Abstract and Applied Analysis

Volume 2014 (2014), Article ID 691632, 9 pages

http://dx.doi.org/10.1155/2014/691632

Research Article

New Difference Sequence Spaces Defined by Musielak-Orlicz Function

M. Mursaleen,¹ Sunil K. Sharma,² S. A. Mohiuddine,³ and A. Kılıçman⁴

¹Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India

²Department of Mathematics, Model Institute of Engineering & Technology, Kot Bhalwal, Jammu and Kashmir 181122, India

³Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

⁴Department of Mathematics, University Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia

Received 17 March 2014; Revised 11 July 2014; Accepted 11 July 2014; Published 22 July 2014

Academic Editor: Feyzi Başar

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.

We introduce new sequence spaces by using Musielak-Orlicz function and a generalized -difference operator on -normed space. Some topological properties and inclusion relations are also examined.

1. Introduction and Preliminaries

The notion of the difference sequence space was introduced by Kızmaz [1]. It was further generalized by Et and Çolak [2] 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 [3] taking , , and .

Dutta [4] introduced the following difference sequence spaces using a new difference operator:

where for all .

In [5], 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 authors [6–14] and references therein. Başar and Altay [15] introduced the generalized difference matrix for all , which is a generalization of -difference operator by

Başarir and Kayikçi [16] defined the matrix which reduced the difference matrix in case , . 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

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

We say that an Orlicz function satsfies the -condition if there exists and such that for all . The -condition is equivalent to for all and for .

Lindenstrauss and Tzafriri [17] 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 [17] that every Orlicz sequence space contains a subspace isomorphic to .

A sequence of Orlicz function is called a Musielak-Orlicz function; see [18, 19]. A sequence defined by

is called the complimentary function of a Musielak-Orlicz function . For a given Musielak-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

By a lacunary sequence , , where , we will mean an increasing sequence of nonnegative integers . The intervals determined by are denoted by and the ratio will be denoted by . The space of lacunary strongly convergent sequences was defined by Freedman et al. [20] as follows:

The concept of 2-normed spaces was initially developed by Gähler [21] in the mid of 1960’s, while that of -normed spaces one can see in Misiak [22]. Since then, many others have studied this concept and obtained various results; see Gunawan [23, 24] and Gunawan and Mashadi [25]. For more details about sequence spaces see [26–33] and references therein. Let and be linear space over the field , where is the field of real or complex numbers of dimension , where .

A real valued function on satisfying the following four conditions:

(1) if and only if are linearly dependent in ;(2) is invariant under permutation;(3) for any ;(4)

is called an -norm on and the pair is called an -normed space over the field . For example, we may take being equipped with the Euclidean -norm the volume of the -dimensional parallelepiped spanned by the vectors which may be given explicitly by the formula

where for each and denotes the Euclidean norm. Let be an -normed space of dimension and linearly independent set in . Then the following function on defined by

defines an norm on with respect to .

A sequence in an -normed space is said to converge to some if

A sequence in a normed space is said to be Cauchy if

If every Cauchy sequence in converges to some then is said to be complete with respect to the -norm. Any complete -normed space is said to be -Banach space.

Let be an -normed space and let denote the space of -valued sequences. Let be any bounded sequence of positive real numbers and a Musielak-Orlicz function. We define the following sequence spaces in this paper:

when , we get

when , for all , we get

The following inequality will be used throughout the paper. If , , then

for all and . Also for all .

2. Main Results

Theorem 1. Let be a Musielak-Orlicz function and a bounded sequence of positive real numbers; the spaces , , and are linear over the field of complex numbers .

Proof. Let , , and . Then there exist positive real numbers and such that

Define . Since is an -norm on and are nondecreasing and convex functions so by using inequality (23) we have

Thus, we have . Hence is a linear space. Similarly, we can prove that and are linear spaces. This completes the proof of the theorem.

Theorem 2. Let be a Musielak-Orlicz function and a bounded sequence of positive real numbers; the space is a topological linear space paranormed by

where .

Proof. Clearly for . Since , we get . Again, if , then

This implies that, for a given , there exist some such that

Thus

for each , and suppose that for each . This implies that for each . Let , then

which is a contradiction. Therefore, for each and thus for each . Let and be such that

Let ; then by using Minkowski’s inequality, we have

Since are nonnegative, we have

Therefore, .

Finally, we prove that the scalar multiplication is continuous. Let be any complex number. By definition,

Then

where . Since , we have

So, the fact that scalar multiplication is continuous follows from the above inequality. This completes the proof of theorem.

Theorem 3. Let be a Musielak-Orlicz function. If for all fixed , then .

Proof. Let . Then there exists some positive number such that

Define . Since is nondecreasing and convex and by using inequality (23), we have

Hence . This completes the proof of the theorem.

Theorem 4. Let be a Musielak-Orlicz function and . Then

if and only if

Proof. Let . Suppose (40) does not hold. Therefore there are a subinterval of the set of intervals and a number , where for all , such that

Let us define as follows:

Thus by (41), . But . Hence (40) must hold.

Conversely, suppose that (40) holds and let . Then,

Suppose that . Then for some number , , there is a number such that, for a subinterval of the set of intervals ,

We have , which contradicts (40) by using (43). Hence we get

This completes the proof.

Theorem 5. Let . For any Musielak-Orlicz function which satisfies -condition, one has

(i)(ii)(iii).

Proof. (i) Let . Then, we have

Let , and choose with such that for . We can write

For the first summation above, we can write

By using continuity of , for the second summation we can write

Since each is nondecreasing and convex and satisfies -condition, it follows that

Taking limit as and , it follows that . Hence . Similarly, we can prove (ii) and (iii). This completes the proof of the theorem.

Theorem 6. Let be a Musielak-Orlicz function. Then the following statements are equivalent:

(i); (ii); (iii) for all , where .

Proof. (i) (ii) Suppose (i) holds. In order to prove (ii) we have to show that

Let . Then for a given there exists such that

Hence there exists such that

This shows that .

(ii) (iii) Suppose (ii) holds and (iii) fails to hold. Then for some ,

and, therefore, we can find a subinterval of the set of intervals such that

Let us define as follows:

Thus . But by (55), which contradicts (ii). Hence (iii) must hold.

(iii) (i) Let (iii) hold. Suppose that . Then for .

Let for each , and then by (57) , which contradicts (iii). Hence (i) must hold. This completes the proof of the theorem.

Theorem 7. Let be a Musielak-Orlicz function. Then the following statements are equivalent:

(i); (ii); (iii) for all .

Proof. (i) (ii) is obvious

(ii) (iii) Let (ii) hold and let (iii) fail to hold. Then

and we can find a subinterval of the set of intervals such that

Let us define as follows:

Thus by (iii), . But which contradict (ii). Hence (iii) must hold.

(iii) (i) Let (iii) hold. Suppose that . Therefore,

Again suppose for some number and a subinterval of the set of intervals , we have

Then, from properties of the Orlicz function, we can write

Consequently, by (61), we have , which contradicts (iii). Hence (i) must hold. This completes the proof of the theorem.

Theorem 8. (i) If for all , then .

(ii) If , then .

Proof. (i) Let . Since , we get

and hence .

(ii) and . Then for each there exists a positive integer such that

This implies that

Therefore . 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.

Acknowledgments

The authors gratefully acknowledge that this research was partially supported by the University Putra Malaysia under the ERGS Grant Scheme having Project no. ERGS 1-2013/5527179. The authors are grateful also to the anonymous referees for a careful checking of the details and for helpful comments that improved the paper.

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