnijxMϵ,Nϵ<ϵFI.

Proof. Let x=xij2MBVσIf. Suppose Ilimx=L. Then, the set

Bϵ=ij:m,n=0fϕmnijxijL<ϵ2FI,forallϵ>0.

Fix Mε,NεBε.Then we have

m,n=0fϕmnijxijϕmnijxMϵ,Nϵm,n=0fϕmnijxMϵ,NϵL+m,n=0fLϕmnijxij<ϵ2+ϵ2=ϵ

which holds for all ijBϵ.

Hence

ij:m,n=0fϕmnijxijϕmnijxMϵ,Nϵ<ϵFI.

Conversely, suppose that

ij:m,n=0fϕmnijxijϕmnijxMϵ,Nϵ<ϵFI.

Then, being fa modulus function and by using basic triangular inequality, we have

ij:m,n=0fϕmnijxijm,n=0fϕmnijxMϵ,Nϵ<ϵFI,forallϵ>0.

Then, the set

Cε=ij:m,n=0fϕmnijxijm,n=0fϕmnijxMϵ,Nϵϵm,n=0fϕmnijxMϵ,Nϵ+ϵFI.

Let Jϵ=m,n=0f(ϕmnijxMϵ,Nϵ)ϵm,n=0f(ϕmnijxMϵ,Nϵ)+ϵ.

If we fix ϵ>0then, we have CϵFIas well as Cϵ2FI.

Hence CϵCϵ2FI. This implies that

J=JϵJϵ2ϕ.

That is

ij:m,n=0fϕmnijxijJFI.

This shows that

diamJdiamJϵ

where the diamJdenotes the length of interval J. In this way, by induction we get the sequence of closed intervals

Jϵ=I0I1I2Ik

with the property that diamIk12diamIk1for k=2,3,4and ij:m,n=0 fϕmnijxijIkFIfor k=1,2,3,4.

Then there exists a ξIkwhere kNsuch that

ξ=Ilimi,jm,n=0fϕmnijxij,

showing that x=xij2MBVσIfis I-convergent. Hence the result holds.

As the reader knows about solid and monotone sequence space now turn to theorem on solid and monotone double sequence spaces of invariant mean defined by ideal and modulus function.

Theorem 4.3 For any modulus function f, the spaces 20BVσIfand 20MBVσIfare solid and monotone.

Proof. We consider 20BVσIfand for 20MBVσIfthe proof shall be similar.

Let x=xij20BVσIfbe an arbitrary element, then the set

ij:m,n=0f(ϕmnijx)ϵI.E34

Let αijbe a sequence of scalars with αij1for all i,jN.

Now, since fis a modulus function. Then the result follows from (2.18) and the inequality

fαijϕmnijxαijfϕmnijxfϕmnijx.

Therefore,

ij:m,n=0f(αijϕmnijx)ϵij:m,n=0f(ϕmnijx)ϵI

implies that

ij:m,n=0f(αijϕmnijx)ϵI.

Thus we have αijxij20BVσIf.Hence 20BVσIfis solid. Therefore 20BVσIfis monotone. Since every solid sequence space is monotone.

Remark 4.1 The space 2BVσIfand 2MBVσIfare neither solid nor monotone in general.

Example 4.1 Here we give counter example for establishment of this result. Let X=2BVσIand 2MBVσI. Let us consider I=Ifand fx=x, for all x=xijand xij0. Consider, the K-step space XKfof Xfdefined as follows:

Let x=xijXfand y=yijXKfbe such that

yij=xij,ifi,jareeven0,otherwise.

Consider the sequence xijdefined by xij=1for all i,jN.

Then, x=xij2BVσIfand 2MBVσIf, but K-step space preimage does not belong to BVσIfand 2MBVσIf. Thus, 2BVσIfand 2MBVσIfare not monotone and hence they are not solid.

After discussing about solid and monotone sequence space now we come to the concept of sequence algebra which will help to understand our further work.

Theorem 4.4 For any modulus function f, the spaces 20BVσIfand 2BVσIfare sequence algebra.

Proof. Let x=xij,y=yij20BVσIfbe any two arbitrary elements.

Then, the sets

ij:m,n=0f(ϕmnijx)ϵI

and

ij:m,n=0f(ϕmnijy)ϵI.

Therefore,

ij:m,n=0f(ϕmnijx.ϕmnijy)ϵI.

Thus, we have xij.yij20BVσIf.Hence 20BVσIfis sequence algebra. And for 2BVσIfthe result can be proved similarly.

Remark 4.2 If Iis not maximal and IIfthen the spaces 2BVσIfand 20BVσIfare not symmetric.

Example 4.2 Let AIbe an infinite set and fx=xfor all x=xijand xij0.If

xij=1,ifijA0,otherwise

Then, it is clearly seen that xij20BVσIf2BVσIf.

Let KN×Nbe such that KIandKcI.Let ϕ:KAandψ:KcAcbe a bijective maps (as all four sets are infinite). Then, the mapping π:N×NN×Ndefined by

πij=ϕij,ifijKψij,otherwise.

is a permutation on N×N.

But xπij2BVσIfand hence xπij20BVσIfshowing that 2BVσIfand 20BVσIfare not symmetric double sequence spaces.

5. Conclusion

In this chapter, we study different forms of BVσdouble sequence spaces of invariant means with the help of ideal, operators and some functions such as Orlicz function and modulus function. The chapter shows the potential of the new theoretical tools to deal with the convergence problems of sequences in sigma bounded variation, occurring in many branches of science, engineering and applied mathematics.

Acknowledgements

The authors express their gratitude to our Chairman Prof. Mohammad Imdad for his advices and continuous support. We are grateful to the learned referees for careful reading of our manuscript and considering the same for publication.

Conflict of interest

The authors declare that they have no competing interests.

More

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

How to cite and reference

Cite this chapter Copy to clipboard

Vakeel Ahmad Khan, Hira Fatima and Mobeen Ahmad (April 23rd 2019). A Study of Bounded Variation Sequence Spaces, Applied Mathematics, Bruno Carpentieri, IntechOpen, DOI: 10.5772/intechopen.81907. Available from:

More statistics for editors and authors

Login to your personal dashboard for more detailed statistics on your publications.

Related Content

Next chapter

Simple Approach to Special Polynomials: Laguerre, Hermite, Legendre, Tchebycheff, and Gegenbauer

By Vicente Aboites and Miguel Ramírez

First chapter