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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 461081, 7 pages
http://dx.doi.org/10.1155/2014/461081
Research Article

New Classes of Generalized Seminormed Difference Sequence Spaces

1Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India
2Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
3Department of Mathematics, Model Institute of Engineering & Technology, Kot Bhalwal 181122, J&K, India

Received 23 March 2014; Revised 2 May 2014; Accepted 4 May 2014; Published 1 June 2014

Academic Editor: Adem Kılıçman

Copyright © 2014 M. Mursaleen et al. 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 purpose of this paper is to introduce new classes of generalized seminormed difference sequence spaces defined by a Musielak-Orlicz function. We also study some topological properties and prove some inclusion relations between resulting sequence spaces.

1. Introduction and Preliminaries

Let denote the space of all real sequences . Let denote the space whose elements are the sets of distinct positive integers. Given any element of , we denote by the sequence such that if , and otherwise. Further the set of those   whose support has cardinality at most , and where .

For , Sargent [1] defined the following sequence space: which was further studied in [24].

The space was extended to by Tripathy and Sen [5] as follows:

The notion of the difference sequence space was introduced by Kızmaz [6] which was generalized by Mursaleen [7]. It was further generalized by Et and Çolak [8] as follows: for , , and , where is a nonnegative integer and or equivalent to the following binomial representation: These sequence spaces were generalized by Et and Basarir [9] for , , and .

Dutta [10] introduced the following difference sequence spaces using a new difference operator: where for all .

In [11], Dutta introduced the sequence spaces , ,   , and , where and and for all , which is equivalent to the following binomial representation:

The difference sequence spaces have been studied by several authors [1219] and references therein. Başar and Altay [20] introduced the generalized difference matrix by

Başarir and Kayikçi [21] defined the matrix which reduces to the difference matrix if . The generalized -difference operator is equivalent to the following binomial representation:

Let be a sequence of nonzero scalars. Then, for a sequence space , the multiplier sequence space , associated with the multiplier sequence , is defined as

Let denote the space of all sequences with elements in , where denotes a seminormed space, seminormed by . The zero sequence is denoted by .

An Orlicz function is a function, , which is continuous nondecreasing and convex with for and as .

Lindenstrauss and Tzafriri [22] used the idea of Orlicz function to define the following sequence space: which is called an Orlicz sequence space. The space is a Banach space with the norm It is shown in [22] that every Orlicz sequence space contains a subspace isomorphic to . The -condition is equivalent to for all values of and for .

A sequence of Orlicz functions is called a Musielak-Orlicz function. A sequence defined by is called the complimentary function of a Musielak-Orlicz function (see [23, 24]). For a given Musiclak-Orlicz function , the Musielak-Orlicz sequence space and its subspace are defined as follows: where is a convex modular defined by We consider equipped with the Luxemburg norm, or equipped with the Orlicz norm, A sequence space is said to be solid if , whenever for all sequences of scalars such that for all .

A sequence space is said to be monotone if contains the canonical preimages of all its step spaces.

Remark 1. It is well known that a sequence space is solid implies that it is monotone (see Kamthan and Gupta [25]).

The sequence space was introduced by Sargent [1]. He studied some of its properties and obtained its relationship with the space . Later on, it was investigated from sequence space point of view and related with summability theory by Bilgin [26], Esi [27], Tripathy and Mahanta [28], and many others.

The main goal of the present paper is to introduce new classes of generalized seminormed difference sequence spaces defined by Musielak-Orlicz function.

For a given infinite matrix . The -transform of a sequence is the sequence (), where provided that the series on the right converges for each .

Let be a seminormed space, a Musielak-Orlicz function, and a bounded sequence of positive real numbers. Then we define the following classes of sequences:

The following inequality will be used throughout the paper. If and then for all and . Also for all .

We study here some topological properties and establish inclusion relations between these sequence spaces.

2. Main Results

Theorem 2. Let be a Musielak-Orlicz function and a bounded sequence of positive real numbers. Then the spaces , , and are linear spaces over the field of complex number .

Proof. Let ,   , and . Then there exist positive real numbers such that Define . Since is a nondecreasing, convex function and so by using inequality (21), we have Thus Thus . Hence is a linear space. Similarly, we can prove that the spaces and are linear spaces. This completes the proof of the theorem.

Theorem 3. Let be a Musielak-Orlicz function and a bounded sequence of positive real numbers. Then .

Proof. Let . Then, for some , we have Since is a monotonic increasing, we have Hence, Thus, . Therefore, .
Next, let . Then, for some , we have Hence, Thus, . Therefore, . This completes the proof of the theorem.

Theorem 4. Let be a Musielak-Orlicz function and a bounded sequence of positive real numbers. Then the space is a seminormed space, seminormed by

Proof. Clearly, for all and . Let , . Then there exist and such that
Let . Thus, we have Since the ’s are nonnegative, so we have Thus, . Next, for , without loss of generality, , then
This completes the proof of the theorem.

Theorem 5. Let be a Musielak-Orlicz function and a bounded sequence of positive real numbers. Then (i)the space is a seminormed space, seminormed by (ii)the space is a seminormed space, seminormed by

Proof. It is easy to prove in view of Theorem 4, so we omit the details.

Theorem 6. Let be a Musielak-Orlicz function and a bounded sequence of positive real numbers. Then if and only if .

Proof. Suppose and . Then, we have for some Thus, Therefore, . Hence, .
Conversely, let . Suppose that . Then there exists a sequence of naturals such that . Let . Then there exists such that Now, we have Therefore, , which is a contradiction. Hence .

We get the following corollary as a consequence of Theorem 6.

Corollary 7. Let be a Musielak-Orlicz function and a bounded sequence of positive real numbers. Then if and only if and for all .

Theorem 8. Let be Musielak-Orlicz functions which satisfy -conditions and a bounded sequence of positive real numbers. Then (i);(ii).

Proof. (i) Let . Then there exists such that Let and such that for . Let and, for any , let , where the first summation is over and the second summation is over . Since satisfies -condition, we have For Since is nondecreasing and convex, so Since also satisfies -condition, so Hence, By (42) and (46), we have . Hence
(ii) Let . Then there exists such that The rest of the proof follows from the equality This completes the proof of the theorem.

Corollary 9. Let be a Musielak-Orlicz function and a bounded sequence of positive real numbers. Then, we have .

Proof. It follows from Theorem 8(i) on considering , for all .

The following result is a consequence of Theorem 8 and Corollary 9.

Corollary 10. Let be a Musielak-Orlicz function and a bounded sequence of positive real numbers. Then if and only if .

Theorem 11. Let be a Musielak-Orlicz function and a bounded sequence of positive real numbers. Then the space is solid.

Proof. Let . Then Let be a sequence of scalars with for all . Then the result follows from (50) and the following inequality
This completes the proof of the theorem.

In view of the above result, we get the following corollaries.

Corollary 12. Let be a Musielak-Orlicz function and a bounded sequence of positive real numbers. Then the space is monotone.

We formulate the following result which can be established following the technique of Theorem 11 and Corollary 12.

Corollary 13. Let be a Musielak-Orlicz function and a bounded sequence of positive real numbers. Then the spaces and are solid and monotone.

Theorem 14. If is complete, then is also complete.

Proof. Let be a Cauchy sequence in , where for each . Let and be fixed. Then for each , there exists a positive integer such that This implies We have for all , and by (53) We can find such that , where is the kernel associated with Musielak-Orlicz function , such that Hence is a Cauchy sequence in , which is complete. Therefore, for each , there exist and such that as . Using the continuity of , so for some , we have Now, taking the infimum of such ’s by (53), we get Since is a linear space and are in , so it follows that . Hence is complete. This completes the proof of the theorem.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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