#### Abstract

The aim of this paper is to introduce some interval valued double difference sequence spaces by means of Musielak-Orlicz function . We also determine some topological properties and inclusion relations between these double difference sequence spaces.

#### 1. Introduction

Interval arithmetic was first suggested by Dwyer [1] in 1951. Development of interval arithmetic as a formal system and evidence of its value as a computational device was provided by Moore [2] in 1959 and also by Moore and Yang [3] in 1962. Further works on interval numbers can be found in Dwyer [4] and Markov [5]. Furthermore, Moore and Yang [6] have developed applications of interval number sequences to differential equations. Chiao in [7] introduced sequences of interval numbers and defined usual convergence of sequences of interval number. Şengönül and Eryilmaz in [8] introduced and studied bounded and convergent sequence spaces of interval numbers and showed that these spaces are complete metric spaces. Recently, Esi in [9, 10] introduced and studied strongly almost -convergence and statistically almost -convergence of interval numbers and lacunary sequence spaces of interval numbers, respectively (also see [11–17]).

A set consisting of a closed interval of real numbers such that is called an interval number. A real interval can also be considered as a set. Thus we can investigate some properties of interval numbers, for instance, arithmetic properties or analysis properties. We denote the set of all real valued closed intervals by . Any elements of are called closed interval and denoted by . That is, . An interval number is a closed subset of real numbers [7]. Let and be first and last points of interval number, respectively. For , we have , . Consider , and if , then and if , then ,

In [2], Moore proved that the set of all interval numbers is a complete metric space defined by . In the special cases and , we obtain usual metric of . Let us define transformation by . Then is called sequence of interval numbers. The is called th term of sequence . We denote the set of all interval numbers with real terms as . The algebraic properties of can be found in [7]. Now we give the basic definitions used in this paper.

*Definition 1 (see [7]). *A sequence of interval numbers is said to be convergent to the interval number if for each there exists a positive integer such that for all and we denote it by . Thus, and .

*Definition 2. *A transformation from to is defined by . Then is called sequence of double interval numbers. Then is called term of sequence .

*Definition 3. *An interval valued double sequence is said to be convergent in Pringsheim’s sense or -convergent to an interval number , if, for every , there exists such that
where is the set of natural numbers, and we denote it also by . The interval number is called the Pringsheim limit of .

More exactly, we say that a double sequence converges to a finite interval number if tend to as both and tend to independently of one another. We denote by the set of all double convergent interval numbers of double interval numbers.

*Definition 4. *An interval valued double sequence is bounded if there exists a positive number such that for all . We will denote all bounded double interval number sequences by . It should be noted that, similar to the case of double sequences, is not the subset of .

*Definition 5. *Let denote a four-dimensional summability method that maps the complex double sequences into the double sequence where the th term to is as follows:
Such a transformation is said to be nonnegative if is nonnegative for all and .

The notion of difference sequence spaces was introduced by Kizmaz [18] who studied the difference sequence spaces , , and . The notion was further generalized by Et and Çolak [19] by introducing the spaces , , and . Let denote the set of all real and complex sequences and let be a nonnegative integer; then for , and , we have sequence spaces where and for all , which is equivalent to the following binomial representation: Taking , we get the spaces studied by Et and Çolak [19]. For more details about sequence spaces see [20–32] and references therein. Quite recently, Et et al. [33] defined and studied the concept of statistical convergence of order involving the notions of and ideal .

*Definition 6. *An Orlicz function is a 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 [34] 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 [34] 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 and is a right differentiable for , , , and is nondecreasing and as .

*Definition 7. *A sequence of Orlicz functions is said to be Musielak-Orlicz function (see [35, 36]). A sequence is defined by
and 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 -condition if there exist constants and a sequence (the positive cone of ) such that the inequality
holds for all and , whenever .

*Definition 8. *Let be a linear metric space. A function : is called paranorm, if (1) for all ;(2) for all ;(3) for all ;(4) is a sequence of scalars with as and is a sequence of vectors with as , then as .

A paranorm for which implies is called total paranorm and the pair is called a total paranormed space. It is well known that the metric of any linear metric space is given by some total paranorm.

Let be a Musielak-Orlicz function and let be a nonnegative four-dimensional bounded regular matrix (see [37, 38]). Let be a bounded double sequence of positive real numbers and be a double sequence of strictly positive real numbers. In the present paper we define the following new double sequence spaces for interval sequences:

*Remark 9. *Let us consider a few special cases of the above sequence spaces.(i)If for all , then we have
(ii)If = = , for all , then we have
(iii)If = = , for all , then we have
(iv)If , that is, the double Cesàro matrix, then the above classes of sequences reduce to the following sequence spaces:
(v)Let = = and for all . If, in addition, and , then the spaces , , and are reduced to , and which were introduced and studied by Esi and Hazarika [39].

The following inequality will be used throughout the paper. If , then for all and . Also for all .

The main purpose of this paper is to introduce interval valued double difference sequence spaces , , and and to study different properties of these spaces like linearity, paranorm, solidity, monotone, and so forth. Some inclusion relations between theses spaces are also established.

#### 2. Main Results

Theorem 10. *If for each and , then we have .*

*Proof. *Let . Then there exists such that
This implies that
for sufficiently large values of and . Since is nondecreasing, we get
Thus . This completes the proof.

Theorem 11. *Suppose that is a Musielak-Orlicz function, a bounded double sequence of positive real numbers, and a double sequence of strictly positive real numbers. Then the following hold.*(i)*If < < ≤ , then .*(ii)*If ≤ ≤ < , then .*

*Proof. *(i) Let . Since , we obtain the following:
and hence .

(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.

Theorem 12. *Let for all and be bounded. Then we have .*

*Proof. *Let . Then
Let and . Since , we have . Take .

Define
. It follows that . since , then
but
Therefore,
Hence . Thus, we get .

Theorem 13. *Let and be two Musielak-Orlicz functions,
*

*Proof. *Let ,. Then
Let . The result follows from the inequality
Thus, . Therefore, .

Theorem 14. *Let be a Musielak-Orlicz function and let be a nonnegative four-dimensional regular summability method. 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 15. *Let . Then for a Musielak-Orlicz function which satisfies the -condition, we have .*

*Proof. *Let ; that is,
Let and choose with such that for . Then
where
by using continuity of . For the second summation, we will make the following procedure. Thus we have
Since is nondecreasing and convex, so we have
Again, since satisfies the -condition, it follows that
Thus, it follows that
Taking the limit as and , it follows that .

Theorem 16. *Suppose that is a Musielak-Orlicz function, a bounded double sequence of positive real numbers, and a double sequence of strictly positive real numbers. If for all fixed , then
*

*Proof. *Let . Then there exists a positive number such that
Define . Since is nondecreasing and convex, for each , so by using (20), we have
Thus . This completes the proof of the theorem.

Theorem 17. *The double sequence space is solid.*

*Proof. *Suppose
Let be a double sequence of scalars such that for all . Then we get
This completes the proof.

Theorem 18. *The double sequence space is monotone.*

*Proof. *The proof is trivial so we omit it.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia.