Abstract

We introduce some new generalized difference sequence spaces by means of ideal convergence, infinite matrix, and a sequence of modulus functions over -normed spaces. We also make an effort to study several properties relevant to topological, algebraic, and inclusion relations between these spaces.

1. Introduction and Preliminaries

The concept of 2-normed spaces was initially developed by Gähler [1] in the middle of 1960s, while that of -normed spaces one can see in Misiak [2]. Since then, many others have studied this concept and obtained various results; see Gunawan [3, 4] and Gunawan and Mashadi [5]. Let and be a linear space over the field of reals 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 a -norm on and the pair is called a -normed space over the field . For more details about -normed spaces, see [6, 7] and references therein.

The notion of difference sequence spaces was introduced by Kızmaz [8], who studied the difference sequence spaces , , and . The notion was further generalized by Et and Çolak [9] by introducing the spaces , , and . Later the concept has been studied by Bektaş et al. [10] and Et and Esi [11]. Another type of generalization of the difference sequence spaces is due to Tripathy and Esi [12] who studied the spaces , , and . Recently, Esi et al. [13] and Tripathy et al. [14] have introduced a new type of generalized difference operators and unified those as follows.

Let and be nonnegative integers; then for , a given sequence space, we havefor , , and where and , for all , which is equivalent to the following binomial representation:Taking , we get the spaces , , and studied by Et and Çolak [9]. Taking , we get the spaces , , and introduced and studied by Kızmaz [8]. For more details about difference sequence spaces, see [15–19].

A modulus function is a function such that(1) if and only if ,(2), for all ,(3) is increasing,(4) is continuous from the right at .It follows that must be continuous everywhere on . The modulus function may be bounded or unbounded. For example, if we take , then is bounded. If , , then the modulus function is unbounded. Subsequently, modulus function has been discussed in [20] and references therein.

In [21], Mursaleen introduced the idea of -statistical convergence by extending the concept of summability. Further, Savaş and Das [22] unified the two approaches and gave new concepts of -statistical convergence, --convergence, and - convergence. Later some pioneer works have been extended in this direction by numerous authors such as Belen and Mohiuddine [23], Das et al. [24], Gürdal and Sarí [25], and references therein. Quite recently, many authors including Et et al. [26] and Maddox [27] have constructed some sequence spaces by using modulus function and difference sequences and investigate their properties.

Let and be two sequence spaces and be an infinite matrix of real or complex numbers , where . Then we say that defines a matrix mapping from into if, for every sequence , the sequence , the -transform of , is in , whereBy , we denote the class of all matrices such that . Thus, if and only if the series on the right-hand side of (3) converges for each and every .

The matrix domain of an infinite matrix in a sequence space is defined byThe approach constructing a new sequence space by means of the matrix domain of a particular limitation method has recently been employed by several authors (see [28]).

Let be a nondecreasing sequence of positive numbers such that , , as , and .

Let be a nonempty set. Then a family of sets (power set of ) is said to be an ideal if is additive; that is, and , . A nonempty family of sets is said to be filtered on if and only if ; for we have and for each and implying .

An ideal is called nontrivial if . A nontrivial ideal is called admissible if . A nontrivial ideal is maximal if there cannot exist any nontrivial ideal containing as a subset. For each ideal , there exist a filter corresponding to ; that is, , where .

Definition 1. A sequence in a -normed space is said to be statistically convergent to some if, for each , the set has its natural density zero.

Definition 2 (see [26]). A sequence is said to be -statistically convergent to the number , if, for every , . In this case one writes .

Definition 3 (see [22]). A sequence is said to be summable to , if, for any , , where .

Definition 4 (see [22]). A sequence is said to be -statistically convergent or convergent to if, for every and , . In this case one writes or .

The following well-known lemma is required for establishing some important results in this paper.

Lemma 5. Let be a sequence of modulus functions and . Then for each one has .

Throughout the paper, will denote the set of all positive integers. By we denote the space of all sequences defined over -normed space .

Let be an admissible ideal, a sequence modulus functions, a bounded sequence of positive (strictly) real numbers, a sequence of positive real numbers, an infinite matrix, and a -normed space. Then, for every and , we define the following sequence spaces:for some , andfor some .

If we take , for all , , and and -normed space is replaced by 2-normed space then we get the sequence spaces defined by Kumar et al. [29].

The following inequality will be used throughout the paper. Let be a sequence of positive real numbers with , and let . Then, for the factorable sequences and in the complex plane, we have

2. Main Results

The main purpose of this section is to study some topological properties and some inclusion relations between the sequence spaces which we have defined above.

Theorem 6. Suppose is a sequence of modulus functions, is a bounded sequence of strictly positive real numbers, is a sequence of positive real numbers, and   is an infinite matrix. Then the spaces , , and are linear spaces over the real field .

Proof. The proof of the theorem is easy so we omit it.

For the next result we shall define the following sequence space:

Theorem 7. Let be a sequence of modulus functions and a bounded sequence of strictly positive real numbers. Then the space is a paranormed space with the paranorm:where and .

Proof. Clearly for . It is trivial that for . Since , we get for . Since , using Minkowski’s inequality, we haveHence . Finally, to check the continuity of scalar multiplication, let be any complex number; therefore by definitionwhere is a positive integer such that . Let for any fixed with . By definition, for , we haveAlso, for , taking small enough, since is coninuous, we haveNow (12) and (13) imply that as . This completes the proof.

Theorem 8. Let be a sequence of positive real numbers. Then for the following inclusions:(i),(ii),are strict.

Proof. We will prove the result for only. The others can be proved similarly.
Suppose ; by definition, for every and , we haveBy the property of modulus function, we haveNow, for given , we havefor each . Since , it follows that the sets on the right-hand side in the above containment belong to . Hence . To show that the inclusion is strict, we give the following example.
We take , , and , for all , , and consider a sequence ; then but does not belong to for . This shows that the inclusion is strict.

Theorem 9. Let and be sequences of modulus functions. If   = , then .

Proof. Let ; then there exists a constant such that for all . Therefore, for each , we haveThen, for every and , we have the following relationship:Since , it follows that the set on left-side of the above containment belongs to , which gives .