Abstract and Applied Analysis

Abstract and Applied Analysis / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 950364 | 9 pages | https://doi.org/10.1155/2011/950364

The Cesáro 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, 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 ℓ∞2.

1. Introduction

A double sequence 𝑥=[𝑥𝑗𝑘]âˆžğ‘—,𝑘=0 is said to be convergent in the Pringsheim sense or 𝑃-convergent if for every 𝜖>0 there exists an 𝑁∈ℕ such that |𝑥𝑗𝑘−ℓ|<𝜀 whenever 𝑗,𝑘>𝑁, [1]. In this case, we write 𝑃−lim𝑥=ℓ. By 𝑐2, we mean the space of all 𝑃-convergent sequences.

A double sequence 𝑥 is bounded if ‖𝑥‖=sup𝑗,𝑘≥0||𝑥𝑗𝑘||<∞.(1.1) By ℓ2∞, 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 ğ‘âˆž2 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: limğ‘˜â†’âˆžğ‘¥ğ‘—ğ‘˜=ℓ𝑗,(𝑗=1,2,3,…),limğ‘—â†’âˆžğ‘¥ğ‘—ğ‘˜=𝑡𝑗,(𝑘=1,2,3,…).(1.2) Let 𝐴=[ğ‘Žğ‘šğ‘›ğ‘—ğ‘˜]âˆžğ‘—,𝑘=0 be a four-dimensional infinite matrix of real numbers for all 𝑚,𝑛=0,1,…. The sums 𝑦𝑚𝑛=âˆžî“âˆžğ‘—=0𝑘=0ğ‘Žğ‘šğ‘›ğ‘—ğ‘˜ğ‘¥ğ‘—ğ‘˜(1.3) are called the 𝐴-transforms of the double sequence 𝑥=[𝑥𝑗𝑘]. We say that a sequence 𝑥=[𝑥𝑗𝑘] is 𝐴-summable to the limit ℓ if the 𝐴-transform of 𝑥=[𝑥𝑗𝑘] exists for all 𝑚,𝑛=0,1,… and is convergent to ℓ in the Pringsheim sense, that is, lim𝑝𝑝,ğ‘žâ†’âˆžî“ğ‘žğ‘—=0𝑘=0ğ‘Žğ‘šğ‘›ğ‘—ğ‘˜ğ‘¥ğ‘—ğ‘˜=𝑦𝑚𝑛,lim𝑚,ğ‘›â†’âˆžğ‘¦ğ‘šğ‘›=ℓ.(1.4)

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 [2]) are lim𝑚,ğ‘›â†’âˆžğ‘Žğ‘šğ‘›ğ‘—ğ‘˜=0,(𝑗,𝑘=0,1,…),limâˆžğ‘š,ğ‘›â†’âˆžî“âˆžğ‘—=0𝑘=0ğ‘Žğ‘šğ‘›ğ‘—ğ‘˜=1,limâˆžğ‘š,ğ‘›â†’âˆžî“ğ‘—=0||ğ‘Žğ‘šğ‘›ğ‘—ğ‘˜||=0,(𝑘=0,1,…),limâˆžğ‘š,ğ‘›â†’âˆžî“ğ‘˜=0||ğ‘Žğ‘šğ‘›ğ‘—ğ‘˜||=0,(𝑗=0,1,…),âˆžâˆ‘âˆžğ‘—=0∑𝑘=0||ğ‘Žğ‘šğ‘›ğ‘—ğ‘˜||.≤𝐶<∞(𝑚,𝑛=0,1,…)(1.5) A double sequence 𝑥=[𝑥𝑗𝑘] is said to be almost convergent (see [3]) to a number 𝐿 if lim𝑝,ğ‘žâ†’âˆžsup𝑠,𝑡≥01ğ‘ğ‘žğ‘î“ğ‘žğ‘—=0𝑘=0𝑥𝑠+𝑗,𝑡+𝑘=𝐿.(1.6) Let ğœŽ be a one-to-one mapping from ℕ into itself. The almost convergence of double sequences has been generalized to the ğœŽ-convergence in [4] as follows: lim𝑝,ğ‘žâ†’âˆžsup𝑠,𝑡≥01ğ‘ğ‘žğ‘î“ğ‘žğ‘—=0𝑘=0ğ‘¥ğœŽğ‘—(𝑠),ğœŽğ‘˜(𝑡)=ℓ,(1.7) where ğœŽğ‘—(𝑠)=ğœŽ(ğœŽğ‘—âˆ’1(𝑠)). In this case, we write ğœŽâˆ’lim𝑥=ℓ. By 𝑉2ğœŽ, we denote the set of all ğ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 ğœŽ-convergent. But every bounded convergent double sequence is ğœŽ-convergent. So, ğ‘âˆž2⊂𝑉2ğœŽâŠ‚â„“âˆž2. In the case ğœŽ(𝑖)=𝑖+1, ğœŽ-convergence of double sequences reduces to the almost convergence. A matrix 𝐴=[ğ‘Žğ‘šğ‘›ğ‘—ğ‘˜]âˆžğ‘—,𝑘=0 is said to be ğœŽ-regular if ğ´ğ‘¥âˆˆğ‘‰ğœŽ2 for 𝑥=[𝑥𝑗𝑘]âˆˆğ‘âˆž2 with ğœŽâˆ’lim𝐴𝑥=lim𝑥, and we denote this by 𝐴∈(ğ‘âˆž2,ğ‘‰ğœŽ2)reg, (see [5, 6]). Mursaleen and Mohiuddine defined and characterized ğœŽ-conservative and ğœŽ-coercive matrices for double sequences [6].

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 𝑥∈𝐶1, where 𝐶1=î‚»ğ‘¥âˆˆâ„“âˆž2∶lim𝑝,ğ‘žâ†’âˆžğ‘‡ğ‘ğ‘ž(𝑥)=𝐿;𝐿=𝐶1,𝑇−limğ‘¥ğ‘ğ‘ž(1𝑥)=(𝑝+1)(ğ‘ž+1)ğ‘î“ğ‘žğ‘—=1𝑘=1𝑥𝑚𝑛𝑗𝑘.(1.8) We shall denote by 𝐶1 the space of Cesáro convergent (𝐶1-convergent) double sequences.

A matrix 𝐴=(ğ‘Žğ‘šğ‘›ğ‘—ğ‘˜) is said to be 𝐶1-multiplicative if 𝐴𝑥∈𝐶1 for 𝑥=[𝑥𝑗𝑘]âˆˆğ‘âˆž2 with 𝐶1−lim𝐴𝑥=𝛼lim𝑥, and in this case we write 𝐴∈(ğ‘âˆž2,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 [ℓ(𝑥),𝐿(𝑥)], where ℓ(𝑥)=liminf𝑥 and 𝐿(𝑥)=limsup𝑥. The well-known Knopp core theorem states (cf., Maddox [7] and Knopp [8]) that in order that 𝐿(𝐴𝑥)≤𝐿(𝑥) for every bounded real sequence 𝑥, it is necessary and sufficient that 𝐴=(ğ‘Žğ‘›ğ‘˜) should be regular and limğ‘›â†’âˆžâˆ‘âˆžğ‘˜=0|ğ‘Žğ‘›ğ‘˜|=1. Patterson [9] 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,ğ‘žâ†’âˆžsup𝑠,𝑡1ğ‘ğ‘žğ‘î“ğ‘žğ‘—=0𝑘=0𝑥𝑗+𝑠,𝑘+𝑡,ğ¶ğœŽ(𝑥)=limsup𝑝,ğ‘žâ†’âˆžsup𝑠,𝑡1ğ‘ğ‘žğ‘î“ğ‘žğ‘—=0𝑘=0ğ‘¥ğœŽğ‘—(𝑠),ğœŽğ‘˜(𝑡).(1.9)

Then, MR- (Moricz-Rhoades) and ğœŽ-core of a double sequence have been introduced by the closed intervals [−𝐿∗(−𝑥),𝐿∗(−𝑥)] and [âˆ’ğ¶ğœŽ(−𝑥),ğ¶ğœŽ(𝑥)], and also the inequalities 𝐿(𝐴𝑥)≤𝐿∗(𝑥),𝐿∗(𝐴𝑥)≤𝐿(𝑥),𝐿∗(𝐴𝑥)≤𝐿∗(𝑥),𝐿(𝐴𝑥)â‰¤ğ¶ğœŽ(𝑥),ğ¶ğœŽ(𝐴𝑥)≤𝐿(𝑥)(1.10) have been studies in [3–5, 11].

In this paper, we introduce the concept of 𝐶1-multiplicative matrices and determine the necessary and sufficient conditions for a matrix 𝐴=(ğ‘Žğ‘šğ‘›ğ‘—ğ‘˜) to belong to the class (ğ‘âˆž2,𝐶1)𝛼. Further we investigate the necessary and sufficient conditions for the inequality 𝐶∗1(𝐴𝑥)≤𝛼𝐿(𝑥)(1.11) for all 𝑥∈ℓ2∞.

2. Main Results

Let us write 𝐶∗1(𝑥)=limsup𝑝,ğ‘žâ†’âˆž1(𝑝+1)(ğ‘ž+1)ğ‘î“ğ‘žğ‘—=0𝑘=0𝑥𝑗𝑘.(2.1) Then, we will define the 𝑃𝐶-core of a realvalued bounded double sequence 𝑥=[𝑥𝑗𝑘] by the closed interval [−𝐶∗1(−𝑥),𝐶∗1(𝑥)]. Since every bounded convergent double sequence is Cesáro convergent, we have 𝐶∗1(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=(ğ‘Žğ‘šğ‘›ğ‘—ğ‘˜) is 𝐶1-multiplicative if and only if lim𝑝,ğ‘žâ†’âˆžğ›½(𝑗,𝑘,𝑝,ğ‘ž)=0(𝑗,𝑘=0,1,…),(2.2)limâˆžğ‘,ğ‘žâ†’âˆžî“âˆžğ‘—=0𝑘=0𝛽(𝑗,𝑘,𝑝,ğ‘ž)=𝛼,(2.3)limâˆžğ‘,ğ‘žâ†’âˆžî“ğ‘—=0||||𝛽(𝑗,𝑘,𝑝,ğ‘ž)=0(𝑘=0,1,…),(2.4)limâˆžğ‘,ğ‘žâ†’âˆžî“ğ‘˜=0||||𝛽(𝑗,𝑘,𝑝,ğ‘ž)=0(𝑗=0,1,…),(2.5)âˆžî“âˆžğ‘—=0𝑘=0||ğ‘Žğ‘šğ‘›ğ‘—ğ‘˜||≤𝐶<∞,(𝑚,𝑛=0,1,…),(2.6) where the lim means 𝑃−lim and 1𝛽(𝑗,𝑘,𝑝,ğ‘ž)=(𝑝+1)(ğ‘ž+1)ğ‘î“ğ‘žğ‘—=0𝑘=0ğ‘Žğ‘šğ‘›ğ‘—ğ‘˜.(2.7)

Proof. Sufficiency. Suppose that the conditions (2.2)-(2.6) hold and 𝑥=[𝑥𝑗𝑘]âˆˆğ‘âˆž2 with 𝑃−lim𝑗,𝑘𝑥𝑗𝑘=𝐿, say. So that for every 𝜖>0 there exists 𝑁>0 such that |𝑥𝑗𝑘|<|ℓ|+𝜖 whenever 𝑗,𝑘>𝑁.
Then, we can write âˆžî“âˆžğ‘—=0𝑘=0𝛽(𝑗,𝑘,𝑝,ğ‘ž)𝑥𝑗𝑘=𝑁𝑁𝑗=0𝑘=0𝛽(𝑗,𝑘,𝑝,ğ‘ž)𝑥𝑗𝑘+âˆžî“ğ‘—=𝑁𝑁−1𝑘=0𝛽(𝑗,𝑘,𝑝,ğ‘ž)𝑥𝑗𝑘+𝑁−1î“âˆžğ‘—=0𝑘=𝑁𝛽(𝑗,𝑘,𝑝,ğ‘ž)𝑥𝑗𝑘+âˆžî“âˆžğ‘—=𝑁+1𝑘=𝑁+1𝛽(𝑗,𝑘,𝑝,ğ‘ž)𝑥𝑗𝑘.(2.8) Therefore, |||||âˆžî“âˆžğ‘—=0𝑘=0𝛽(𝑗,𝑘,𝑝,ğ‘ž)𝑥𝑗𝑘|||||≤‖𝑥‖𝑁𝑁𝑗=0𝑘=0||||𝛽(𝑗,𝑘,𝑝,ğ‘ž)+â€–ğ‘¥â€–âˆžî“ğ‘—=𝑁𝑁−1𝑘=0||𝛽(𝑗,𝑘,𝑝,ğ‘ž)𝑥𝑗𝑘||+‖𝑥‖𝑁−1î“âˆžğ‘—=0𝑘=𝑁||||+||𝐿|||||||𝛽(𝑗,𝑘,𝑝,ğ‘ž)+ğœ–âˆžî“âˆžğ‘—=𝑁+1𝑘=𝑁+1|||||.𝛽(𝑗,𝑘,𝑝,ğ‘ž)(2.9) Letting 𝑝,ğ‘žâ†’âˆž and using the conditions (2.2)–(2.6), we get |||||limâˆžğ‘,ğ‘žâ†’âˆžî“âˆžğ‘—=0𝑘=0𝛽(𝑗,𝑘,𝑝,ğ‘ž)𝑥𝑗𝑘|||||≤||𝐿||+𝜖𝛼.(2.10) Since 𝜖 is arbitrary, 𝐶1−lim𝐴𝑥=𝛼𝐿. Hence 𝐴∈(ğ‘âˆž2,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 𝐴𝑥∈𝐶1 for each ğ‘¥âˆˆğ‘âˆž2. Therefore, 𝐴𝑥 is also bounded. Then, we can write supâˆžğ‘š,ğ‘›î“âˆžğ‘—=0𝑘=0||ğ‘Žğ‘šğ‘›ğ‘—ğ‘˜ğ‘¥ğ‘—ğ‘˜||<𝑀<∞,(2.11) for each ğ‘¥âˆˆğ‘âˆž2. Now, let us define a sequence 𝑦=[𝑦𝑗𝑘] by 𝑦𝑗𝑘=sgnğ‘Žğ‘šğ‘›ğ‘—ğ‘˜,0≤𝑗≤𝑟,0≤𝑘≤𝑟,0,otherwise,(2.12)  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 limâˆžğ‘,ğ‘žâ†’âˆžî“âˆžğ‘—=0𝑘=0𝛽(𝑗,𝑘,𝑝,ğ‘ž)𝑥𝑗𝑘=𝛼lim𝑗,ğ‘˜â†’âˆžğ‘¥ğ‘—ğ‘˜.(2.13) Now let us define the sequence 𝑒𝑖𝑙 as follows: 𝑒𝑖𝑙=1,(𝑗,𝑘)=(𝑖,𝑙),0,otherwise,(2.14) and write 𝑠𝑙=∑𝑖𝑒𝑖𝑙(𝑙∈ℕ), 𝑟𝑖=∑𝑙𝑒𝑖𝑙(𝑖∈ℕ). 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∈(ğ‘âˆž2,𝐶1)reg. Now, we are ready to construct our main theorem.

Theorem 2.2. For every bounded double sequence x, 𝐶∗1(𝐴𝑥)≤𝛼𝐿(𝑥),(2.15) or (𝑃𝐶−𝑐𝑜𝑟𝑒{𝐴𝑥}⊆𝛼(𝑃−𝑐𝑜𝑟𝑒{𝑥})) if and only if 𝐴 is 𝐶1-multiplicative and limsupâˆžğ‘,ğ‘žâ†’âˆžî“âˆžğ‘—=0𝑘=0||||𝛽(𝑗,𝑘,𝑝,ğ‘ž)=𝛼.(2.16)

Proof. Necessity. Let (2.15) hold and for all 𝑥∈ℓ2∞. So, since ğ‘âˆž2⊂ℓ2∞, then, we get 𝛼(−𝐿(−𝑥))≤−𝐶∗1(−𝐴𝑥)≤𝐶∗1(𝐴𝑥)≤𝛼𝐿(𝑥).(2.17) That is, 𝛼liminf𝑥≤−𝐶∗1(−𝐴𝑥)≤𝐶∗1(𝐴𝑥)≤𝛼limsup𝑥,(2.18) where −𝐶∗1(−𝐴𝑥)=liminfâˆžğ‘,ğ‘žâ†’âˆžî“âˆžğ‘—=0𝑘=0𝛽(𝑗,𝑘,𝑝,ğ‘ž)𝑥𝑗𝑘.(2.19) By choosing 𝑥=[𝑥𝑗𝑘]∈𝑐2∞, we get from (2.17) that −𝐶∗1(−𝐴𝑥)=𝐶∗1(𝐴𝑥)=𝐶1−lim𝐴𝑥=𝛼lim𝑥.(2.20) This means that 𝐴 is 𝐶1-multiplicative.
By Lemma  3.1 of Patterson [9], there exists a 𝑦∈ℓ2∞ with ||𝑦||≤1 such that 𝐶∗1(𝐴𝑦)=limsupâˆžğ‘,ğ‘žâ†’âˆžî“âˆžğ‘—=0𝑘=0𝛽(𝑗,𝑘,𝑝,ğ‘ž).(2.21) If we choose 𝑦=𝑣=[𝑣𝑗𝑘], it follows 𝑣𝑗𝑘=1if𝑗=𝑘,0,elsewhere.(2.22) Since ‖𝑣𝑗𝑘‖≤1, we have from (2.15) that 𝛼=𝐶∗1(𝐴𝑣)=limsupâˆžğ‘,ğ‘žâ†’âˆžî“âˆžğ‘—=0𝑘=0||||𝑣𝛽(𝑗,𝑘,𝑝,ğ‘ž)≤𝛼𝐿𝑗𝑘≤𝛼‖𝑣‖≤𝛼.(2.23) 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 𝑥𝑗𝑘≤𝐾 for all 𝑗,𝑘≥0. Now, we can write the following inequality: |||||âˆžî“âˆžğ‘—=0𝑘=0𝛽(𝑗,𝑘,𝑝,ğ‘ž)𝑥𝑗𝑘|||||=|||||âˆžî“âˆžğ‘—=0𝑘=0||𝛽||(𝑗,𝑘,𝑝,ğ‘ž)+𝛽(𝑗,𝑘,𝑝,ğ‘ž)2−||||𝛽(𝑗,𝑘,𝑝,ğ‘ž)−𝛽(𝑗,𝑘,𝑝,ğ‘ž)2𝑥𝑗𝑘||||â‰¤âˆžî“âˆžğ‘—=0𝑘=0||𝛽||||𝑥(𝑗,𝑘,𝑝,ğ‘ž)𝑗𝑘||+âˆžî“âˆžğ‘—=0𝑘=0||||𝛽||𝑥(𝑗,𝑘,𝑝,ğ‘ž)−𝛽(𝑗,𝑘,𝑝,ğ‘ž)𝑗𝑘||≤‖𝑥‖𝑀𝑁𝑗=0𝑘=0||||𝛽(𝑗,𝑘,𝑝,ğ‘ž)+â€–ğ‘¥â€–âˆžî“ğ‘ğ‘—=𝑀+1𝑘=0||||𝛽(𝑗,𝑘,𝑝,ğ‘ž)+â€–ğ‘¥â€–ğ‘€î“âˆžğ‘—=0𝑘=𝑁+1||||𝛽(𝑗,𝑘,𝑝,ğ‘ž)+sup𝑗,𝑘≥𝑀,𝑁||𝑥𝑗𝑘||âˆžî“âˆžğ‘—=𝑀+1𝑘=𝑁+1||||𝛽(𝑗,𝑘,𝑝,ğ‘ž)+â€–ğ‘¥â€–âˆžî“âˆžğ‘—=0𝑘=0||||.𝛽(𝑗,𝑘,𝑝,ğ‘ž)−𝛽(𝑗,𝑘,𝑝,ğ‘ž)(2.24) Using the condition of 𝐶1-multiplicative and condition (2.16), we get 𝐶∗1(𝐴𝑥)≤𝛼𝐿(𝑥).(2.25) This completes the proof of the theorem.

Theorem 2.3. For 𝑥,ğ‘¦âˆˆâ„“âˆž2, 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 𝐶∗1(𝑥−𝑦)=−𝐶∗1(−(𝑥−𝑦))=0. Since 𝐶∗1 is sublinear, 0=−𝐶∗1(−(𝑥−𝑦))≤−𝐶∗1(−𝑥)−𝐶∗1(𝑦).(2.26) Therefore, 𝐶∗1(𝑦)≤−𝐶∗1(−𝑥). Since −𝐶∗1(−𝑥)≤𝐶∗1(𝑥), this implies that 𝐶∗1(𝑦)≤𝐶∗1(𝑥).By an argument similar as above, we can show that 𝐶∗1(𝑥)≤𝐶∗1(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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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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