Abstract and Applied AnalysisVolume 2011 (2011), Article ID 741382, 9 pagesdoi:10.1155/2011/741382

Research Article

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

E. Savaş

Department of Mathematics, Istanbul Commerce University, Üsküdar, Istanbul, Turkey

Received 2 March 2011; Revised 17 April 2011; Accepted 20 April 2011

Academic Editor: Ondřej Došlý

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.

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 [2–5].

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, [9–12]).

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, 18–23].

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.

References

P. Kostyrko, T. Šalát, and W. Wilczyński, “I-convergence,” Real Analysis Exchange, vol. 26, no. 2, pp. 669–685, 2000.

B. K. Lahiri and P. Das, “I and -convergence in topological spaces,” Mathematica Bohemica, vol. 130, no. 2, pp. 153–160, 2005.

P. Kostyrko, M. Mačaj, T. Šalát, and M. Sleziak, “I-convergence and extremal I-limit points,” Mathematica Slovaca, vol. 55, no. 4, pp. 443–464, 2005.

P. Das, P. Kostyrko, W. Wilczyński, and P. Malik, “I and -convergence of double sequences,” Mathematica Slovaca, vol. 58, no. 5, pp. 605–620, 2008.

P. Das and P. Malik, “On the statistical and I- variation of double sequences,” Real Analysis Exchange, vol. 33, no. 2, pp. 351–363, 2008.

S. Gähler, “2-metrische Räume und ihre topologische Struktur,” Mathematische Nachrichten, vol. 26, pp. 115–148, 1963.

H. Gunawan and Mashadi, “On finite-dimensional 2-normed spaces,” Soochow Journal of Mathematics, vol. 27, no. 3, pp. 321–329, 2001.

R. W. Freese and Y. J. Cho, Geometry of Linear 2-Normed Spaces, Nova Science Publishers, Hauppauge, NY, USA, 2001.

A. Şahiner, M. Gürdal, S. Saltan, and H. Gunawan, “Ideal convergence in 2-normed spaces,” Taiwanese Journal of Mathematics, vol. 11, no. 5, pp. 1477–1484, 2007.

E. Savaş, “On some new sequence spaces in n-normed spaces using ideal convergence and an Orlicz function,” Journal of Inequalities and Applications, vol. 2010, Article ID 482392, 8 pages, 2010.

B. C. Tripathy and B. Hazarika, “I-convergent sequence spaces associated with multiplier sequences,” Mathematical Inequalities & Applications, vol. 11, no. 3, pp. 543–548, 2008.

B. C. Tripathy and B. Hazarika, “Paranorm I-convergent sequence spaces,” Mathematica Slovaca, vol. 59, no. 4, pp. 485–494, 2009.

M. Gürdal, A. Şahiner, and I. Açık, “Approximation theory in 2-Banach spaces,” Nonlinear Analysis, vol. 71, no. 5-6, pp. 1654–1661, 2009.

M. A. Krasnoselskii and Y. B. Rutisky, Convex Function and Orlicz Spaces, P. Noordhoff, Groningen, The Netherlands, 1961.

S. D. Parashar and B. Choudhary, “Sequence spaces defined by Orlicz functions,” Indian Journal of Pure and Applied Mathematics, vol. 25, no. 4, pp. 419–428, 1994.

B. C. Tripathy, M. ET, and Y. Altin, “Generalized difference sequence spaces defined by Orlicz function in a locally convex space,” Journal of Analysis and Applications, vol. 1, no. 3, pp. 175–192, 2003.

A. Sahiner and M. Gurdal, “New sequence spaces in n-spaces with respect to an Orlicz function,” The Aligarh Bulletin of Mathematics, vol. 27, no. 1, pp. 53–58, 2008.

W. H. Ruckle, “FK spaces in which the sequence of coordinate vectors is bounded,” Canadian Journal of Mathematics, vol. 25, pp. 973–978, 1973.

I. J. Maddox, “Sequence spaces defined by a modulus,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 100, no. 1, pp. 161–166, 1986.

R. Çolak, M. Et, and E. Malkowsky, “Strongly almost (w, )-summable sequences defined by Orlicz functions,” Hokkaido Mathematical Journal, vol. 34, no. 2, pp. 265–276, 2005.

E. Savaş and R. F. Patterson, “An Orlicz extension of some new sequence spaces,” Rendiconti dell'Istituto di Matematica dell'Università di Trieste, vol. 37, no. 1-2, pp. 145–154, 2005.

B. C. Tripathy and P. Chandra, “On some generalized difference paranormed sequence spaces associated with multiplier sequence defined by modulus function,” Analysis in Theory and Applications, vol. 27, no. 1, pp. 21–27, 2011.

B. C. Tripathy and S. Mahanta, “On a class of generalized lacunary difference sequence spaces defined by Orlicz functions,” Acta Mathematicae Applicatae Sinica, vol. 20, no. 2, pp. 231–238, 2004.

E. Savaş, “-strongly summable sequences spaces in 2-normed spaces defined by ideal convergence and an Orlicz function,” Applied Mathematics and Computation, vol. 217, no. 1, pp. 271–276, 2010.

I. J. Maddox, Elements of Functional Analysis, Cambridge University Press, London, UK, 1970.