#### 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 [6–8], 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 [16–18].

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 [21–27] 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 *(RH

_{1})

*for each and ,*(RH

_{2})

*,*(RH

_{3})

*for each ,*(RH

_{4})

*for each ,*(RH

_{5})

*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