Abstract

The aim of this paper is to introduce some new double difference sequence spaces with the help of the Musielak-Orlicz function and four-dimensional bounded-regular (shortly, RH-regular) matrices . We also make an effort to study some topological properties and inclusion relations between these double difference sequence spaces.

1. Introduction, Notations, and Preliminaries

In [1], Hardy introduced the concept of regular convergence for double sequences. Some important work on double sequences is also found by Bromwich [2]. Later on, it was studied by various authors, for example, Móricz [3], Móricz and Rhoades [4], Başarır and Sonalcan [5], Mursaleen and Mohiuddine [68], and many others. Mursaleen [9] has defined and characterized the notion of almost strong regularity of four-dimensional matrices and applied these matrices to establish a core theorem (also see [10, 11]). Altay and Başar [12] have recently introduced the double sequence spaces , , , , , and consisting of all double series whose sequence of partial sums are in the spaces , , , , , and , respectively. Başar and Sever [13] extended the well-known space from single sequence to double sequences, denoted by , and established its interesting properties. The authors of [14] defined some convex and paranormed sequences spaces and presented some interesting characterization. Most recently, Mohiuddine and Alotaibi [15] introduced some new double sequences spaces for -convergence of double sequences and invariant mean and also determined some inclusion results for these spaces. For more details on these concepts, one can be referred to [1618].

The notion of difference sequence spaces was introduced by Kızmaz [19], who studied the difference sequence spaces , , and . The notion was further generalized by Et and Çolak [20] by introducing the spaces , , and .

Let be the space of all complex or real sequences and let and be two nonnegative integers. Then for , we have the following sequence spaces: where and for all , which is equivalent to the following binomial representation:

We remark that for and , we obtain the sequence spaces which were introduced and studied by Et and Çolak [20] and Kızmaz [19], respectively. For more details about sequence spaces see [2127] and references therein.

An Orlicz function is continuous, nondecreasing, and convex such that , for and as . If convexity of Orlicz function is replaced by , then this function is called modulus function. Lindenstrauss and Tzafriri [28] used the idea of Orlicz function to define the following sequence space: which is known as an Orlicz sequence space. The space is a Banach space with the norm

Also it was shown in [28] that every Orlicz sequence space contains a subspace isomorphic to . An Orlicz function can always be represented in the following integral form: where is known as the kernel of , is a right differentiable for is nondecreasing, and as .

A sequence of Orlicz functions is said to be a Musielak-Orlicz function (see [29, 30]). A sequence is defined by which is called the complementary 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

A Musielak-Orlicz function is said to satisfy if there exist constants and a sequence (the positive cone of ) such that the inequality holds for all and , whenever .

A double sequence is said to be bounded if . We denote by the space of all bounded double sequences.

By the convergence of double sequence we mean the convergence in the Pringsheim sense; that is, a double sequence is said to converge to the limit in Pringsheim sense (denoted by, ) provided that given there exists such that whenever (see [31]). We will write more briefly as -convergent. If, in addition, , then is said to be boundedly P-convergent to . We will denote the space of all bounded convergent double sequences (or boundedly -convergent) by .

Let and let be given. By , we denote the characteristic function of the set .

Let be a four-dimensional infinite matrix of scalers. For all , where , the sum is called the - of the double sequence . A double sequence is said to be - to the limit if the -means exist for all in the sense of Pringsheim’s convergence:

A four-dimensional matrix is said to be bounded-regular (or RH-regular) if every bounded -convergent sequence is -summable to the same limit and the -means are also bounded.

The following is a four-dimensional analogue of the well-known Silverman-Toeplitz theorem [32].

Theorem 1 (Robison [33] and Hamilton [34]). The four-dimensional matrix is RH-regular if and only if (RH1) for each and ,(RH2),(RH3) for each ,(RH4) for each ,(RH5) for all .

2. The Double Difference Sequence Spaces

In this section, we define some new paranormed double difference sequence spaces with the help of Musielak-Orlicz functions and four-dimensional bounded-regular matrices. Before proceeding further, first we recall the notion of paranormed space as follows.

A linear topological space over the real field (the set of real numbers) is said to be a paranormed space if there is a subadditive function such that , , and scalar multiplication is continuous; that is, and imply for all ’s in and all ’s in , where is the zero vector in the linear space .

The linear spaces , , and were defined by Maddox [35] (also, see Simons [36]).

Let be a Musielak-Orlicz function; that is, is a sequence of Orlicz functions and let be a nonnegative four-dimensional bounded-regular matrix. Then, we define the following: where is a double sequence of real numbers such that for , , and is a double sequence of strictly positive real numbers.

Remark 2. If we take in and , then we have the following spaces:

Remark 3. Let for all . Then and are reduced to respectively.

Remark 4. Let for all . Then, the spaces and are reduced to respectively.

Remark 5. If we take in and , then we have the following spaces:

Remark 6. If we take and in and , then we have the following spaces:

Remark 7. Let for all . If, in addition, and , then the spaces and are reduced to and which were introduced and studied by Yurdakadim and Tas [37] as below:
Throughout the paper, we will use the following inequality: let and be two double sequences. Then where and (see [15]). We will also assume throughout this paper that the symbol will denote the sublinear Musielak-Orlicz function.

3. Main Results

Theorem 8. Let be a sublinear Musielak-Orlicz function, a nonnegative four-dimensional -regular matrix, a bounded sequence of positive real numbers, and a sequence of strictly positive real numbers. Then and are linear spaces over the complex field .

Proof. Let and . Then there exist integers and such that and .
Since is a nondecreasing function, so by inequality (21), we have
Thus . This proves that is a linear space. Similarly we can prove that is also a linear space.

Theorem 9. Let be a sublinear Musielak-Orlicz function, a nonnegative four-dimensional -regular matrix, a bounded sequence of positive real numbers, and a sequence of strictly positive real numbers. Then and are paranormed spaces with the paranorm where and .

Proof. We will prove the result for . Let . Then for each , exists. Also it is clear that , and .
We now show that the scalar multiplication is continuous. First observe the following: where denotes the integer part of . It is also clear that if and implies . For fixed , if , then . We need to show that for fixed implies . Let . Thus
Then, for there exists such that for . Also, for each with , since there exists an integer such that
Let . We have for each with
Also from (26), for , we have
Thus is an integer independent of such that
Since , therefore
For each and by the continuity of as , we have the following: in Pringsheim’s sense. Now choose such that implies
In the same manner, we have
It follows from (31), (34), (35), and (36) that
Thus as . Therefore is a paranormed space. Similarly, we can prove that is a paranormed space. This completes the proof.

Theorem 10. Let be a sublinear Musielak-Orlicz function, a nonnegative four-dimensional -regular matrix, a bounded sequence of positive real numbers, and a sequence of strictly positive real numbers. Then and are complete topological linear spaces.

Proof. Let be a Cauchy sequence in ; that is, as . Then, we have
Thus for each fixed and as , since is nonnegative, we are granted that and by continuity of , is a Cauchy sequence in for each fixed and .
Since is complete as , we have for each . Now from (36), we have that, for , there exists a natural number such that
Since for any fixed natural number , from (38) we have By letting in the above expression we obtain
Since is arbitrary, by letting we obtain
Thus as . This proves that is a complete topological linear space.
Now we will show that is a complete topological linear space. For this, since is also a sequence in by definition of , for each , there exists with whence from the fact that and from the definition of Musielak-Orlicz function, we have as and so converges to . Thus
Hence and this completes the proof.

Theorem 11. Let be a sublinear Musielak-Orlicz function which satisfies the -condition. Then .

Proof. Let ; that is,
Let and choose with such that for . Write and consider
For , we use the fact that . Hence
Since satisfies the -condition, we have and hence
Since is -regular and , we get .

Theorem 12. Let be a sublinear Musielak-Orlicz function and let be a nonnegative four-dimensional RH-regular matrix. Suppose that . Then

Proof. In order to prove that , it is sufficient to show that . Now, let . By definition of , we have for all . Since , we have for all . Let . Thus, we have which implies that . This completes the proof.

Theorem 13. (i) Let . Then
(ii) Let . Then

Proof. (i) Let . Then since , we obtain the following:
Thus .
(ii) Let for each and and . Let . Then for each there exists a positive integer such that
This implies that
Therefore . This completes the proof.

Lemma 14. Let be a sublinear Musielak-Orlicz function which satisfies the -condition and let be a nonnegative four-dimensional -regular matrix. Then is an ideal in .

Proof. Let and . We need to show that . Since , there exists such that . In this case for all . Since is nondecreasing and satisfies -condition, we have for all and . Therefore . Thus . This completes the proof.

Lemma 15. Let be an ideal in and let . Then is in the closure of in if and only if for all .

Proof. Let be in the closure of and let be given. Suppose that such that and observe that . Define a double sequence by
Clearly . Since and , hence .
Conversely, if then . It follows that for all ; then is in the closure of .

Lemma 16. If is a nonnegative four-dimensional -regular matrix, then is a closed ideal in .

Proof. We have and it is clear that . For , we get . Now, we have by the -condition and the convexity of . Since where , so .
Let and . Thus, there exists a positive integer , so that, for every , we have . Therefore and so
Hence . So is an ideal in for a Musielak-Orlicz function which satisfies the -condition.
Now, we have to show that is closed. Let ; there exists such that . For every there exists such that, for all , . Now, for , we have
Since and is RH-regular, we get