Abstract and Applied Analysis

Abstract and Applied Analysis / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 741382 | 9 pages | https://doi.org/10.1155/2011/741382

𝐴 -Sequence Spaces in 2-Normed Space Defined by Ideal Convergence and an Orlicz Function

Academic Editor: Ondřej Došlý
Received02 Mar 2011
Revised17 Apr 2011
Accepted20 Apr 2011
Published30 Jun 2011

Abstract

We study some new -sequence spaces using ideal convergence and an Orlicz function in 2-normed space and we give some relations related to these sequence spaces.

1. Introduction

Let and be two nonempty subsets of the space of complex sequences. Let be an infinite matrix of complex numbers. We write if converges for each . If we say that defines a (matrix) transformation from to , and we denote it by .

The notion of ideal convergence was introduced first by Kostyrko et al. [1] as a generalization of statistical convergence. More applications of ideals can be seen in [25].

The concept of 2-normed space was initially introduced by Gähler [6] as an interesting nonlinear generalization of a normed linear space which was subsequently studied by many authors (see, [7, 8]). Recently a lot of activities have started to study summability, sequence spaces, and related topics in these nonlinear spaces (see, [912]).

Let be a normed space. Recall that a sequence of elements of is called statistically convergent to if the set has natural density zero for each .

A family of subsets a nonempty set is said to be an ideal in if(i) imply ;(ii), imply , while an admissible ideal of further satisfies for each , (see [7, 13]).

Given a nontrivial ideal in . The sequence in is said to be -convergent to , if for each the set belongs to , (see, [1, 3]).

Let be a real vector space of dimension , where . A 2-norm on is a function which satisfies(i) if and only if and are linearly dependent; (ii); (iii), ; (iv).

The pair is then called a 2-normed space [7]. As an example of a normed space we may take being equipped with the norm the area of the parallelogram spanned by the vectors and , which may be given explicitly by the formula Recall that is a 2-Banach space if every Cauchy sequence in is convergent to some in .

Recall in [14] that an Orlicz function is a continuous, convex, nondecreasing function such that and for , and as .

Subsequently Orlicz function was used to define sequence spaces by Parashar and Choudhary [15] and others [16, 17].

If convexity of Orlicz function is replaced by then this function is called modulus function, which was presented and discussed by Ruckle [18] and Maddox [19]. It should be mentioned that notable works involving Orlicz function and modulus function were done in [16, 1823].

In this article, we define some new sequence spaces in 2-normed spaces by using Orlicz function, infinite matrix, generalized difference sequences, and ideals. We introduce and examine certain new sequence spaces using the above tools as also the 2-norm.

2. Main Results

Let be an admissible ideal of , be an Orlicz function, be a 2-normed space, and be a nonnegative matrix method. Further, let be a bounded sequence of positive real numbers. By , we denote the space of all sequences defined over Now we define the following sequence spaces: where .

Let us consider a few special cases of the above sets. (1)If , for all , then the above classes of sequences are denoted by ,,, and , respectively.(2)If for all , then we denote the above classes of sequences by , , , and , respectively.(3)If , for all , and for all , then we denote the above spaces by , , , and , respectively.(4)If we take as then the above classes of sequences are denoted by , , , and respectively, which were defined and studied by Savaş [24](5)If we take is a de la Vallée poussin mean, that is, where is a nondecreasing sequence of positive numbers tending to and , then the above classes of sequences are denoted by ,,, and.(6)By a lacunary ; where , we will mean an increasing sequence of nonnegative integers with as . The intervals determined by will be denoted by and . As a final illustration let Then we denote the above classes of sequences by , , , and.

The following well-known inequality (see [25, p. 190]) will be used in the study.

If then for all and . Also for all .

Theorem 2.1. , , and are linear spaces.

Proof. We will prove the assertion for only, and the others can be proved similarly. Assume that and In order to prove the result we need to find some such that Since , there exist some positive and such that Define . Since is nondecreasing and convex and also is a 2-norm, is linear where . From the above inequality we get Two sets on the right-hand side belong to , and this completes the proof.

It is also easy to verify that the space is also a linear space and moreover we have the following.

Theorem 2.2. For any fixed , is paranormed space with respect to the paranorm defined by

Proof. The proof is parallel to the proof of the Theorem 2 in [24] and so is omitted.

Theorem 2.3. Let stand for, , or and . Then the inclusion is strict. In general for all and the inclusion is strict.

Proof. We shall give the proof for only. It can be proved in a similar way for , and . Let . Then given we have
Since is nondecreasing and convex it follows that Hence we have Since the set on the right hand side belongs to so does the left hand side. The inclusion is strict as the sequence , for example, belongs to but does not belong to for , Cesàro matrix and for all .

Theorem 2.4. (i) Let . Then .
(ii). Then .

Proof. (i) Let . Since , we have So
(ii) Let for each , and . Let . Then for each there exists a positive integer such that for all . This implies that So we have This completes the proof.

The following corollary follows immediately from the above theorem.

Corollary 2.5. Let Cesàro matrix and let be an Orlicz function.
(1) If , then .
(2) If , then .

Definition 2.6. Let be a sequence space. Then is called solid if whenever for all sequences of scalars with for all .

Theorem 2.7. The sequence spaces and are solid.

Proof. We give the proof for only. Let , and let be a sequence of scalars such that for all . Then we have where . Hence for all sequences of scalars with for all whenever .

Remark 2.8. In general it is difficult to predict the solidity of and when . For this, consider the following example.

Example 2.9. Let for all , Cesro matrix and . Then but when for all . Hence is not solid.

Acknowledgment

The authors wish to thank the referees for their careful reading of the paper and for their helpful suggestions.

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Copyright © 2011 E. Savaş. 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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