A Note on Lacunary Sequence Spaces of Fractional Difference Operator of Order

Abstract

In the present paper, we defined lacunary sequence spaces of fractional difference operator of order over -normed spaces via Musielak-Orlicz function . Our aim in this paper is to study some topological properties and inclusion relation between the spaces , , and .

1. Introduction and Preliminaries

The concept of statistical convergence was introduced by Fast [1] and Schoenberg [2] independently. Many authors studied the concept of statistical convergence from the past few years we may refer to ([3–19]) and references therein.

The sequence is statistically convergent of order to (see Çolak) if there is a complex number such that

Let . We define the -density of the subset of by provided the limit exists, where denotes the th power of number of elements of not exceeding ([20–22]).

By a lacunary sequence , we mean a sequence of positive integers such that , , and as . The intervals determined by will be denoted by and . Freedman et al. [23] defined the space of lacunary strongly convergent sequences by

Definition 1. Let be a lacunary sequence. The sequence is -statistically convergent (or lacunary statistically convergent of order ) (see [20]) if there is a real number such that where and denotes the th power of , that is, . In this case, we write The set of all -statistically convergent sequences is denoted by . If and , then, we will write instead of .
A family of subsets of a nonempty set is said to be an ideal in if (1)(2) imply (3), imply while an admissible ideal of further satisfies for each (see [24]).

A sequence in is said to be -convergent to (see [24]), if for each

A sequence in is said to be -bounded to if there exists an such that . Many authors studied the topological properties and applications of ideal, we refer to ([25–37]) and references therein.

The concept of difference sequence spaces was introduced in [38] and further generalized in [39].

In [40], Baliarsingh defined the fractional difference operator as follows:

Let and be a real number, then, the fractional difference operator is defined by where denotes the Pochhammer symbol defined as

The concept of difference sequences, Orlicz function, Musielak-Orlicz function, and -normed spaces was used by many authors and proves some topological properties (see [41–50]) and references therein. For details about -normed spaces, we refer to ([51–55]), difference sequence spaces ([38, 39]), Orlicz function ([56–58]). Ideal convergence and fractional difference operator has been studied in [59, 60]. We continue in this connection and construct new sequence spaces as follows.

Let be a Musielak-Orlicz function, be a bounded sequence of positive real numbers, and . We define the following sequence spaces in the present paper

If we take , the above spaces reduces to , , and .

If we take , the above spaces reduces to , , and .

The following inequality will be used in the proceeding results. If , , then for all and . Also for all .

2. Main Results

In this section, we study topological properties and prove some inclusion relations. In what follows, we will take a Musielak-Orlicz function and a bounded sequence of positive real numbers.

Theorem 2. The spaces , , and are linear spaces.

Proof. Let and let be scalars. Then, there exist two positive numbers and for Let and by inequality (1), we have Now by (11) and (12), we get Therefore, . Hence, is a linear space. On a similar way, we can prove that and are linear spaces.

Theorem 3. The inclusions hold.

Proof. The inclusion is obvious. We prove . For this, let . Then, there exists such that for every We put and is a Musielak-Orlicz function, we have Suppose that . Hence by above inequality and (1), we have By using , we have Put It follows that This shows that , which completes the proof.

Theorem 4. The space is a paranormed space with paranorm defined by

Proof. Since and , we have . Let . Let Let and and , we have Thus, and Let where and let as . We have to show that as . Let If and ; then, we have From the above inequality, it follows that and consequently, which completes the proof.

Theorem 5. Let and be Musielak-Orlicz functions that satisfy the -condition. Then

Proof. (i) Let . Then, there exists such that for . Since is a Musielak-Orlicz function which satisfies -condition, we have for . By continuity of , we have Suppose . Then, by using (34) and (35), we have Hence, and so which implies . This shows that . Hence, . Similarly, we can prove (ii) and (iii) part.

Corollary 6. Let satisfy -condition. Then,

Proof. If we put and in Theorem 5, the result follows.

Theorem 7. Let and be Musielak-Orlicz functions that satisfy the -condition. Then,

Proof. (i) Let . Then, there exists and such that for some . Let . Then, we have . We have and so which implies . This shows that . Hence, . Similarly, we can prove (ii) and (iii) part of the theorem.

Theorem 8. Let and be bounded. Then, the following inclusions hold

Proof. (i) Let . Write and , so that . By using Hölder inequality, we have Hence, for every , we have This implies that and so . Hence, . Similarly, we can prove .

Corollary 9. If . Then, the following inclusions hold:

Proof. The proof follows from Theorem 8.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to writing this paper. All authors read and approved the manuscript.

Acknowledgments

Corresponding author is supported by JADD program (by UPM-UoN) while the first author is supported by the Natural Science Foundation of Fujian Province of China (Grant no. 2020J01783), the Project for High-level Talent Innovation and Entrepreneurship of Quanzhou (Grant no. 2018C087R), and the Program for New Century Excellent Talents in Fujian Province University.