AAAAbstract and Applied Analysis1687-04091085-3375Hindawi Publishing Corporation95036410.1155/2011/950364950364Research ArticleThe Cesáro Core of Double SequencesKayadumanKuddusi1ÇakanCelal2ZizovicMalisa R.1Department of MathematicsFaculty of Science and ArtsGaziantep University27310 GaziantepTurkeygantep.edu.tr2Faculty of EducationInönü University44280 MalatyaTurkeyinonu.edu.tr20112162011201124032011240520112011Copyright © 2011 Kuddusi Kayaduman and Celal Çakan.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We have characterized a new type of core for double sequences, PC-core, and determined the necessary and sufficient conditions on a four-dimensional matrix A to yield PC-core{Ax}⊆α(P-core{x}) for all 2.

1. Introduction

A double sequence x=[xjk]j,k=0 is said to be convergent in the Pringsheim sense or P-convergent if for every ϵ>0 there exists an N such that |xjk-|<ε whenever j,k>N, . In this case, we write P-limx=. By c2, we mean the space of all P-convergent sequences.

A double sequence x is bounded if x=supj,k0|xjk|<. By 2, we denote the space of all bounded double sequences.

Note that, in contrast to the case for single sequences, a convergent double sequence need not be bounded. So, we denote by c2 the space of double sequences which are bounded and convergent.

A double sequence x=[xjk] is said to converge regularly if it converges in Pringsheim's sense and, in addition, the following finite limits exist: limkxjk=lj,(j=1,2,3,),limjxjk=tj,(k=1,2,3,). Let A=[ajkmn]j,k=0 be a four-dimensional infinite matrix of real numbers for all m,n=0,1,. The sums ymn=j=0k=0ajkmnxjk are called the A-transforms of the double sequence x=[xjk]. We say that a sequence x=[xjk] is A-summable to the limit if the A-transform of x=[xjk] exists for all m,n=0,1, and is convergent to in the Pringsheim sense, that is, limp,qj=0pk=0qajkmnxjk=ymn,limm,nymn=l.

We say that a matrix A is bounded-regular if every bounded-convergent sequence x is A-summable to the same limit and the A-transforms are also bounded. The necessary and sufficient conditions for A to be bounded-regular or RH-regular (cf., Robison ) are limm,najkmn=0,(j,k=0,1,),limm,nj=0k=0ajkmn=1,limm,nj=0|ajkmn|=0,(k=0,1,),limm,nk=0|ajkmn|=0,(j=0,1,),j=0k=0|ajkmn|C<(m,n=0,1,). A double sequence x=[xjk] is said to be almost convergent (see ) to a number L if limp,qsups,t01pqj=0pk=0qxs+j,t+k=L. Let σ be a one-to-one mapping from into itself. The almost convergence of double sequences has been generalized to the σ-convergence in  as follows: limp,qsups,t01pqj=0pk=0qxσj(s),σk(t)=l, where σj(s)=σ(σj-1(s)). In this case, we write σ-limx=. By Vσ2, we denote the set of all σ-convergent and bounded double sequences. One can see that in contrast to the case for single sequences, a convergent double sequence need not be σ-convergent. But every bounded convergent double sequence is σ-convergent. So, c2Vσ22. In the case σ(i)=i+1, σ-convergence of double sequences reduces to the almost convergence. A matrix A=[ajkmn]j,k=0 is said to be σ-regular if AxV2σ for x=[xjk]c2 with σ-limAx=limx, and we denote this by A(c2,V2σ)reg, (see [5, 6]). Mursaleen and Mohiuddine defined and characterized σ-conservative and σ-coercive matrices for double sequences .

A double sequence x=[xjk] of real numbers is said to be Cesáro convergent (or C1-convergent) to a number L if and only if xC1, where C1={xl2:limp,qTpq(x)=L;L=C1-limx},Tpq(x)=1(p+1)(q+1)j=1pk=1qxjkmn. We shall denote by C1 the space of Cesáro convergent (C1-convergent) double sequences.

A matrix A=(ajkmn) is said to be C1-multiplicative if AxC1 for x=[xjk]c2 with C1-limAx=αlimx, and in this case we write A(c2,C1)α. Note that if α=1, then C1-multiplicative matrices are said to be C1-regular matrices.

Recall that the Knopp core (or K-core) of a real number single sequence x=(xk) is defined by the closed interval [(x),L(x)], where (x)=liminfx and L(x)=limsupx. The well-known Knopp core theorem states (cf., Maddox  and Knopp ) that in order that L(Ax)L(x) for every bounded real sequence x, it is necessary and sufficient that A=(ank) should be regular and limnk=0|ank|=1. Patterson  extended this idea for double sequences by defining the Pringsheim core (or P-core) of a real bounded double sequence x=[xjk] as the closed interval [P-liminfx,P-limsupx]. Some inequalities related to the these concepts have been studied in [5, 9, 10]. Let L*(x)  =limsupp,qsups,t1pqj=0pk=0qxj+s,k+t,Cσ(x)=limsupp,qsups,t1pqj=0pk=0qxσj(s),σk(t).

Then, MR- (Moricz-Rhoades) and σ-core of a double sequence have been introduced by the closed intervals [-L*(-x),L*(-x)] and [-Cσ(-x),Cσ(x)], and also the inequalities L(Ax)L*(x),L*(Ax)L(x),L*(Ax)L*(x),L(Ax)Cσ(x),Cσ(Ax)L(x) have been studies in [35, 11].

In this paper, we introduce the concept of C1-multiplicative matrices and determine the necessary and sufficient conditions for a matrix A=(ajkmn) to belong to the class (c2,C1)α. Further we investigate the necessary and sufficient conditions for the inequality C1*(Ax)αL(x) for all x2.

2. Main Results

Let us write C1*(x)=limsupp,q1(p+1)(q+1)j=0pk=0qxjk. Then, we will define the PC-core of a realvalued bounded double sequence x=[xjk] by the closed interval [-C1*(-x),C1*(x)]. Since every bounded convergent double sequence is Cesáro convergent, we have C1*(x)P-limsupx, and hence it follows that PC-core(x)P-core(x) for a bounded double sequence x=[xjk].

Lemma 2.1 ..

A matrix A=(ajkmn) is C1-multiplicative if and only if limp,qβ(j,k,p,q)=0(j,k=0,1,),limp,qj=0k=0β(j,k,p,q)=α,limp,qj=0|β(j,k,p,q)|=0(k=0,1,),limp,qk=0|β(j,k,p,q)|=0(j=0,1,),j=0k=0|ajkmn|C<,(m,n=0,1,), where the lim means P-lim and β(j,k,p,q)=1(p+1)(q+1)j=0pk=0qajkmn.

Proof.

Sufficiency. Suppose that the conditions (2.2)-(2.6) hold and x=[xjk]c2 with P-limj,kxjk=L, say. So that for every ϵ>0 there exists N>0 such that |xjk|<||+ϵ whenever j,k>N.

Then, we can write j=0k=0β(j,k,p,q)xjk=j=0Nk=0Nβ(j,k,p,q)xjk+j=Nk=0N-1β(j,k,p,q)xjk+j=0N-1k=Nβ(j,k,p,q)xjk+j=N+1k=N+1β(j,k,p,q)xjk. Therefore, |j=0k=0β(j,k,p,q)xjk|xj=0Nk=0N|β(j,k,p,q)|+xj=Nk=0N-1|β(j,k,p,q)xjk|+xj=0N-1k=N|β(j,k,p,q)|+(|L|+ϵ)|j=N+1k=N+1β(j,k,p,q)|. Letting p,q and using the conditions (2.2)–(2.6), we get |limp,qj=0k=0β(j,k,p,q)xjk|(|L|+ϵ)α. Since ϵ is arbitrary, C1-limAx=αL. Hence A(c2,C1)α, that is, A is C1-multiplicative.

Necessity.

Suppose that A is C1-multiplicative. Then, by the definition, the A-transform of x exists and AxC1 for each xc2. Therefore, Ax is also bounded. Then, we can write supm,nj=0k=0|ajkmnxjk|<M<, for each xc2. Now, let us define a sequence y=[yjk] by yjk={sgnajkmn,0jr,0kr,0,otherwise,m,n=0,1,2,. Then, the necessity of (10) follows by considering the sequence y=[yjk] in (2.11).

Also, by the assumption, we have limp,qj=0k=0β(j,k,p,q)xjk=αlimj,kxjk. Now let us define the sequence eil as follows: eil={1,(j,k)=(i,l),0,otherwise, and write sl=ieil(l), ri=leil(i). Then, the necessity of (2.2), (2.4), and (2.5) follows from C1-limAeil, C1-limArj and C1-limAsk, respectively.

Note that when α=1, the above theorem gives the characterization of A(c2,C1)reg. Now, we are ready to construct our main theorem.

Theorem 2.2.

For every bounded double sequence x, C1*(Ax)αL(x), or (PC-core{Ax}α(P-core{x})) if and only if A is C1-multiplicative and limsupp,qj=0k=0|β(j,k,p,q)|=α.

Proof.

Necessity. Let (2.15) hold and for all x2. So, since c22, then, we get α(-L(-x))-C1*(-Ax)C1*(Ax)αL(x). That is, αliminfx-C1*(-Ax)C1*(Ax)αlimsupx, where -C1*(-Ax)=liminfp,qj=0k=0β(j,k,p,q)xjk. By choosing x=[xjk]c2, we get from (2.17) that -C1*(-Ax)=C1*(Ax)=C1-limAx=αlimx. This means that A is C1-multiplicative.

By Lemma  3.1 of Patterson , there exists a y2 with ||y||1 such that C1*(Ay)=limsupp,qj=0k=0β(j,k,p,q). If we choose y=v=[vjk], it follows vjk={1        if  j=k,0,        elsewhere. Since vjk1, we have from (2.15) that α=C1*(Av)=limsupp,qj=0k=0|β(j,k,p,q)|αL(vjk)αvα. This gives the necessity of (2.16).

Sufficiency.

Suppose that A is C1-regular and (2.16) holds. Let x=[xjk] be an arbitrary bounded sequence. Then, there exist M,N>0 such that xjkK for all j,k0. Now, we can write the following inequality: |j=0k=0β(j,k,p,q)xjk|=|j=0k=0(|β(j,k,p,q)|+β(j,k,p,q)2-|β(j,k,p,q)|-β(j,k,p,q)2)xjk|j=0k=0|β(j,k,p,q)||xjk|+j=0k=0|(|β(j,k,p,q)|-β(j,k,p,q))xjk|xj=0Mk=0N|β(j,k,p,q)|+xj=M+1k=0N|β(j,k,p,q)|+xj=0Mk=N+1|β(j,k,p,q)|+supj,kM,N|xjk|j=M+1k=N+1|β(j,k,p,q)|+xj=0k=0(|β(j,k,p,q)|-β(j,k,p,q)). Using the condition of C1-multiplicative and condition (2.16), we get C1*(Ax)αL(x). This completes the proof of the theorem.

Theorem 2.3.

For x,y2, if C1-lim|x-y|=0, then C1-core{x}=C1-core{y}.

Proof.

Since C2-lim|x-y|=0, we have C1-lim(x-y)=0 and C1-lim(-(x-y))=0. Using definition of C1-core, we take C1*(x-y)=-C1*(-(x-y))=0. Since C1* is sublinear,   0=-C1*(-(x-y))-C1*(-x)-C1*(y). Therefore, C1*(y)-C1*(-x). Since -C1*(-x)C1*(x), this implies that C1*(y)C1*(x).By an argument similar as above, we can show that C1*(x)C1*(y). This completes the proof.

Acknowledgment

The authors would like to state their deep thanks to the referees for their valuable suggestions improving the paper.

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