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Research Article | Open Access

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

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

# The Cesáro Core of Double Sequences

Accepted24 May 2011
Published21 Aug 2011

#### Abstract

We have characterized a new type of core for double sequences, -core, and determined the necessary and sufficient conditions on a four-dimensional matrix to yield -core-core for all .

#### 1. Introduction

A double sequence is said to be convergent in the Pringsheim sense or -convergent if for every there exists an such that whenever , . In this case, we write . By , we mean the space of all -convergent sequences.

A double sequence is bounded if By , 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 the space of double sequences which are bounded and convergent.

A double sequence is said to converge regularly if it converges in Pringsheim's sense and, in addition, the following finite limits exist: Let be a four-dimensional infinite matrix of real numbers for all . The sums are called the -transforms of the double sequence . We say that a sequence is -summable to the limit if the -transform of exists for all and is convergent to in the Pringsheim sense, that is,

We say that a matrix is bounded-regular if every bounded-convergent sequence is -summable to the same limit and the -transforms are also bounded. The necessary and sufficient conditions for to be bounded-regular or RH-regular (cf., Robison ) are A double sequence is said to be almost convergent (see ) to a number if Let be a one-to-one mapping from into itself. The almost convergence of double sequences has been generalized to the -convergence in  as follows: where . In this case, we write . By , we denote the set of all -convergent and bounded double sequences. One can see that in contrast to the case for single sequences, a convergent double sequence need not be -convergent. But every bounded convergent double sequence is -convergent. So, . In the case , -convergence of double sequences reduces to the almost convergence. A matrix is said to be -regular if for with , and we denote this by , (see [5, 6]). Mursaleen and Mohiuddine defined and characterized -conservative and -coercive matrices for double sequences .

A double sequence of real numbers is said to be Cesáro convergent (or -convergent) to a number L if and only if , where We shall denote by the space of Cesáro convergent (-convergent) double sequences.

A matrix is said to be -multiplicative if for with , and in this case we write . Note that if , then -multiplicative matrices are said to be -regular matrices.

Recall that the Knopp core (or K-core) of a real number single sequence is defined by the closed interval , where and . The well-known Knopp core theorem states (cf., Maddox  and Knopp ) that in order that for every bounded real sequence , it is necessary and sufficient that should be regular and . Patterson  extended this idea for double sequences by defining the Pringsheim core (or P-core) of a real bounded double sequence as the closed interval . Some inequalities related to the these concepts have been studied in [5, 9, 10]. Let

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

In this paper, we introduce the concept of -multiplicative matrices and determine the necessary and sufficient conditions for a matrix to belong to the class . Further we investigate the necessary and sufficient conditions for the inequality for all .

#### 2. Main Results

Let us write Then, we will define the -core of a realvalued bounded double sequence by the closed interval . Since every bounded convergent double sequence is Cesáro convergent, we have , and hence it follows that -core-core for a bounded double sequence .

Lemma 2.1 .. A matrix is -multiplicative if and only if where the means and

Proof. Sufficiency. Suppose that the conditions (2.2)-(2.6) hold and with , say. So that for every there exists such that whenever .
Then, we can write Therefore, Letting and using the conditions (2.2)–(2.6), we get Since is arbitrary, . Hence , that is, is -multiplicative.

Necessity. Suppose that A is -multiplicative. Then, by the definition, the A-transform of exists and for each . Therefore, is also bounded. Then, we can write for each . Now, let us define a sequence by . Then, the necessity of (10) follows by considering the sequence in (2.11).
Also, by the assumption, we have Now let us define the sequence as follows: and write , . Then, the necessity of (2.2), (2.4), and (2.5) follows from , and , respectively.

Note that when , the above theorem gives the characterization of . Now, we are ready to construct our main theorem.

Theorem 2.2. For every bounded double sequence , or if and only if is -multiplicative and

Proof. Necessity. Let (2.15) hold and for all . So, since , then, we get That is, where By choosing , we get from (2.17) that This means that is -multiplicative.
By Lemma  3.1 of Patterson , there exists a with such that If we choose , it follows Since , we have from (2.15) that This gives the necessity of (2.16).

Sufficiency. Suppose that is -regular and (2.16) holds. Let be an arbitrary bounded sequence. Then, there exist such that for all . Now, we can write the following inequality: Using the condition of -multiplicative and condition (2.16), we get This completes the proof of the theorem.

Theorem 2.3. For , if , then .

Proof. Since , we have and . Using definition of , we take . Since is sublinear, Therefore, . Since , this implies that .By an argument similar as above, we can show that . 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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