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Abstract and Applied Analysis
Volume 2012, Article ID 426357, 17 pages
http://dx.doi.org/10.1155/2012/426357
Research Article

On Some Lacunary Almost Convergent Double Sequence Spaces and Banach Limits

Department of Mathematics, Sakarya University, 54187 Sakarya, Turkey

Received 13 June 2012; Accepted 9 July 2012

Academic Editor: Chaitan Gupta

Copyright © 2012 Metin Başarır and Şükran Konca. 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.

Abstract

The object of this paper is to introduce some new sequence spaces related with the concept of lacunary strong almost convergence for double sequences and also to characterize these spaces through sublinear functionals that both dominate and generate Banach limits and to establish some inclusion relations.

1. Introduction and Preliminaries

Let be the set of all real or complex double sequences. We mean the convergence in the Pringsheim sense, that is, a double sequence has a Pringsheim limit denoted by   provided that given and there exists such that whenever [1]. We denote by , the space of -convergent sequences. A double sequence is bounded if . Let and be the set of all real or complex bounded double sequences and the set of bounded and convergent double sequences, respectively. Moricz and Rhoades [2] defined the almost convergence of double sequences that is said to be almost convergent to a number if that is, the average value of taken over any rectangle tends to as both and tend to and this convergence is uniform in and . We denote the space of almost convergent double sequences by , as where

The notion of almost convergence for single sequences was introduced by Lorentz [3] and for double sequences by Moricz and Rhoades [2] and some further studies are in [414].

A double sequence is called strongly almost convergent to a number if

By , we denote the space of all strongly almost convergent double sequences. It is easy to see that the inclusions strictly hold. As in the case of single sequences, every almost convergent double sequence is bounded. But a convergent double sequence need not be bounded. Thus, a convergent double sequence need not be almost convergent. However every bounded convergent double sequence is almost convergent.

The notion of strong almost convergence for single sequences has been introduced by Maddox [15, 16] and for double sequences by Başarir [17].

A linear functional on is said to be Banach limit if it has the following properties [7], (1) if (i.e., for all ,  ),(2), where with for all ,   and(3) where the shift operators are defined by .

Let be the set of all Banach limits on . A double sequence is said to be almost convergent to a number if for all . If is any sublinear functional on , then we write to denote the set of all linear functionals on , such that , that is, . A sublinear functional is said to generate Banach limits if implies that is a Banach limit; is said to dominate Banach limits if is a Banach limit implies that . Then if both generates and dominates Banach limits, then is the set of all Banach limits.

Using the notations for single sequences, we present the notations for double-lacunary sequences that can be seen in [10]. The double sequence is called a double-lacunary if there exist two increasing sequences of nonnegative integers such that as and as . Let is determined by . Also and is determined by , where and and and with and .

Das and Mishra [18] introduced the space of lacunary almost convergent sequences by combining the space of lacunary convergent sequences and the space of almost convergent sequences. Savaş and Patterson [10] extended the notions of lacunary almost convergence and lacunary strongly almost convergence to double-lacunary -convergence and double-lacunary strongly almost -convergence. They also established multidimensional analogues of Das and Patel's results.

We will use the following definition which may be called convergence in Pringsheim's sense with a bound: and also we will use the following definition which may be called convergence in Pringsheim's sense as follows:

The following sequence spaces were introduced and examined by Başarir [19]: with respect to sublinear functionals on (the set of all real or complex bounded single sequences) by where and .

It can be easily seen that each of the above functionals are finite, well defined, and sublinear on . There is a very close connection among these sequence spaces with the sublinear functionals which were given by Başarir [19]. Recently Mursaleen and Mohiuddine [7] generalized the sequence spaces which were studied by Das and Sahoo [20] for single sequences, to the double sequences as follows: by using (1.4).

The object of the present paper is to determine some new sublinear functionals involving double-lacunary sequence that both dominates and generates Banach limits. We also extend the sequence spaces which were introduced for single sequences by Başarir [19] to the double sequences with respect to these sublinear functionals. Furthermore, we present some inclusion relations with these new sequence spaces between the sequence spaces which were introduced by Mursaleen and Mohiuddine [7], earlier.

2. Sublinear Functionals and Double-Lacunary Sequence Spaces

In this section, we introduce the following sequence spaces:

It may be noted that almost convergent double sequences are necessarily bounded but the sequence spaces and may contain unbounded sequences. Now we define the following functionals on for a double-lacunary sequence by, It is easy to see that each of the above functionals are finite, well defined, and sublinear on .

Throughout the paper we will write for and by this notation we shall mean the convergence in the Pringsheim sense. In the following theorem, we demonstrate that is the set of all Banach limits on and characterize the space in terms of the sublinear functional .

Theorem 2.1. One has the following.
(1) The sublinear functional both dominates and generates Banach limits, that is, , for all .
(2)

Proof. From the definition of , for given there exist such that for and for all . This implies that for all . Since is arbitrary, so that , for all and hence that is, generates Banach limits.
Conversely, suppose that . As is the shift invariant, that is, and using the properties of , we obtain
It follows from the definition of , that for given there exist such that for and for all . Hence by (2.8) and properties (1) and (2) of Banach limits, we have for and for all ; where with for all ,  . Since is arbitrary, it follows from (2.7) and (2.9) that Hence That is, dominates Banach limits. Combining (2.6) and (2.11), we get this implies that dominates and generates Banach limits and for all .
As a consequence of Hahn-Banach theorem, is non empty and a linear functional is not necessarily uniquely defined at any particular value of . This is evident in the manner the linear functionals are constructed. But in order that all the functionals coincide at , it is necessary and sufficient that we have But (2.14) holds if and only if Hence, . But (2.13) is equivalent to , this holds if and only if . This completes the proof of the theorem.

If for all , then we say that is -convergent to . Similarly we define the -convergent sequences. In the following theorem we characterize the spaces and in terms of the sublinear functionals.

Theorem 2.2. One has the following:(1)  =  , for  some   =  , for all   , for  some   (2)  =  , for  some   =  , for all  , for  some .

Proof. It can be easily verified that if and only if Since then (2.16) reduces to Now if then from (2.17) and linearity of , we have Conversely, suppose that for all and hence by Hahn-Banach theorem, there exists such that . Hence
The proof is similar to the proof of (1), above.

3. Inclusion Relations

We establish here some inclusion relations between the sequence spaces defined in Section 2.

Theorem 3.1. We have the following proper inclusions and the limit is preserved in each case.(1). (2). (3).

Proof. Let with . Then This implies that This proves that and . Since this implies that and . Since converges uniformly in as , implies the convergence for . It follows that and . This completes the proof of .
It is easy to see the proof of and . So we omit them.

Theorem 3.2. One has the following proper inclusions;

Proof. The proof of the theorem is similar as in [7, Theorem 4.2]. So we omit it.

Prior to giving Lemmas 3.3 and 3.5, we need the following notations used in [10]:

Lemma 3.3. Suppose there exist , and such that for and . Then .

Proof. Let be given. Choose and such that for . We need only to show that given there exist and such that for and . If we take and , then (3.9) holds for and for all and , which gives the result. Once and have been chosen, they are fixed, so is finite. Now taking , and , we have from (3.8) and (3.10) Therefore taking and sufficiently large, we can make which gives (3.9) and hence the result.

Theorem 3.4. We have for every .

Proof. Let ; then given there exist and such that for , and where . Let such that where is an integer. Also let such that where is an integer and , . Since for and for we have, which gives the result. Therefore by Lemma 3.3, . It is clear that for every . This completes the proof.

Lemma 3.5. Suppose, for a given , there exist , and such that for all and . Then .

Proof. Let be given and choose and such that for all , and . As in Lemma 3.3, it is enough to show that there exist and such that for implies for all and with and . Since and are fixed, Now, let , and and consider the following; Let , then for . Also if we let , then for . Therefore from (3.15) From (3.18) and (3.19) for sufficiently large values of and . Hence the result.

Theorem 3.6. For every , One has .

Proof. Let . For , there exist and such that for and with where and . Let and where and . Then Since for all  and , there exists such that . From (3.21) and (3.22), we have the following: Thus for and sufficiently large, we have the following: for and . Thus by Lemma 3.5, we have . It is clear that . This completes the proof of the theorem.

Corollary 3.7. .

Proof. It is easy to see by combining Theorem 3.4, Theorem 3.6 with [7, Theorems 4.1 and 3.1(ii)]. So we omit it.

A paranormed space is a topological linear space with the topology given by the paranorm . It may be recalled that a paranorm is a real subadditive function on such that and scalar multiplication is continuous, that is, imply that where are scalars and .

Let be a bounded double sequence of positive real numbers, that is, for all with . Let If is constant we write in place of . If we take with for all and , then is reduced to which is defined in Section 2.

Theorem 3.8. Let be a bounded sequence of positive real numbers with . Then is a complete linear topological space paranormed by where . In the case is constant, is a Banach space if and is a -normed space if  .

Proof. It is easy to see that is a linear space with coordinatewise addition and scalar multiplication. Clearly and is subadditive. To prove the continuity of multiplication, assume that . Since is bounded and positive there exists a constant such that for all . Now for ,   and hence . This proves the fact that is a paranorm on .
To prove that is complete, assume that is a Cauchy sequence in , that is, as . Since it follows that as for each , and . In particular Hence, is a Cauchy sequence in or . Since or is complete, there exists or such that coordinate wise as . It follows from (3.27) that given , there exists such that for ,  . Now making and then taking supremum with respect to and in (3.29) we obtain for . This proves that and . Hence is complete. When is constant, it is easy to derive the rest of the theorem.

Theorem 3.9. Let for each and . Then .

Proof. Let . By the definition of , that for given there exist such that for and for all . Since as , then for and for all . This implies that for sufficiently large values of and for all . Then we get, as and for all . Hence, is obtained and consequently we have . This completes the proof.

Theorem 3.10. One has the following.(1)Let for each and . Then .(2)Let for each and . Then .

Proof. It is clear from the above theorem. If we take and for each and , then we have .
From the above theorem, if we take and for each and , then we have .
This completes the proof.

Acknowledgment

The authors are grateful to anonymous referees for their careful reading of the paper which improved it greatly.

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