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 Γ‡akan, "The CesΓ‘ro Core of Double Sequences", Abstract and Applied Analysis, vol. 2011, Article ID 950364, 9 pages, 2011. https://doi.org/10.1155/2011/950364

The CesΓ‘ro 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, 𝑃𝐢-core, and determined the necessary and sufficient conditions on a four-dimensional matrix 𝐴 to yield 𝑃𝐢-core{𝐴π‘₯}βŠ†π›Ό(𝑃-core{π‘₯}) for all β„“βˆž2.

1. Introduction

A double sequence π‘₯=[π‘₯π‘—π‘˜]βˆžπ‘—,π‘˜=0 is said to be convergent in the Pringsheim sense or 𝑃-convergent if for every πœ–>0 there exists an π‘βˆˆβ„• such that |π‘₯π‘—π‘˜βˆ’β„“|<πœ€ whenever 𝑗,π‘˜>𝑁, [1]. In this case, we write π‘ƒβˆ’limπ‘₯=β„“. By 𝑐2, we mean the space of all 𝑃-convergent sequences.

A double sequence π‘₯ is bounded if β€–π‘₯β€–=sup𝑗,π‘˜β‰₯0||π‘₯π‘—π‘˜||<∞.(1.1) By β„“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 π‘βˆž2 the space of double sequences which are bounded and convergent.

A double sequence π‘₯=[π‘₯π‘—π‘˜] is said to converge regularly if it converges in Pringsheim's sense and, in addition, the following finite limits exist: limπ‘˜β†’βˆžπ‘₯π‘—π‘˜=ℓ𝑗,(𝑗=1,2,3,…),limπ‘—β†’βˆžπ‘₯π‘—π‘˜=𝑑𝑗,(π‘˜=1,2,3,…).(1.2) Let 𝐴=[π‘Žπ‘šπ‘›π‘—π‘˜]βˆžπ‘—,π‘˜=0 be a four-dimensional infinite matrix of real numbers for all π‘š,𝑛=0,1,…. The sums π‘¦π‘šπ‘›=βˆžξ“βˆžπ‘—=0ξ“π‘˜=0π‘Žπ‘šπ‘›π‘—π‘˜π‘₯π‘—π‘˜(1.3) are called the 𝐴-transforms of the double sequence π‘₯=[π‘₯π‘—π‘˜]. We say that a sequence π‘₯=[π‘₯π‘—π‘˜] is 𝐴-summable to the limit β„“ if the 𝐴-transform of π‘₯=[π‘₯π‘—π‘˜] exists for all π‘š,𝑛=0,1,… and is convergent to β„“ in the Pringsheim sense, that is, lim𝑝𝑝,π‘žβ†’βˆžξ“π‘žπ‘—=0ξ“π‘˜=0π‘Žπ‘šπ‘›π‘—π‘˜π‘₯π‘—π‘˜=π‘¦π‘šπ‘›,limπ‘š,π‘›β†’βˆžπ‘¦π‘šπ‘›=β„“.(1.4)

We say that a matrix 𝐴 is bounded-regular if every bounded-convergent sequence π‘₯ 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 limπ‘š,π‘›β†’βˆžπ‘Žπ‘šπ‘›π‘—π‘˜=0,(𝑗,π‘˜=0,1,…),limβˆžπ‘š,π‘›β†’βˆžξ“βˆžπ‘—=0ξ“π‘˜=0π‘Žπ‘šπ‘›π‘—π‘˜=1,limβˆžπ‘š,π‘›β†’βˆžξ“π‘—=0||π‘Žπ‘šπ‘›π‘—π‘˜||=0,(π‘˜=0,1,…),limβˆžπ‘š,π‘›β†’βˆžξ“π‘˜=0||π‘Žπ‘šπ‘›π‘—π‘˜||=0,(𝑗=0,1,…),βˆžβˆ‘βˆžπ‘—=0βˆ‘π‘˜=0||π‘Žπ‘šπ‘›π‘—π‘˜||.≀𝐢<∞(π‘š,𝑛=0,1,…)(1.5) A double sequence π‘₯=[π‘₯π‘—π‘˜] is said to be almost convergent (see [3]) to a number 𝐿 if lim𝑝,π‘žβ†’βˆžsup𝑠,𝑑β‰₯01π‘π‘žπ‘ξ“π‘žπ‘—=0ξ“π‘˜=0π‘₯𝑠+𝑗,𝑑+π‘˜=𝐿.(1.6) Let 𝜎 be a one-to-one mapping from β„• into itself. The almost convergence of double sequences has been generalized to the 𝜎-convergence in [4] as follows: lim𝑝,π‘žβ†’βˆžsup𝑠,𝑑β‰₯01π‘π‘žπ‘ξ“π‘žπ‘—=0ξ“π‘˜=0π‘₯πœŽπ‘—(𝑠),πœŽπ‘˜(𝑑)=β„“,(1.7) where πœŽπ‘—(𝑠)=𝜎(πœŽπ‘—βˆ’1(𝑠)). In this case, we write πœŽβˆ’limπ‘₯=β„“. By 𝑉2𝜎, 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, π‘βˆž2βŠ‚π‘‰2πœŽβŠ‚β„“βˆž2. In the case 𝜎(𝑖)=𝑖+1, 𝜎-convergence of double sequences reduces to the almost convergence. A matrix 𝐴=[π‘Žπ‘šπ‘›π‘—π‘˜]βˆžπ‘—,π‘˜=0 is said to be 𝜎-regular if 𝐴π‘₯βˆˆπ‘‰πœŽ2 for π‘₯=[π‘₯π‘—π‘˜]βˆˆπ‘βˆž2 with πœŽβˆ’lim𝐴π‘₯=limπ‘₯, and we denote this by 𝐴∈(π‘βˆž2,π‘‰πœŽ2)reg, (see [5, 6]). Mursaleen and Mohiuddine defined and characterized 𝜎-conservative and 𝜎-coercive matrices for double sequences [6].

A double sequence π‘₯=[π‘₯π‘—π‘˜] of real numbers is said to be CesΓ‘ro convergent (or 𝐢1-convergent) to a number L if and only if π‘₯∈𝐢1, where 𝐢1=ξ‚»π‘₯βˆˆβ„“βˆž2∢lim𝑝,π‘žβ†’βˆžπ‘‡π‘π‘ž(π‘₯)=𝐿;𝐿=𝐢1ξ‚Ό,π‘‡βˆ’limπ‘₯π‘π‘ž(1π‘₯)=(𝑝+1)(π‘ž+1)π‘ξ“π‘žπ‘—=1ξ“π‘˜=1π‘₯π‘šπ‘›π‘—π‘˜.(1.8) We shall denote by 𝐢1 the space of CesΓ‘ro convergent (𝐢1-convergent) double sequences.

A matrix 𝐴=(π‘Žπ‘šπ‘›π‘—π‘˜) is said to be 𝐢1-multiplicative if 𝐴π‘₯∈𝐢1 for π‘₯=[π‘₯π‘—π‘˜]βˆˆπ‘βˆž2 with 𝐢1βˆ’lim𝐴π‘₯=𝛼limπ‘₯, and in this case we write 𝐴∈(π‘βˆž2,𝐢1)𝛼. Note that if 𝛼=1, then 𝐢1-multiplicative matrices are said to be 𝐢1-regular matrices.

Recall that the Knopp core (or K-core) of a real number single sequence π‘₯=(π‘₯π‘˜) is defined by the closed interval [β„“(π‘₯),𝐿(π‘₯)], where β„“(π‘₯)=liminfπ‘₯ and 𝐿(π‘₯)=limsupπ‘₯. The well-known Knopp core theorem states (cf., Maddox [7] and Knopp [8]) that in order that 𝐿(𝐴π‘₯)≀𝐿(π‘₯) for every bounded real sequence π‘₯, it is necessary and sufficient that 𝐴=(π‘Žπ‘›π‘˜) should be regular and limπ‘›β†’βˆžβˆ‘βˆžπ‘˜=0|π‘Žπ‘›π‘˜|=1. Patterson [9] extended this idea for double sequences by defining the Pringsheim core (or P-core) of a real bounded double sequence π‘₯=[π‘₯π‘—π‘˜] as the closed interval [π‘ƒβˆ’liminfπ‘₯,π‘ƒβˆ’limsupπ‘₯]. Some inequalities related to the these concepts have been studied in [5, 9, 10]. Let πΏβˆ—(π‘₯)=limsup𝑝,π‘žβ†’βˆžsup𝑠,𝑑1π‘π‘žπ‘ξ“π‘žπ‘—=0ξ“π‘˜=0π‘₯𝑗+𝑠,π‘˜+𝑑,𝐢𝜎(π‘₯)=limsup𝑝,π‘žβ†’βˆžsup𝑠,𝑑1π‘π‘žπ‘ξ“π‘žπ‘—=0ξ“π‘˜=0π‘₯πœŽπ‘—(𝑠),πœŽπ‘˜(𝑑).(1.9)

Then, MR- (Moricz-Rhoades) and 𝜎-core of a double sequence have been introduced by the closed intervals [βˆ’πΏβˆ—(βˆ’π‘₯),πΏβˆ—(βˆ’π‘₯)] and [βˆ’πΆπœŽ(βˆ’π‘₯),𝐢𝜎(π‘₯)], and also the inequalities 𝐿(𝐴π‘₯)β‰€πΏβˆ—(π‘₯),πΏβˆ—(𝐴π‘₯)≀𝐿(π‘₯),πΏβˆ—(𝐴π‘₯)β‰€πΏβˆ—(π‘₯),𝐿(𝐴π‘₯)β‰€πΆπœŽ(π‘₯),𝐢𝜎(𝐴π‘₯)≀𝐿(π‘₯)(1.10) have been studies in [3–5, 11].

In this paper, we introduce the concept of 𝐢1-multiplicative matrices and determine the necessary and sufficient conditions for a matrix 𝐴=(π‘Žπ‘šπ‘›π‘—π‘˜) to belong to the class (π‘βˆž2,𝐢1)𝛼. Further we investigate the necessary and sufficient conditions for the inequality πΆβˆ—1(𝐴π‘₯)≀𝛼𝐿(π‘₯)(1.11) for all π‘₯βˆˆβ„“2∞.

2. Main Results

Let us write πΆβˆ—1(π‘₯)=limsup𝑝,π‘žβ†’βˆž1(𝑝+1)(π‘ž+1)π‘ξ“π‘žπ‘—=0ξ“π‘˜=0π‘₯π‘—π‘˜.(2.1) Then, we will define the 𝑃𝐢-core of a realvalued bounded double sequence π‘₯=[π‘₯π‘—π‘˜] by the closed interval [βˆ’πΆβˆ—1(βˆ’π‘₯),πΆβˆ—1(π‘₯)]. Since every bounded convergent double sequence is CesΓ‘ro convergent, we have πΆβˆ—1(π‘₯)β‰€π‘ƒβˆ’limsupπ‘₯, and hence it follows that 𝑃𝐢-core(π‘₯)βŠ†π‘ƒ-core(π‘₯) for a bounded double sequence π‘₯=[π‘₯π‘—π‘˜].

Lemma 2.1 .. A matrix 𝐴=(π‘Žπ‘šπ‘›π‘—π‘˜) is 𝐢1-multiplicative if and only if lim𝑝,π‘žβ†’βˆžπ›½(𝑗,π‘˜,𝑝,π‘ž)=0(𝑗,π‘˜=0,1,…),(2.2)limβˆžπ‘,π‘žβ†’βˆžξ“βˆžπ‘—=0ξ“π‘˜=0𝛽(𝑗,π‘˜,𝑝,π‘ž)=𝛼,(2.3)limβˆžπ‘,π‘žβ†’βˆžξ“π‘—=0||||𝛽(𝑗,π‘˜,𝑝,π‘ž)=0(π‘˜=0,1,…),(2.4)limβˆžπ‘,π‘žβ†’βˆžξ“π‘˜=0||||𝛽(𝑗,π‘˜,𝑝,π‘ž)=0(𝑗=0,1,…),(2.5)βˆžξ“βˆžπ‘—=0ξ“π‘˜=0||π‘Žπ‘šπ‘›π‘—π‘˜||≀𝐢<∞,(π‘š,𝑛=0,1,…),(2.6) where the lim means π‘ƒβˆ’lim and 1𝛽(𝑗,π‘˜,𝑝,π‘ž)=(𝑝+1)(π‘ž+1)π‘ξ“π‘žπ‘—=0ξ“π‘˜=0π‘Žπ‘šπ‘›π‘—π‘˜.(2.7)

Proof. Sufficiency. Suppose that the conditions (2.2)-(2.6) hold and π‘₯=[π‘₯π‘—π‘˜]βˆˆπ‘βˆž2 with π‘ƒβˆ’lim𝑗,π‘˜π‘₯π‘—π‘˜=𝐿, say. So that for every πœ–>0 there exists 𝑁>0 such that |π‘₯π‘—π‘˜|<|β„“|+πœ– whenever 𝑗,π‘˜>𝑁.
Then, we can write βˆžξ“βˆžπ‘—=0ξ“π‘˜=0𝛽(𝑗,π‘˜,𝑝,π‘ž)π‘₯π‘—π‘˜=𝑁𝑁𝑗=0ξ“π‘˜=0𝛽(𝑗,π‘˜,𝑝,π‘ž)π‘₯π‘—π‘˜+βˆžξ“π‘—=π‘π‘βˆ’1ξ“π‘˜=0𝛽(𝑗,π‘˜,𝑝,π‘ž)π‘₯π‘—π‘˜+π‘βˆ’1ξ“βˆžπ‘—=0ξ“π‘˜=𝑁𝛽(𝑗,π‘˜,𝑝,π‘ž)π‘₯π‘—π‘˜+βˆžξ“βˆžπ‘—=𝑁+1ξ“π‘˜=𝑁+1𝛽(𝑗,π‘˜,𝑝,π‘ž)π‘₯π‘—π‘˜.(2.8) Therefore, |||||βˆžξ“βˆžπ‘—=0ξ“π‘˜=0𝛽(𝑗,π‘˜,𝑝,π‘ž)π‘₯π‘—π‘˜|||||≀‖π‘₯‖𝑁𝑁𝑗=0ξ“π‘˜=0||||𝛽(𝑗,π‘˜,𝑝,π‘ž)+β€–π‘₯β€–βˆžξ“π‘—=π‘π‘βˆ’1ξ“π‘˜=0||𝛽(𝑗,π‘˜,𝑝,π‘ž)π‘₯π‘—π‘˜||+β€–π‘₯β€–π‘βˆ’1ξ“βˆžπ‘—=0ξ“π‘˜=𝑁||||+ξ€·||𝐿||ξ€Έ|||||𝛽(𝑗,π‘˜,𝑝,π‘ž)+πœ–βˆžξ“βˆžπ‘—=𝑁+1ξ“π‘˜=𝑁+1|||||.𝛽(𝑗,π‘˜,𝑝,π‘ž)(2.9) Letting 𝑝,π‘žβ†’βˆž and using the conditions (2.2)–(2.6), we get |||||limβˆžπ‘,π‘žβ†’βˆžξ“βˆžπ‘—=0ξ“π‘˜=0𝛽(𝑗,π‘˜,𝑝,π‘ž)π‘₯π‘—π‘˜|||||≀||𝐿||ξ€Έ+πœ–π›Ό.(2.10) Since πœ– is arbitrary, 𝐢1βˆ’lim𝐴π‘₯=𝛼𝐿. Hence 𝐴∈(π‘βˆž2,𝐢1)𝛼, that is, 𝐴 is 𝐢1-multiplicative.

Necessity. Suppose that A is 𝐢1-multiplicative. Then, by the definition, the A-transform of π‘₯ exists and 𝐴π‘₯∈𝐢1 for each π‘₯βˆˆπ‘βˆž2. Therefore, 𝐴π‘₯ is also bounded. Then, we can write supβˆžπ‘š,π‘›ξ“βˆžπ‘—=0ξ“π‘˜=0||π‘Žπ‘šπ‘›π‘—π‘˜π‘₯π‘—π‘˜||<𝑀<∞,(2.11) for each π‘₯βˆˆπ‘βˆž2. Now, let us define a sequence 𝑦=[π‘¦π‘—π‘˜] by π‘¦π‘—π‘˜=ξƒ―sgnπ‘Žπ‘šπ‘›π‘—π‘˜,0β‰€π‘—β‰€π‘Ÿ,0β‰€π‘˜β‰€π‘Ÿ,0,otherwise,(2.12) β€‰π‘š,𝑛=0,1,2,…. Then, the necessity of (10) follows by considering the sequence 𝑦=[π‘¦π‘—π‘˜] in (2.11).
Also, by the assumption, we have limβˆžπ‘,π‘žβ†’βˆžξ“βˆžπ‘—=0ξ“π‘˜=0𝛽(𝑗,π‘˜,𝑝,π‘ž)π‘₯π‘—π‘˜=𝛼lim𝑗,π‘˜β†’βˆžπ‘₯π‘—π‘˜.(2.13) Now let us define the sequence 𝑒𝑖𝑙 as follows: 𝑒𝑖𝑙=ξƒ―1,(𝑗,π‘˜)=(𝑖,𝑙),0,otherwise,(2.14) and write 𝑠𝑙=βˆ‘π‘–π‘’π‘–π‘™(π‘™βˆˆβ„•), π‘Ÿπ‘–=βˆ‘π‘™π‘’π‘–π‘™(π‘–βˆˆβ„•). Then, the necessity of (2.2), (2.4), and (2.5) follows from 𝐢1βˆ’lim𝐴𝑒𝑖𝑙, 𝐢1βˆ’limπ΄π‘Ÿπ‘— and 𝐢1βˆ’limπ΄π‘ π‘˜, respectively.

Note that when 𝛼=1, the above theorem gives the characterization of 𝐴∈(π‘βˆž2,𝐢1)reg. Now, we are ready to construct our main theorem.

Theorem 2.2. For every bounded double sequence x, πΆβˆ—1(𝐴π‘₯)≀𝛼𝐿(π‘₯),(2.15) or (π‘ƒπΆβˆ’π‘π‘œπ‘Ÿπ‘’{𝐴π‘₯}βŠ†π›Ό(π‘ƒβˆ’π‘π‘œπ‘Ÿπ‘’{π‘₯})) if and only if 𝐴 is 𝐢1-multiplicative and limsupβˆžπ‘,π‘žβ†’βˆžξ“βˆžπ‘—=0ξ“π‘˜=0||||𝛽(𝑗,π‘˜,𝑝,π‘ž)=𝛼.(2.16)

Proof. Necessity. Let (2.15) hold and for all π‘₯βˆˆβ„“2∞. So, since π‘βˆž2βŠ‚β„“2∞, then, we get 𝛼(βˆ’πΏ(βˆ’π‘₯))β‰€βˆ’πΆβˆ—1(βˆ’π΄π‘₯)β‰€πΆβˆ—1(𝐴π‘₯)≀𝛼𝐿(π‘₯).(2.17) That is, 𝛼liminfπ‘₯β‰€βˆ’πΆβˆ—1(βˆ’π΄π‘₯)β‰€πΆβˆ—1(𝐴π‘₯)≀𝛼limsupπ‘₯,(2.18) where βˆ’πΆβˆ—1(βˆ’π΄π‘₯)=liminfβˆžπ‘,π‘žβ†’βˆžξ“βˆžπ‘—=0ξ“π‘˜=0𝛽(𝑗,π‘˜,𝑝,π‘ž)π‘₯π‘—π‘˜.(2.19) By choosing π‘₯=[π‘₯π‘—π‘˜]βˆˆπ‘2∞, we get from (2.17) that βˆ’πΆβˆ—1(βˆ’π΄π‘₯)=πΆβˆ—1(𝐴π‘₯)=𝐢1βˆ’lim𝐴π‘₯=𝛼limπ‘₯.(2.20) This means that 𝐴 is 𝐢1-multiplicative.
By Lemma  3.1 of Patterson [9], there exists a π‘¦βˆˆβ„“2∞ with ||𝑦||≀1 such that πΆβˆ—1(𝐴𝑦)=limsupβˆžπ‘,π‘žβ†’βˆžξ“βˆžπ‘—=0ξ“π‘˜=0𝛽(𝑗,π‘˜,𝑝,π‘ž).(2.21) If we choose 𝑦=𝑣=[π‘£π‘—π‘˜], it follows π‘£π‘—π‘˜=ξƒ―1if𝑗=π‘˜,0,elsewhere.(2.22) Since β€–π‘£π‘—π‘˜β€–β‰€1, we have from (2.15) that 𝛼=πΆβˆ—1(𝐴𝑣)=limsupβˆžπ‘,π‘žβ†’βˆžξ“βˆžπ‘—=0ξ“π‘˜=0||||𝑣𝛽(𝑗,π‘˜,𝑝,π‘ž)β‰€π›ΌπΏπ‘—π‘˜ξ€Έβ‰€π›Όβ€–π‘£β€–β‰€π›Ό.(2.23) This gives the necessity of (2.16).

Sufficiency. Suppose that 𝐴 is 𝐢1-regular and (2.16) holds. Let π‘₯=[π‘₯π‘—π‘˜] be an arbitrary bounded sequence. Then, there exist 𝑀,𝑁>0 such that π‘₯π‘—π‘˜β‰€πΎ for all 𝑗,π‘˜β‰₯0. Now, we can write the following inequality: |||||βˆžξ“βˆžπ‘—=0ξ“π‘˜=0𝛽(𝑗,π‘˜,𝑝,π‘ž)π‘₯π‘—π‘˜|||||=|||||βˆžξ“βˆžπ‘—=0ξ“π‘˜=0ξ‚΅||𝛽||(𝑗,π‘˜,𝑝,π‘ž)+𝛽(𝑗,π‘˜,𝑝,π‘ž)2βˆ’||||𝛽(𝑗,π‘˜,𝑝,π‘ž)βˆ’π›½(𝑗,π‘˜,𝑝,π‘ž)2ξ‚Άπ‘₯π‘—π‘˜||||β‰€βˆžξ“βˆžπ‘—=0ξ“π‘˜=0||𝛽||||π‘₯(𝑗,π‘˜,𝑝,π‘ž)π‘—π‘˜||+βˆžξ“βˆžπ‘—=0ξ“π‘˜=0||ξ€·||𝛽||ξ€Έπ‘₯(𝑗,π‘˜,𝑝,π‘ž)βˆ’π›½(𝑗,π‘˜,𝑝,π‘ž)π‘—π‘˜||≀‖π‘₯‖𝑀𝑁𝑗=0ξ“π‘˜=0||||𝛽(𝑗,π‘˜,𝑝,π‘ž)+β€–π‘₯β€–βˆžξ“π‘π‘—=𝑀+1ξ“π‘˜=0||||𝛽(𝑗,π‘˜,𝑝,π‘ž)+β€–π‘₯β€–π‘€ξ“βˆžπ‘—=0ξ“π‘˜=𝑁+1||||𝛽(𝑗,π‘˜,𝑝,π‘ž)+sup𝑗,π‘˜β‰₯𝑀,𝑁||π‘₯π‘—π‘˜||βˆžξ“βˆžπ‘—=𝑀+1ξ“π‘˜=𝑁+1||||𝛽(𝑗,π‘˜,𝑝,π‘ž)+β€–π‘₯β€–βˆžξ“βˆžπ‘—=0ξ“π‘˜=0ξ€·||||ξ€Έ.𝛽(𝑗,π‘˜,𝑝,π‘ž)βˆ’π›½(𝑗,π‘˜,𝑝,π‘ž)(2.24) Using the condition of 𝐢1-multiplicative and condition (2.16), we get πΆβˆ—1(𝐴π‘₯)≀𝛼𝐿(π‘₯).(2.25) This completes the proof of the theorem.

Theorem 2.3. For π‘₯,π‘¦βˆˆβ„“βˆž2, if 𝐢1βˆ’lim|π‘₯βˆ’π‘¦|=0, then 𝐢1βˆ’core{π‘₯}=𝐢1βˆ’core{𝑦}.

Proof. Since 𝐢2βˆ’lim|π‘₯βˆ’π‘¦|=0, we have 𝐢1βˆ’lim(π‘₯βˆ’π‘¦)=0 and 𝐢1βˆ’lim(βˆ’(π‘₯βˆ’π‘¦))=0. Using definition of 𝐢1βˆ’core, we take πΆβˆ—1(π‘₯βˆ’π‘¦)=βˆ’πΆβˆ—1(βˆ’(π‘₯βˆ’π‘¦))=0. Since πΆβˆ—1 is sublinear, 0=βˆ’πΆβˆ—1(βˆ’(π‘₯βˆ’π‘¦))β‰€βˆ’πΆβˆ—1(βˆ’π‘₯)βˆ’πΆβˆ—1(𝑦).(2.26) Therefore, πΆβˆ—1(𝑦)β‰€βˆ’πΆβˆ—1(βˆ’π‘₯). Since βˆ’πΆβˆ—1(βˆ’π‘₯)β‰€πΆβˆ—1(π‘₯), this implies that πΆβˆ—1(𝑦)β‰€πΆβˆ—1(π‘₯).By an argument similar as above, we can show that πΆβˆ—1(π‘₯)β‰€πΆβˆ—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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