Abstract and Applied Analysis

Abstract and Applied Analysis / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 950364 | https://doi.org/10.1155/2011/950364

Kuddusi Kayaduman, Celal ร‡akan, "The Cesรกro Core of Double Sequences", Abstract and Applied Analysis, vol. 2011, Article ID 950364, 9 pages, 2011. https://doi.org/10.1155/2011/950364

The Cesรกro Core of Double Sequences

Academic Editor: Malisa R. Zizovic
Received24 Mar 2011
Accepted24 May 2011
Published21 Aug 2011

Abstract

We have characterized a new type of core for double sequences, ๐‘ƒ๐ถ-core, and determined the necessary and sufficient conditions on a four-dimensional matrix ๐ด to yield ๐‘ƒ๐ถ-core{๐ด๐‘ฅ}โІ๐›ผ(๐‘ƒ-core{๐‘ฅ}) for all โ„“โˆž2.

1. Introduction

A double sequence ๐‘ฅ=[๐‘ฅ๐‘—๐‘˜]โˆž๐‘—,๐‘˜=0 is said to be convergent in the Pringsheim sense or ๐‘ƒ-convergent if for every ๐œ–>0 there exists an ๐‘โˆˆโ„• such that |๐‘ฅ๐‘—๐‘˜โˆ’โ„“|<๐œ€ whenever ๐‘—,๐‘˜>๐‘, [1]. In this case, we write ๐‘ƒโˆ’lim๐‘ฅ=โ„“. By ๐‘2, we mean the space of all ๐‘ƒ-convergent sequences.

A double sequence ๐‘ฅ is bounded if โ€–๐‘ฅโ€–=sup๐‘—,๐‘˜โ‰ฅ0||๐‘ฅ๐‘—๐‘˜||<โˆž.(1.1) By โ„“2โˆž, 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 ๐‘โˆž2 the space of double sequences which are bounded and convergent.

A double sequence ๐‘ฅ=[๐‘ฅ๐‘—๐‘˜] is said to converge regularly if it converges in Pringsheim's sense and, in addition, the following finite limits exist: lim๐‘˜โ†’โˆž๐‘ฅ๐‘—๐‘˜=โ„“๐‘—,(๐‘—=1,2,3,โ€ฆ),lim๐‘—โ†’โˆž๐‘ฅ๐‘—๐‘˜=๐‘ก๐‘—,(๐‘˜=1,2,3,โ€ฆ).(1.2) Let ๐ด=[๐‘Ž๐‘š๐‘›๐‘—๐‘˜]โˆž๐‘—,๐‘˜=0 be a four-dimensional infinite matrix of real numbers for all ๐‘š,๐‘›=0,1,โ€ฆ. The sums ๐‘ฆ๐‘š๐‘›=โˆž๎“โˆž๐‘—=0๎“๐‘˜=0๐‘Ž๐‘š๐‘›๐‘—๐‘˜๐‘ฅ๐‘—๐‘˜(1.3) are called the ๐ด-transforms of the double sequence ๐‘ฅ=[๐‘ฅ๐‘—๐‘˜]. We say that a sequence ๐‘ฅ=[๐‘ฅ๐‘—๐‘˜] is ๐ด-summable to the limit โ„“ if the ๐ด-transform of ๐‘ฅ=[๐‘ฅ๐‘—๐‘˜] exists for all ๐‘š,๐‘›=0,1,โ€ฆ and is convergent to โ„“ in the Pringsheim sense, that is, lim๐‘๐‘,๐‘žโ†’โˆž๎“๐‘ž๐‘—=0๎“๐‘˜=0๐‘Ž๐‘š๐‘›๐‘—๐‘˜๐‘ฅ๐‘—๐‘˜=๐‘ฆ๐‘š๐‘›,lim๐‘š,๐‘›โ†’โˆž๐‘ฆ๐‘š๐‘›=โ„“.(1.4)

We say that a matrix ๐ด is bounded-regular if every bounded-convergent sequence ๐‘ฅ is ๐ด-summable to the same limit and the ๐ด-transforms are also bounded. The necessary and sufficient conditions for ๐ด to be bounded-regular or RH-regular (cf., Robison [2]) are lim๐‘š,๐‘›โ†’โˆž๐‘Ž๐‘š๐‘›๐‘—๐‘˜=0,(๐‘—,๐‘˜=0,1,โ€ฆ),limโˆž๐‘š,๐‘›โ†’โˆž๎“โˆž๐‘—=0๎“๐‘˜=0๐‘Ž๐‘š๐‘›๐‘—๐‘˜=1,limโˆž๐‘š,๐‘›โ†’โˆž๎“๐‘—=0||๐‘Ž๐‘š๐‘›๐‘—๐‘˜||=0,(๐‘˜=0,1,โ€ฆ),limโˆž๐‘š,๐‘›โ†’โˆž๎“๐‘˜=0||๐‘Ž๐‘š๐‘›๐‘—๐‘˜||=0,(๐‘—=0,1,โ€ฆ),โˆžโˆ‘โˆž๐‘—=0โˆ‘๐‘˜=0||๐‘Ž๐‘š๐‘›๐‘—๐‘˜||.โ‰ค๐ถ<โˆž(๐‘š,๐‘›=0,1,โ€ฆ)(1.5) A double sequence ๐‘ฅ=[๐‘ฅ๐‘—๐‘˜] is said to be almost convergent (see [3]) to a number ๐ฟ if lim๐‘,๐‘žโ†’โˆžsup๐‘ ,๐‘กโ‰ฅ01๐‘๐‘ž๐‘๎“๐‘ž๐‘—=0๎“๐‘˜=0๐‘ฅ๐‘ +๐‘—,๐‘ก+๐‘˜=๐ฟ.(1.6) Let ๐œŽ be a one-to-one mapping from โ„• into itself. The almost convergence of double sequences has been generalized to the ๐œŽ-convergence in [4] as follows: lim๐‘,๐‘žโ†’โˆžsup๐‘ ,๐‘กโ‰ฅ01๐‘๐‘ž๐‘๎“๐‘ž๐‘—=0๎“๐‘˜=0๐‘ฅ๐œŽ๐‘—(๐‘ ),๐œŽ๐‘˜(๐‘ก)=โ„“,(1.7) where ๐œŽ๐‘—(๐‘ )=๐œŽ(๐œŽ๐‘—โˆ’1(๐‘ )). In this case, we write ๐œŽโˆ’lim๐‘ฅ=โ„“. By ๐‘‰2๐œŽ, we denote the set of all ๐œŽ-convergent and bounded double sequences. One can see that in contrast to the case for single sequences, a convergent double sequence need not be ๐œŽ-convergent. But every bounded convergent double sequence is ๐œŽ-convergent. So, ๐‘โˆž2โŠ‚๐‘‰2๐œŽโŠ‚โ„“โˆž2. In the case ๐œŽ(๐‘–)=๐‘–+1, ๐œŽ-convergence of double sequences reduces to the almost convergence. A matrix ๐ด=[๐‘Ž๐‘š๐‘›๐‘—๐‘˜]โˆž๐‘—,๐‘˜=0 is said to be ๐œŽ-regular if ๐ด๐‘ฅโˆˆ๐‘‰๐œŽ2 for ๐‘ฅ=[๐‘ฅ๐‘—๐‘˜]โˆˆ๐‘โˆž2 with ๐œŽโˆ’lim๐ด๐‘ฅ=lim๐‘ฅ, and we denote this by ๐ดโˆˆ(๐‘โˆž2,๐‘‰๐œŽ2)reg, (see [5, 6]). Mursaleen and Mohiuddine defined and characterized ๐œŽ-conservative and ๐œŽ-coercive matrices for double sequences [6].

A double sequence ๐‘ฅ=[๐‘ฅ๐‘—๐‘˜] of real numbers is said to be Cesรกro convergent (or ๐ถ1-convergent) to a number L if and only if ๐‘ฅโˆˆ๐ถ1, where ๐ถ1=๎‚ป๐‘ฅโˆˆโ„“โˆž2โˆถlim๐‘,๐‘žโ†’โˆž๐‘‡๐‘๐‘ž(๐‘ฅ)=๐ฟ;๐ฟ=๐ถ1๎‚ผ,๐‘‡โˆ’lim๐‘ฅ๐‘๐‘ž(1๐‘ฅ)=(๐‘+1)(๐‘ž+1)๐‘๎“๐‘ž๐‘—=1๎“๐‘˜=1๐‘ฅ๐‘š๐‘›๐‘—๐‘˜.(1.8) We shall denote by ๐ถ1 the space of Cesรกro convergent (๐ถ1-convergent) double sequences.

A matrix ๐ด=(๐‘Ž๐‘š๐‘›๐‘—๐‘˜) is said to be ๐ถ1-multiplicative if ๐ด๐‘ฅโˆˆ๐ถ1 for ๐‘ฅ=[๐‘ฅ๐‘—๐‘˜]โˆˆ๐‘โˆž2 with ๐ถ1โˆ’lim๐ด๐‘ฅ=๐›ผlim๐‘ฅ, and in this case we write ๐ดโˆˆ(๐‘โˆž2,๐ถ1)๐›ผ. Note that if ๐›ผ=1, then ๐ถ1-multiplicative matrices are said to be ๐ถ1-regular matrices.

Recall that the Knopp core (or K-core) of a real number single sequence ๐‘ฅ=(๐‘ฅ๐‘˜) is defined by the closed interval [โ„“(๐‘ฅ),๐ฟ(๐‘ฅ)], where โ„“(๐‘ฅ)=liminf๐‘ฅ and ๐ฟ(๐‘ฅ)=limsup๐‘ฅ. The well-known Knopp core theorem states (cf., Maddox [7] and Knopp [8]) that in order that ๐ฟ(๐ด๐‘ฅ)โ‰ค๐ฟ(๐‘ฅ) for every bounded real sequence ๐‘ฅ, it is necessary and sufficient that ๐ด=(๐‘Ž๐‘›๐‘˜) should be regular and lim๐‘›โ†’โˆžโˆ‘โˆž๐‘˜=0|๐‘Ž๐‘›๐‘˜|=1. Patterson [9] extended this idea for double sequences by defining the Pringsheim core (or P-core) of a real bounded double sequence ๐‘ฅ=[๐‘ฅ๐‘—๐‘˜] as the closed interval [๐‘ƒโˆ’liminf๐‘ฅ,๐‘ƒโˆ’limsup๐‘ฅ]. Some inequalities related to the these concepts have been studied in [5, 9, 10]. Let ๐ฟโˆ—(๐‘ฅ)=limsup๐‘,๐‘žโ†’โˆžsup๐‘ ,๐‘ก1๐‘๐‘ž๐‘๎“๐‘ž๐‘—=0๎“๐‘˜=0๐‘ฅ๐‘—+๐‘ ,๐‘˜+๐‘ก,๐ถ๐œŽ(๐‘ฅ)=limsup๐‘,๐‘žโ†’โˆžsup๐‘ ,๐‘ก1๐‘๐‘ž๐‘๎“๐‘ž๐‘—=0๎“๐‘˜=0๐‘ฅ๐œŽ๐‘—(๐‘ ),๐œŽ๐‘˜(๐‘ก).(1.9)

Then, MR- (Moricz-Rhoades) and ๐œŽ-core of a double sequence have been introduced by the closed intervals [โˆ’๐ฟโˆ—(โˆ’๐‘ฅ),๐ฟโˆ—(โˆ’๐‘ฅ)] and [โˆ’๐ถ๐œŽ(โˆ’๐‘ฅ),๐ถ๐œŽ(๐‘ฅ)], and also the inequalities ๐ฟ(๐ด๐‘ฅ)โ‰ค๐ฟโˆ—(๐‘ฅ),๐ฟโˆ—(๐ด๐‘ฅ)โ‰ค๐ฟ(๐‘ฅ),๐ฟโˆ—(๐ด๐‘ฅ)โ‰ค๐ฟโˆ—(๐‘ฅ),๐ฟ(๐ด๐‘ฅ)โ‰ค๐ถ๐œŽ(๐‘ฅ),๐ถ๐œŽ(๐ด๐‘ฅ)โ‰ค๐ฟ(๐‘ฅ)(1.10) have been studies in [3โ€“5, 11].

In this paper, we introduce the concept of ๐ถ1-multiplicative matrices and determine the necessary and sufficient conditions for a matrix ๐ด=(๐‘Ž๐‘š๐‘›๐‘—๐‘˜) to belong to the class (๐‘โˆž2,๐ถ1)๐›ผ. Further we investigate the necessary and sufficient conditions for the inequality ๐ถโˆ—1(๐ด๐‘ฅ)โ‰ค๐›ผ๐ฟ(๐‘ฅ)(1.11) for all ๐‘ฅโˆˆโ„“2โˆž.

2. Main Results

Let us write ๐ถโˆ—1(๐‘ฅ)=limsup๐‘,๐‘žโ†’โˆž1(๐‘+1)(๐‘ž+1)๐‘๎“๐‘ž๐‘—=0๎“๐‘˜=0๐‘ฅ๐‘—๐‘˜.(2.1) Then, we will define the ๐‘ƒ๐ถ-core of a realvalued bounded double sequence ๐‘ฅ=[๐‘ฅ๐‘—๐‘˜] by the closed interval [โˆ’๐ถโˆ—1(โˆ’๐‘ฅ),๐ถโˆ—1(๐‘ฅ)]. Since every bounded convergent double sequence is Cesรกro convergent, we have ๐ถโˆ—1(๐‘ฅ)โ‰ค๐‘ƒโˆ’limsup๐‘ฅ, and hence it follows that ๐‘ƒ๐ถ-core(๐‘ฅ)โІ๐‘ƒ-core(๐‘ฅ) for a bounded double sequence ๐‘ฅ=[๐‘ฅ๐‘—๐‘˜].

Lemma 2.1 .. A matrix ๐ด=(๐‘Ž๐‘š๐‘›๐‘—๐‘˜) is ๐ถ1-multiplicative if and only if lim๐‘,๐‘žโ†’โˆž๐›ฝ(๐‘—,๐‘˜,๐‘,๐‘ž)=0(๐‘—,๐‘˜=0,1,โ€ฆ),(2.2)limโˆž๐‘,๐‘žโ†’โˆž๎“โˆž๐‘—=0๎“๐‘˜=0๐›ฝ(๐‘—,๐‘˜,๐‘,๐‘ž)=๐›ผ,(2.3)limโˆž๐‘,๐‘žโ†’โˆž๎“๐‘—=0||||๐›ฝ(๐‘—,๐‘˜,๐‘,๐‘ž)=0(๐‘˜=0,1,โ€ฆ),(2.4)limโˆž๐‘,๐‘žโ†’โˆž๎“๐‘˜=0||||๐›ฝ(๐‘—,๐‘˜,๐‘,๐‘ž)=0(๐‘—=0,1,โ€ฆ),(2.5)โˆž๎“โˆž๐‘—=0๎“๐‘˜=0||๐‘Ž๐‘š๐‘›๐‘—๐‘˜||โ‰ค๐ถ<โˆž,(๐‘š,๐‘›=0,1,โ€ฆ),(2.6) where the lim means ๐‘ƒโˆ’lim and 1๐›ฝ(๐‘—,๐‘˜,๐‘,๐‘ž)=(๐‘+1)(๐‘ž+1)๐‘๎“๐‘ž๐‘—=0๎“๐‘˜=0๐‘Ž๐‘š๐‘›๐‘—๐‘˜.(2.7)

Proof. Sufficiency. Suppose that the conditions (2.2)-(2.6) hold and ๐‘ฅ=[๐‘ฅ๐‘—๐‘˜]โˆˆ๐‘โˆž2 with ๐‘ƒโˆ’lim๐‘—,๐‘˜๐‘ฅ๐‘—๐‘˜=๐ฟ, say. So that for every ๐œ–>0 there exists ๐‘>0 such that |๐‘ฅ๐‘—๐‘˜|<|โ„“|+๐œ– whenever ๐‘—,๐‘˜>๐‘.
Then, we can write โˆž๎“โˆž๐‘—=0๎“๐‘˜=0๐›ฝ(๐‘—,๐‘˜,๐‘,๐‘ž)๐‘ฅ๐‘—๐‘˜=๐‘๎“๐‘๐‘—=0๎“๐‘˜=0๐›ฝ(๐‘—,๐‘˜,๐‘,๐‘ž)๐‘ฅ๐‘—๐‘˜+โˆž๎“๐‘—=๐‘๐‘โˆ’1๎“๐‘˜=0๐›ฝ(๐‘—,๐‘˜,๐‘,๐‘ž)๐‘ฅ๐‘—๐‘˜+๐‘โˆ’1๎“โˆž๐‘—=0๎“๐‘˜=๐‘๐›ฝ(๐‘—,๐‘˜,๐‘,๐‘ž)๐‘ฅ๐‘—๐‘˜+โˆž๎“โˆž๐‘—=๐‘+1๎“๐‘˜=๐‘+1๐›ฝ(๐‘—,๐‘˜,๐‘,๐‘ž)๐‘ฅ๐‘—๐‘˜.(2.8) Therefore, |||||โˆž๎“โˆž๐‘—=0๎“๐‘˜=0๐›ฝ(๐‘—,๐‘˜,๐‘,๐‘ž)๐‘ฅ๐‘—๐‘˜|||||โ‰คโ€–๐‘ฅโ€–๐‘๎“๐‘๐‘—=0๎“๐‘˜=0||||๐›ฝ(๐‘—,๐‘˜,๐‘,๐‘ž)+โ€–๐‘ฅโ€–โˆž๎“๐‘—=๐‘๐‘โˆ’1๎“๐‘˜=0||๐›ฝ(๐‘—,๐‘˜,๐‘,๐‘ž)๐‘ฅ๐‘—๐‘˜||+โ€–๐‘ฅโ€–๐‘โˆ’1๎“โˆž๐‘—=0๎“๐‘˜=๐‘||||+๎€ท||๐ฟ||๎€ธ|||||๐›ฝ(๐‘—,๐‘˜,๐‘,๐‘ž)+๐œ–โˆž๎“โˆž๐‘—=๐‘+1๎“๐‘˜=๐‘+1|||||.๐›ฝ(๐‘—,๐‘˜,๐‘,๐‘ž)(2.9) Letting ๐‘,๐‘žโ†’โˆž and using the conditions (2.2)โ€“(2.6), we get |||||limโˆž๐‘,๐‘žโ†’โˆž๎“โˆž๐‘—=0๎“๐‘˜=0๐›ฝ(๐‘—,๐‘˜,๐‘,๐‘ž)๐‘ฅ๐‘—๐‘˜|||||โ‰ค๎€ท||๐ฟ||๎€ธ+๐œ–๐›ผ.(2.10) Since ๐œ– is arbitrary, ๐ถ1โˆ’lim๐ด๐‘ฅ=๐›ผ๐ฟ. Hence ๐ดโˆˆ(๐‘โˆž2,๐ถ1)๐›ผ, that is, ๐ด is ๐ถ1-multiplicative.

Necessity. Suppose that A is ๐ถ1-multiplicative. Then, by the definition, the A-transform of ๐‘ฅ exists and ๐ด๐‘ฅโˆˆ๐ถ1 for each ๐‘ฅโˆˆ๐‘โˆž2. Therefore, ๐ด๐‘ฅ is also bounded. Then, we can write supโˆž๐‘š,๐‘›๎“โˆž๐‘—=0๎“๐‘˜=0||๐‘Ž๐‘š๐‘›๐‘—๐‘˜๐‘ฅ๐‘—๐‘˜||<๐‘€<โˆž,(2.11) for each ๐‘ฅโˆˆ๐‘โˆž2. Now, let us define a sequence ๐‘ฆ=[๐‘ฆ๐‘—๐‘˜] by ๐‘ฆ๐‘—๐‘˜=๎ƒฏsgn๐‘Ž๐‘š๐‘›๐‘—๐‘˜,0โ‰ค๐‘—โ‰ค๐‘Ÿ,0โ‰ค๐‘˜โ‰ค๐‘Ÿ,0,otherwise,(2.12) โ€‰๐‘š,๐‘›=0,1,2,โ€ฆ. Then, the necessity of (10) follows by considering the sequence ๐‘ฆ=[๐‘ฆ๐‘—๐‘˜] in (2.11).
Also, by the assumption, we have limโˆž๐‘,๐‘žโ†’โˆž๎“โˆž๐‘—=0๎“๐‘˜=0๐›ฝ(๐‘—,๐‘˜,๐‘,๐‘ž)๐‘ฅ๐‘—๐‘˜=๐›ผlim๐‘—,๐‘˜โ†’โˆž๐‘ฅ๐‘—๐‘˜.(2.13) Now let us define the sequence ๐‘’๐‘–๐‘™ as follows: ๐‘’๐‘–๐‘™=๎ƒฏ1,(๐‘—,๐‘˜)=(๐‘–,๐‘™),0,otherwise,(2.14) and write ๐‘ ๐‘™=โˆ‘๐‘–๐‘’๐‘–๐‘™(๐‘™โˆˆโ„•), ๐‘Ÿ๐‘–=โˆ‘๐‘™๐‘’๐‘–๐‘™(๐‘–โˆˆโ„•). Then, the necessity of (2.2), (2.4), and (2.5) follows from ๐ถ1โˆ’lim๐ด๐‘’๐‘–๐‘™, ๐ถ1โˆ’lim๐ด๐‘Ÿ๐‘— and ๐ถ1โˆ’lim๐ด๐‘ ๐‘˜, respectively.

Note that when ๐›ผ=1, the above theorem gives the characterization of ๐ดโˆˆ(๐‘โˆž2,๐ถ1)reg. Now, we are ready to construct our main theorem.

Theorem 2.2. For every bounded double sequence x, ๐ถโˆ—1(๐ด๐‘ฅ)โ‰ค๐›ผ๐ฟ(๐‘ฅ),(2.15) or (๐‘ƒ๐ถโˆ’๐‘๐‘œ๐‘Ÿ๐‘’{๐ด๐‘ฅ}โІ๐›ผ(๐‘ƒโˆ’๐‘๐‘œ๐‘Ÿ๐‘’{๐‘ฅ})) if and only if ๐ด is ๐ถ1-multiplicative and limsupโˆž๐‘,๐‘žโ†’โˆž๎“โˆž๐‘—=0๎“๐‘˜=0||||๐›ฝ(๐‘—,๐‘˜,๐‘,๐‘ž)=๐›ผ.(2.16)

Proof. Necessity. Let (2.15) hold and for all ๐‘ฅโˆˆโ„“2โˆž. So, since ๐‘โˆž2โŠ‚โ„“2โˆž, then, we get ๐›ผ(โˆ’๐ฟ(โˆ’๐‘ฅ))โ‰คโˆ’๐ถโˆ—1(โˆ’๐ด๐‘ฅ)โ‰ค๐ถโˆ—1(๐ด๐‘ฅ)โ‰ค๐›ผ๐ฟ(๐‘ฅ).(2.17) That is, ๐›ผliminf๐‘ฅโ‰คโˆ’๐ถโˆ—1(โˆ’๐ด๐‘ฅ)โ‰ค๐ถโˆ—1(๐ด๐‘ฅ)โ‰ค๐›ผlimsup๐‘ฅ,(2.18) where โˆ’๐ถโˆ—1(โˆ’๐ด๐‘ฅ)=liminfโˆž๐‘,๐‘žโ†’โˆž๎“โˆž๐‘—=0๎“๐‘˜=0๐›ฝ(๐‘—,๐‘˜,๐‘,๐‘ž)๐‘ฅ๐‘—๐‘˜.(2.19) By choosing ๐‘ฅ=[๐‘ฅ๐‘—๐‘˜]โˆˆ๐‘2โˆž, we get from (2.17) that โˆ’๐ถโˆ—1(โˆ’๐ด๐‘ฅ)=๐ถโˆ—1(๐ด๐‘ฅ)=๐ถ1โˆ’lim๐ด๐‘ฅ=๐›ผlim๐‘ฅ.(2.20) This means that ๐ด is ๐ถ1-multiplicative.
By Lemmaโ€‰โ€‰3.1 of Patterson [9], there exists a ๐‘ฆโˆˆโ„“2โˆž with ||๐‘ฆ||โ‰ค1 such that ๐ถโˆ—1(๐ด๐‘ฆ)=limsupโˆž๐‘,๐‘žโ†’โˆž๎“โˆž๐‘—=0๎“๐‘˜=0๐›ฝ(๐‘—,๐‘˜,๐‘,๐‘ž).(2.21) If we choose ๐‘ฆ=๐‘ฃ=[๐‘ฃ๐‘—๐‘˜], it follows ๐‘ฃ๐‘—๐‘˜=๎ƒฏ1if๐‘—=๐‘˜,0,elsewhere.(2.22) Since โ€–๐‘ฃ๐‘—๐‘˜โ€–โ‰ค1, we have from (2.15) that ๐›ผ=๐ถโˆ—1(๐ด๐‘ฃ)=limsupโˆž๐‘,๐‘žโ†’โˆž๎“โˆž๐‘—=0๎“๐‘˜=0||||๎€ท๐‘ฃ๐›ฝ(๐‘—,๐‘˜,๐‘,๐‘ž)โ‰ค๐›ผ๐ฟ๐‘—๐‘˜๎€ธโ‰ค๐›ผโ€–๐‘ฃโ€–โ‰ค๐›ผ.(2.23) This gives the necessity of (2.16).

Sufficiency. Suppose that ๐ด is ๐ถ1-regular and (2.16) holds. Let ๐‘ฅ=[๐‘ฅ๐‘—๐‘˜] be an arbitrary bounded sequence. Then, there exist ๐‘€,๐‘>0 such that ๐‘ฅ๐‘—๐‘˜โ‰ค๐พ for all ๐‘—,๐‘˜โ‰ฅ0. Now, we can write the following inequality: |||||โˆž๎“โˆž๐‘—=0๎“๐‘˜=0๐›ฝ(๐‘—,๐‘˜,๐‘,๐‘ž)๐‘ฅ๐‘—๐‘˜|||||=|||||โˆž๎“โˆž๐‘—=0๎“๐‘˜=0๎‚ต||๐›ฝ||(๐‘—,๐‘˜,๐‘,๐‘ž)+๐›ฝ(๐‘—,๐‘˜,๐‘,๐‘ž)2โˆ’||||๐›ฝ(๐‘—,๐‘˜,๐‘,๐‘ž)โˆ’๐›ฝ(๐‘—,๐‘˜,๐‘,๐‘ž)2๎‚ถ๐‘ฅ๐‘—๐‘˜||||โ‰คโˆž๎“โˆž๐‘—=0๎“๐‘˜=0||๐›ฝ||||๐‘ฅ(๐‘—,๐‘˜,๐‘,๐‘ž)๐‘—๐‘˜||+โˆž๎“โˆž๐‘—=0๎“๐‘˜=0||๎€ท||๐›ฝ||๎€ธ๐‘ฅ(๐‘—,๐‘˜,๐‘,๐‘ž)โˆ’๐›ฝ(๐‘—,๐‘˜,๐‘,๐‘ž)๐‘—๐‘˜||โ‰คโ€–๐‘ฅโ€–๐‘€๎“๐‘๐‘—=0๎“๐‘˜=0||||๐›ฝ(๐‘—,๐‘˜,๐‘,๐‘ž)+โ€–๐‘ฅโ€–โˆž๎“๐‘๐‘—=๐‘€+1๎“๐‘˜=0||||๐›ฝ(๐‘—,๐‘˜,๐‘,๐‘ž)+โ€–๐‘ฅโ€–๐‘€๎“โˆž๐‘—=0๎“๐‘˜=๐‘+1||||๐›ฝ(๐‘—,๐‘˜,๐‘,๐‘ž)+sup๐‘—,๐‘˜โ‰ฅ๐‘€,๐‘||๐‘ฅ๐‘—๐‘˜||โˆž๎“โˆž๐‘—=๐‘€+1๎“๐‘˜=๐‘+1||||๐›ฝ(๐‘—,๐‘˜,๐‘,๐‘ž)+โ€–๐‘ฅโ€–โˆž๎“โˆž๐‘—=0๎“๐‘˜=0๎€ท||||๎€ธ.๐›ฝ(๐‘—,๐‘˜,๐‘,๐‘ž)โˆ’๐›ฝ(๐‘—,๐‘˜,๐‘,๐‘ž)(2.24) Using the condition of ๐ถ1-multiplicative and condition (2.16), we get ๐ถโˆ—1(๐ด๐‘ฅ)โ‰ค๐›ผ๐ฟ(๐‘ฅ).(2.25) This completes the proof of the theorem.

Theorem 2.3. For ๐‘ฅ,๐‘ฆโˆˆโ„“โˆž2, if ๐ถ1โˆ’lim|๐‘ฅโˆ’๐‘ฆ|=0, then ๐ถ1โˆ’core{๐‘ฅ}=๐ถ1โˆ’core{๐‘ฆ}.

Proof. Since ๐ถ2โˆ’lim|๐‘ฅโˆ’๐‘ฆ|=0, we have ๐ถ1โˆ’lim(๐‘ฅโˆ’๐‘ฆ)=0 and ๐ถ1โˆ’lim(โˆ’(๐‘ฅโˆ’๐‘ฆ))=0. Using definition of ๐ถ1โˆ’core, we take ๐ถโˆ—1(๐‘ฅโˆ’๐‘ฆ)=โˆ’๐ถโˆ—1(โˆ’(๐‘ฅโˆ’๐‘ฆ))=0. Since ๐ถโˆ—1 is sublinear, 0=โˆ’๐ถโˆ—1(โˆ’(๐‘ฅโˆ’๐‘ฆ))โ‰คโˆ’๐ถโˆ—1(โˆ’๐‘ฅ)โˆ’๐ถโˆ—1(๐‘ฆ).(2.26) Therefore, ๐ถโˆ—1(๐‘ฆ)โ‰คโˆ’๐ถโˆ—1(โˆ’๐‘ฅ). Since โˆ’๐ถโˆ—1(โˆ’๐‘ฅ)โ‰ค๐ถโˆ—1(๐‘ฅ), this implies that ๐ถโˆ—1(๐‘ฆ)โ‰ค๐ถโˆ—1(๐‘ฅ).By an argument similar as above, we can show that ๐ถโˆ—1(๐‘ฅ)โ‰ค๐ถโˆ—1(๐‘ฆ). 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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Copyright © 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.

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