Journal of Function Spaces
Volume 2018 (2018), Article ID 4312817, 11 pages
https://doi.org/10.1155/2018/4312817

Research Article

Some Seminormed Difference Sequence Spaces over -Normed Spaces Defined by a Musielak-Orlicz Function of Order

S. A. Mohiuddine,¹ Sunil K. Sharma,² and Dina A. Abuzaid1,3

¹Operator Theory and Applications Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
²Department of Mathematics, Cluster University of Jammu, Jammu 180005, India
³Department of Mathematics, Sciences Faculty for Girls, King Abdulaziz University, P.O. Box 4087, Jeddah 21491, Saudi Arabia

Correspondence should be addressed to S. A. Mohiuddine; mohiuddine@gmail.com

Received 2 December 2017; Accepted 12 March 2018; Published 22 April 2018

Academic Editor: Maria Alessandra Ragusa

Copyright © 2018 S. A. Mohiuddine 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 first define the notion of lacunary statistical convergence of order , and taking this notion into consideration, we introduce some seminormed difference sequence spaces over -normed spaces with the help of Musielak-Orlicz function of order . We also examine some topological properties and prove inclusion relations between the resulting sequence spaces.

1. Introduction and Preliminaries

Gähler [1] extended the usual notion of normed spaces into -normed spaces, while the notion was again extended to -normed spaces by Misiak [2]. Assume that is a linear space over the field of real or complex numbers of dimension , ( denotes the set of natural numbers). A real valued function on satisfying the conditions,

(N1) if and only if are linearly dependent in ,(N2) is invariant under permutation,(N3) for any ,(N4),

is called a -norm on , and the pair is called a -normed space over .

A sequence in a -normed space is said to be

(i)convergent to if holds for every ,(ii)Cauchy if holds for every

An -Banach space is a complete -normed space, where completeness means that, for every in with , there exists such that .

Kızmaz [3] was the first who introduced the idea of difference sequence spaces and studied , where for all , is the space of all complex or real sequences, and the standard notations , , and denote bounded, convergent, and null sequences, respectively. Et and Çolak [4] presented a generalization of these difference sequence spaces and introduced the space ; in this case, is given by for . Another generation is given by Tripathy and Esi [5] by considering instead as in Kızmaz spaces and is defined by . In view of and , Tripathy et al. [6] introduced the difference sequence space as follows:

where and , which is equivalent to the following binomial representation:

We remark that if we take , then difference sequence space is reduced to and for the choice of , we obtain the difference sequence space . For recent work related to various kinds of difference sequence spaces, we refer to [7–13] and references therein.

If is a linear space and such that (i) , (ii) , (iii) , and (iv) if and in the sense that for scalars and the vectors , then in the sense that ; then is said to be a paranorm on and the pair is called a paranormed space.

An Orlicz function is a function which is continuous, nondecreasing, and convex with , as and . Clearly, if is a convex function and , then for all . An Orlicz function is said to satisfy -condition for all values of , if there exists a constant such that for all values of (see Krasnoselskii and Rutitsky [14] and, more recent, Giannetti et al. [15]). Using the idea of Orlicz function, Lindenstrauss and Tzafriri [16] constructed the sequence space

which is called Orlicz sequence space and showed that is a Banach space with the following norm:

The space is closely related to the space which is an Orlicz sequence space with for

A sequence of Orlicz functions is said to be a Musielak-Orlicz function [17]. A sequence is defined by which is said to be complementary function of . For a given , the Musielak-Orlicz sequence space and its subspace are defined by

where denotes the convex modular and is defined by

It is noted that can be considered equipped with the Luxemburg norm or equipped with the Orlicz norm, where Luxemburg and Orlicz norms are given by

respectively.

A sequence is said to be almost convergent if all of its Banach limits coincide. The space of almost convergent sequences was defined by Lorentz [18] (also see [19]) as follows:

Recall that is strongly almost convergent [20] to some number if

The idea of statistical convergence first appeared, under the name of almost convergence, in the first edition of Zygmund [21]. Later, this idea was introduced by Fast [22] and Steinhaus [23], independently, and some of its basic properties were studied by Schoenberg [24], Šalát [25], and Fridy [26]. Some useful results, applications, and various developments on this topic have been presented by many authors; we refer to [27–35]. Çolak [36] extended this notion with the help of -density (for , -density reduced to natural density) and called it statistical convergence of order . In the recent past, Şenül [37] presented an interesting generalization of this notion by taking into account instead of and defined statistical convergence of order as follows.

The sequence is said to be statistically convergent of order , briefly -convergence, to if for each , we have

where denotes the th power of number of elements of not exceeding , and we write .

2. Construction of Difference Sequence Spaces

Before defining some difference sequence spaces, let us first introduce the notion of lacunary statistical convergence of order . Recall that the standard notation denotes the lacunary sequence, where is a sequence of positive integers such that , , and , throughout the article, the intervals determined by the lacunary sequence will be denoted by and the ratio by (see [38]).

Suppose that and . We define the -density of by

in case this limit exists, where denotes the th power of and denotes the th power of number of elements of in .

Suppose that and let be a lacunary sequence. Then, we say that the sequence is lacunary statistically convergent of order (shortly, -convergence) to the number and we write if for each , the set has -density zero; that is,

Example 1. Consider a sequence defined by if and otherwise. Also, let . Then for any ,

We see that is -convergent to zero but not convergent.

Remark 2. For , the notion of -convergence coincides with -convergence in [37]. Also, if we take and , then -convergence reduces to the notion of statistical convergence of order due to Çolak [36]. Further, the choice of and in the definition of lacunary statistically convergent of order gives us the notion of statistical convergence due to Fast [22]. Moreover, if in the definition of -convergence, then we obtain lacunary statistical convergence introduced by Fridy and Orhan [39].

We denote by the space of all sequences defined over . Let be a seminormed space, seminormed by , and let be any bounded sequence of positive real numbers. We are now ready to define the following sequence spaces:

Note that if we consider ; then the above difference sequence spaces reduce to the following spaces:

It is to further notice that if we take for all , then the spaces , , and reduce to the following spaces:

Remark 3. If , , and in our difference sequence spaces defined above, then these spaces reduce to the difference sequence spaces defined by Raj et al. [40]. Moreover, if we take in our difference sequence spaces and take for all in the difference sequence spaces defined by Raj and Sharma [41], then we observe that difference sequence spaces of both the manuscripts coincide.

The following inequality will be used to prove some of our results in the next section. If , , then

for all and . Also for all .

3. Main Results

In this section, we are going to study some topological properties and to obtain some inclusion relations between the difference sequence spaces defined in the previous section.

Theorem 4. Let = be a Musielak-Orlicz function and let be a bounded sequence of positive real numbers. Then the spaces , and are linear over the field of complex numbers

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

Let Since is nondecreasing convex function, by (17), we obtain

Thus, we have Hence, is a linear space. In a similar way, one can prove that and are linear spaces.

Theorem 5. For any 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 obtain . Again, if , then we have

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

Thus,

for each and . Assume that for each . This implies that for each Let ; then It follows that

which is a contradiction. Consequently, we have for each and so for each . Suppose that and be such that

for each and . Assume that Then, by Minkowski’s inequality, we have

Since ’s are nonnegative, one obtains

Consequently, . It is only left to show that the scalar multiplication is continuous. Let us consider a complex number . Then, by definition, we obtain

Then

where Since , we have

So, the fact that scalar multiplication is continuous follows from the above inequality.

Theorem 6. Let be a Musielak-Orlicz function. If for all fixed , then

Proof. Let There exists some positive such that

Define Since is nondecreasing and convex, by using (17), we obtain

which gives

Hence,

Theorem 7. Let and let and be two Musielak-Orlicz functions satisfying condition. Then

(i),(ii),(iii)

Proof. Let Then, we have

uniformly in and for some . Let and choose with such that for Let

We can write

Since satisfies -condition, we have

For , we obtain

It follows from the fact that is nondecreasing and convex that

Since satisfies -condition, we can write

Hence,

Equation (37) together with (41) gives

This completes the proof of (i). In a similar way, using -condition for , we can prove that

Corollary 8. Let and let be Musielak-Orlicz function satisfying -condition. Then

Proof. The proof follows by taking in the above Theorem 7.

Theorem 9. If be the Musielak-Orlicz function, then the following statements are equivalent:

(i),(ii),(iii)

Proof. (i) (ii). It is obvious in view of the following fact that

(ii) (iii). Let Assume that (iii) does not hold. Then, for some , we obtain

and so one can find a subinterval of the set of interval such that

Let us define the sequence by

Then but, by (47), which contradicts (ii). Hence, the condition (iii) must hold.
(iii) (i). Suppose that (iii) holds and Also assume that Then

Let for each and fixed . It follows from (49) that

which contradicts (iii). Hence, the condition (i) holds.

Theorem 10. Let and let be a Musielak-Orlicz function. Then the following statements are equivalent:

(i),(ii),(iii)

Proof. It is trivial to obtain that (i) implies (ii). We are now proving that (ii) implies (iii). Let the condition (ii) hold. Assume that (iii) does not hold. Therefore,

and so we can find a subinterval of such that

For all , define if and if . If follows from (52) that but which contradicts (ii). Thus, the condition (iii) holds. It is left to prove that (iii) implies (i). Suppose that (iii) holds and also assume that . Therefore,

Again, assume that . Then, for some number and a subinterval of , one obtains for all and for some . Therefore, in view of the properties of the Orlicz function, one writes

By taking into account (53), we obtain

which contradicts (iii). Hence, (i) must hold.

Theorem 11. Let for all and let be bounded. Then

Proof. Suppose that . Write

and for all . Then for all . Take for . Consider two sequences and defined by and if and and for . Clearly

It follows that and Therefore,

Now, for each , we obtain

By taking the above two inequalities into account, we get

Hence, we conclude that .

Theorem 12.

(i)If , then

(ii)If , then

Proof. (i) Assume that . Then

Since , we have

Thus, we have

which gives . Therefore, we conclude that

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

Since , we have

Hence,

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under Grant no. (RG-14-130-38). The authors, therefore, acknowledge with thanks DSR for technical and financial support.

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