Abstract

We generalize the lacunary statistical convergence by introducing the generalized difference operator of fractional order, where is a proper fraction and is any fixed sequence of nonzero real or complex numbers. We study some properties of this operator and investigate the topological structures of related sequence spaces. Furthermore, we introduce some properties of the strongly Cesaro difference sequence spaces of fractional order involving lacunary sequences and examine various inclusion relations of these spaces. We also determine the relationship between lacunary statistical and strong Cesaro difference sequence spaces of fractional order.

1. Introduction

By , we denote the space of all real valued sequences and any subspace of is called a sequence space. Let , , and be the linear spaces of bounded, convergent, and null sequences with real or complex terms, respectively, normed by , where , the set of positive integers. With this norm, it is proved that these are all Banach spaces. Also by , , , and , we denote the spaces of all bounded, convergent, absolutely summable, and -absolutely summable series, respectively.

The concept of difference sequence space was determined by Kızmaz [1]. Et and Colak generalized difference sequence spaces [2]. Later on Et and Esi [3] generalized these sequence spaces to the following sequence spaces. Let be any fixed sequence of nonzero complex numbers and let be a nonnegative integer. Then, for or , where , , , and so . These are Banach spaces with the norm defined by Furthermore Et and Basarir [4] have generalized difference sequence spaces. Aydın and Başar [5] have introduced some new difference sequence spaces. Also the notion of difference sequence has been extended by Mursaleen [6], Mursaleen and Noman [7], Malkowsky et al. [8], and Bektaş et al. [9]. Also Tripathy et al. [10, 11] have generalized difference sequences by Orlicz functions.

Let be the sequence of positive integers such that , , and as . Then is called a lacunary sequence. The intervals determined by will be denoted by . Das and Mishra [12] introduced lacunary strong almost convergence. Colak et al. [13] have studied lacunary strongly summable sequences. Also some geometric properties of sequence spaces involving lacunary sequence have been examined by Karakaya [14]. Et has generalized Cesaro difference sequence spaces involving lacunary sequences [15].

Besides the Cesaro sequence spaces and have been introduced by Shiue [16]. Jagers [17] has determined the Köthe duals of the sequence space Later on the Cesaro sequence spaces and of nonabsolute type are defined by Ng and Lee [18, 19].

The main focus of the present paper is to generalize strong Cesaro and lacunary statistical difference sequence spaces and investigate their topological structures as well as some interesting results concerning the operator .

The rest of this paper is organized, as follows: in Section 2, some required definitions and consequences related to the difference operator are given. Also some new classes of difference sequences of fractional order involving lacunary sequences are determined and some topological properties are investigated. Section 3 is devoted to the strong Cesaro difference sequence spaces of fractional order. Prior to stating and proving the main results concerning these spaces, we give some theorems about the notion of linearity and -space. In final section, we present some theorems related to the lacunary statistical convergence of difference sequences of fractional order and examine some inclusion relations of these spaces.

2. Some New Difference Sequence Spaces with Fractional Order

By , we denote the Euler gamma function of a real number . Using the definition, with can be expressed as an improper integral as follows:For a positive proper fraction , Baliarsingh and Dutta [20, 21] (also see [22]) have defined the generalized fractional difference operator asIn particular, we have (i) − ,(ii) + + ,(iii) − − .

Theorem 1 (see [20]). (a) For proper fraction , defined by (2) is a linear operator.
(b) For , and .

Now, we determine the new classes of difference sequence spaces as follows:where and is any sequence spaces.

Theorem 2. For a proper fraction , if is a linear space, then is also a linear space.

Proof. The proof is straightforward (see [20]).

Theorem 3. If is a Banach space with the norm , then is also a Banach space with the norm defined bywhere .

Proof. Proof of this theorem is a routine verification, hence omitted.

Remark 4. Without loss of generality, we assume throughout that each series given in (4) is convergent. Furthermore, if is a positive integer, then these infinite sums in (4) reduce to finite sums; that is, and .

Lemma 5. Let be a proper fraction. If , then .

Proof. Let and . It is trivial that implies . Hence and .

Theorem 6. Let be a Banach space and a closed subset of . Then is also closed subset of .

Proof. By using Lemma 5, it is trivial that . Now we prove that . Let ; then there exists a sequence such that Thus, we observe that as in . This implies that .
Conversely, let ; then . Since is closed , hence is a closed subset of .

Theorem 7. If is a -space with the norm , then is also a -space with the norm given in (4).

Proof. It is clear that is a Banach space (see Theorem 3). Suppose that for each . One can conclude that as for each and implies as . This follows from the fact that where ; since the series represented by is finite. Hence as for each . Therefore is a Banach space with the continuous coordinates. This completes the proof.

Definition 8. A sequence is said to be -strongly Cesaro convergent if there is a real or complex number such thatwhere is a fixed positive number and is a proper fraction. The number is unique when it exists. By , one denotes the set of all strongly -Cesaro convergent sequences. In this case, one writes .

Theorem 9. The sequence space is a Banach space for normed by and a complete -normed space with the -norm for .

Proof. Proof follows by using Theorem 3.

3. Cesaro Difference Sequence Spaces of Fractional Order

In this section by using the operator , we introduce some new sequence spaces , , , , and involving lacunary sequences and arbitrary sequence of strictly positive real numbers.

We define the sequence spaces as follows: where is a fixed sequence of nonzero real or complex numbers.

Theorem 10. Assume that is a bounded sequence. Then the sequence spaces , , , , and are linear spaces.

Proof. Because the linearity may be proved in a similar way for each of the sets of sequences, hence it is omitted.

Theorem 11. If for all , then the sequence space is a -space with the norm defined by Also if for all , then the sequence spaces and are -spaces with the norm defined by

Proof. We give the proof for the space and that of others followed by using similar techniques.
Suppose is a Cauchy sequence in , where and are two elements in . Then there exists a positive integer such that as , for all , and for each . Therefore and are Cauchy sequences in complex field and , respectively. By using the completeness of and , we have that they are convergent and suppose that in and in for each as . Then we can find a sequence such that for each . These ’s can be interpreted as for sufficiently large ; that is, . Then converges to for each as . Thus as . Since is a Banach space with continuous coordinates, that is, implies for each , as , this shows that is a -space.

Theorem 12. If for all , then the sequence space is a -space with the norm defined by Also if for all , then the sequence space is a -space with the norm defined by

Proof. The proof follows from Theorem 11.

Now, we can present the following theorem, determining some inclusion relations without proof, since it is a routine verification.

Theorem 13. Let two positive proper fractions and for each be given. Then the following inclusions are satisfied: (i),(ii),(iii), .

4. Lacunary Statistical Convergence of Difference Sequences

The concept of statistical convergence from different aspects has been studied by various mathematicians. The notion of statistical convergence was independently introduced by Fast [23] and Schoenberg [24].

Let be a subset of the set of natural numbers . Then the asymptotic density of denoted by is defined as , where the vertical bars denote the cardinality of the enclosed set.

A number sequence said to be statistically convergent to the number if, for each , the set has asymptotic density zero; that is, .

Definition 14. Let be a lacunary sequence and a proper fraction. Then the sequence is said to be -lacunary statistically convergent to a real or complex number if, for each ,where the vertical bars denote the cardinality of the enclosed set. The set of -lacunary statistically convergent sequences will be denoted by . In this case we write .