On Some Classes of Double Difference Sequences of Interval Numbers
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.
Interval arithmetic was first suggested by Dwyer  in 1951. Development of interval arithmetic as a formal system and evidence of its value as a computational device was provided by Moore  in 1959 and also by Moore and Yang  in 1962. Further works on interval numbers can be found in Dwyer  and Markov . Furthermore, Moore and Yang  have developed applications of interval number sequences to differential equations. Chiao in  introduced sequences of interval numbers and defined usual convergence of sequences of interval number. Şengönül and Eryilmaz in  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 . Let and be first and last points of interval number, respectively. For , we have , . Consider , and if , then and if , then ,
In , 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 . Now we give the basic definitions used in this paper.
Definition 1 (see ). 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  who studied the difference sequence spaces , , and . The notion was further generalized by Et and Çolak  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 . For more details about sequence spaces see [20–32] and references therein. Quite recently, Et et al.  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  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  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 .
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.
The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia.
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