The aim of this paper is to introduce some interval valued double difference sequence spaces by means of Musielak-Orlicz function M=(Mij). 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 λ-convergence and statistically almost λ-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 x such that a≤x≤b 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 R. Any elements of R are called closed interval and denoted by x-. That is, x-={x∈R:a≤x≤b}. An interval number x- is a closed subset of real numbers [7]. Let xl and xr be first and last points of x- interval number, respectively. For x-1,x-2∈R, we have x-1=x-2⇔x1l=x2l, x1r=x2r. Consider x-1+x-2={x∈R:x1l+x2l≤x≤x1r+x2r}, and if α≥0, then αx-={x∈R:αx1l≤x≤αx1r} and if α<0, then αx-={x∈R:αx1r≤x≤αx1l},
(1)x-1·x-2={x∈R:min{x1l·x2l,x1l·x2r,x1r·x2l,x1r·x2r}≤x≤min{x1l·x2l,x1l·x2r,x1r·x2l,x1r·x2r}}.
In [2], Moore proved that the set of all interval numbers R is a complete metric space defined by d(x-1,x-2)=max{|x1l-x2l|,|x1r·x2r|}. In the special cases x-1=[a,a] and x-2=[b,b], we obtain usual metric of R. Let us define transformation f:N→R by k→f(k)=x-,x-=(x-k). Then x-=(x-k) is called sequence of interval numbers. The x-k is called kth term of sequence x-=(x-k). We denote the set of all interval numbers with real terms as wi. The algebraic properties of wi can be found in [7]. Now we give the basic definitions used in this paper.
Definition 1 (see [7]).
A sequence x-=(x-k) of interval numbers is said to be convergent to the interval number x-0 if for each ϵ>0 there exists a positive integer k0 such that d(x-k,x-0)<ϵ for all k≥k0 and we denote it by limkx-k=x-0. Thus, limkx-k=x-0⇔limkxkl=x0l and limkxkr=x0r.
Definition 2.
A transformation f from N×N to R is defined by i,j→f(i,j)=x-,x-=(x-ij). Then x-=(x-ij) is called sequence of double interval numbers. Then x-ij is called ijth term of sequence x-=(x-ij).
Definition 3.
An interval valued double sequence x-=(x-ij) is said to be convergent in Pringsheim’s sense or P-convergent to an interval number x-0, if, for every ϵ>0, there exists N∈N such that
(2)d(x-ij,x-0)<ϵ∀i,j>N,
where N is the set of natural numbers, and we denote it also by P-limx-ij=x-0. The interval number x-0 is called the Pringsheim limit of x-=(x-ij).
More exactly, we say that a double sequence x-=(x-ij) converges to a finite interval number x-0 if x-ij tend to x-0 as both i and j tend to ∞ independently of one another. We denote by c-2 the set of all double convergent interval numbers of double interval numbers.
Definition 4.
An interval valued double sequence x-=(x-ij) is bounded if there exists a positive number M such that d(x-ij,x-0)≤M for all i,j∈N. We will denote all bounded double interval number sequences by l-∞2. It should be noted that, similar to the case of double sequences, c-2 is not the subset of l-∞2.
Definition 5.
Let A=(amnij) denote a four-dimensional summability method that maps the complex double sequences x into the double sequence Ax where the mnth term to Ax is as follows:
(3)(Ax)mn=∑i,j=1,1∞,∞amnijxij.
Such a transformation is said to be nonnegative if amnij is nonnegative for all m,n,i and j.
The notion of difference sequence spaces was introduced by Kizmaz [18] who studied the difference sequence spaces l∞(Δ), c(Δ), and c0(Δ). The notion was further generalized by Et and Çolak [19] by introducing the spaces l∞(Δn), c(Δn), and c0(Δn). Let w denote the set of all real and complex sequences and let n be a nonnegative integer; then for Z=c,c0, and l∞, we have sequence spaces
(4)Z(Δn)={x=(xk)∈w:(Δnxk)∈Z},
where Δnx=(Δnxk)=(Δn-1xk-Δn-1xk+1) and Δ0xk=xk for all k∈N, which is equivalent to the following binomial representation:
(5)Δnxk=∑v=0n(-1)v(nv)xk+v.
Taking n=1, 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 α involving the notions of Δ and ideal I.
Definition 6.
An Orlicz function M:[0,∞)→[0,∞) is a continuous, nondecreasing, and convex such that M(0)=0, M(x)>0 for x>0 and M(x)→∞ as x→∞. If convexity of Orlicz function is replaced by M(x+y)≤M(x)+M(y), then this function is called modulus function. Lindenstrauss and Tzafriri [34] used the idea of Orlicz function to define the following sequence space:
(6)lM={x=(xk)∈w:∑k=1∞M(|xk|ρ)<∞,forsomeρ>0}
which is known as an Orlicz sequence space. The space lM is a Banach space with the norm
(7)∥x∥=inf{ρ>0:∑k=1∞M(|xk|ρ)≤1}.
Also it was shown in [34] that every Orlicz sequence space lM contains a subspace isomorphic to lp(p≥1). An Orlicz function M can always be represented in the following integral form:
(8)M(x)=∫0xη(t)dt,
where η is known as the kernel of M and is a right differentiable for t≥0, η(0)=0, η(t)>0, and η is nondecreasing and η(t)→∞ as t→∞.
Definition 7.
A sequence M=(Mk) of Orlicz functions is said to be Musielak-Orlicz function (see [35, 36]). A sequence N=(Nk) is defined by
(9)Nk(v)=sup{|v|u-Mk(u):u≥0},k=1,2,…,
and is called the complementary function of a Musielak-Orlicz function M. For a given Musielak-Orlicz function M, the Musielak-Orlicz sequence space tM and its subspace hM are defined as follows:
(10)tM={x∈w:IM(cx)<∞forsomec>0},hM={x∈w:IM(cx)<∞∀c>0},
where IM is a convex modular defined by
(11)IM(x)=∑k=1∞Mk(xk),x=(xk)∈tM.
We consider tM equipped with the Luxemburg norm
(12)∥x∥=inf{k>0:IM(xk)≤1}
or equipped with the Orlicz norm
(13)∥x∥0=inf{1k(1+IM(kx)):k>0}.
A Musielak-Orlicz function M=(Mk) is said to satisfy Δ2-condition if there exist constants a,K>0 and a sequence c=(ck)k=1∞∈l+1 (the positive cone of l1) such that the inequality
(14)Mk(2u)≤KMk(u)+ck
holds for all k∈N and u∈R+, whenever Mk(u)≤a.
Definition 8.
Let X be a linear metric space. A function p: X→R is called paranorm, if
p(x)≥0 for all x∈X;
p(-x)=p(x) for all x∈X;
p(x+y)≤p(x)+p(y) for all x,y∈X;
(λn) is a sequence of scalars with λn→λ as n→∞ and (xn) is a sequence of vectors with p(xn-x)→0 as n→∞, then p(λnxn-λx)→0 as n→∞.
A paranorm p for which p(x)=0 implies x=0 is called total paranorm and the pair (X,p) 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 M=(Mij) be a Musielak-Orlicz function and let A=(amnij) be a nonnegative four-dimensional bounded regular matrix (see [37, 38]). Let p=(pij) be a bounded double sequence of positive real numbers and u=(uij) be a double sequence of strictly positive real numbers. In the present paper we define the following new double sequence spaces for interval sequences:
(15)w-2(M,p,u,Δr,A)={∑i,j=1,1m,nx-=(x-ij):P-limmn1mn×∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,x-0)ρ)]pij=0,forsomeρ>0∑i,j=1,1m,n},w-02(M,p,u,Δr,A)={∑i,j=1,1m,nix-=(x-ij):P-limmn1mn×∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,0-)ρ)]pij=0,forsomeρ>0∑i,j=1,1m,no},w-∞2(M,p,u,Δr,A)={∑i,j=1,1m,npx-=(x-ij):supmn1mn×∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,0-)ρ)]pij<∞,forsomeρ>0∑i,j=1,1m,no}.
Remark 9.
Let us consider a few special cases of the above sequence spaces.
If M=Mij(x)=x for all i,j∈N, then we have
(16)2w-(M,p,u,Δr,A)=w-2(p,u,Δr,A),2w-0(M,p,u,Δr,A)=w-20(p,u,Δr,A),w-∞2(M,p,u,Δr,A)=w-2∞(p,u,Δr,A).
If p = (pij) = 1, for all i,j, then we have
(17)w-2(M,p,u,Δr,A)=w-2(M,u,Δr,A),w-02(M,p,u,Δr,A)=w-02(M,u,Δr,A),w-∞2(M,p,u,Δr,A)=w-∞2(M,u,Δr,A).
If u = (uij) = 1, for all i,j, then we have
(18)w-2(M,p,u,Δr,A)=w-2(M,p,Δr,A),w-02(M,p,u,Δr,A)=w-20(M,p,Δr,A),w-2∞(M,p,u,Δr,A)=w-2∞(M,p,Δr,A).
If A=(C,1,1)=1, that is, the double Cesàro matrix, then the above classes of sequences reduce to the following sequence spaces:
(19)w-2(M,p,u,Δr,A)=w-2(M,p,u,Δr),w-20(M,p,u,Δr,A)=w-20(M,p,u,Δr),w-2∞(M,p,u,Δr,A)=w-2∞(M,p,u,Δr).
Let A = (C,1,1) = 1 and uij=1 for all i,j. If, in addition, M(x)=M(x) and r=0, then the spaces w-2(M,p,u,Δr,A), w-20(M,p,u,Δr,A), and w-2∞(M,p,u,Δr,A) are reduced to w-2(M,p),w-20(M,p), and w-2∞(M,p) which were introduced and studied by Esi and Hazarika [39].
The following inequality will be used throughout the paper. If 0≤pij≤suppij=H, K=max(1,2H-1) then
(20)|aij+bij|pij≤K(|aij|pij+|bij|pij)
for all i,j and aij,bij∈C. Also |a|pij≤max(1,|a|H) for all a∈C.
The main purpose of this paper is to introduce interval valued double difference sequence spaces w-2(M,p,u,Δr,A), w-02(M,p,u,Δr,A), and w-∞2(M,p,u,Δr,A) 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 ResultsTheorem 10.
If 0<pij<qij for each i and j, then we have w-2∞(M,p,u,Δr,A)⊂w-2∞(M,q,u,Δr,A).
Proof.
Let x¯=(x-ij)∈w-2∞(M,p,u,Δr,A). Then there exists ρ>0 such that
(21)supmn1mn∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,0-)ρ)]pij<∞.
This implies that
(22)amnij[Mij(uijd(Δrx-ij,0-)ρ)]pij<1,
for sufficiently large values of i and j. Since Mij is nondecreasing, we get
(23)supmn1mn∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,0-)ρ)]qij≤supmn1mn∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,0-)ρ)]pij<∞.
Thus x-=(x-ij)∈w-2∞(M,q,u,Δr,A). This completes the proof.
Theorem 11.
Suppose that M=(Mij) is a Musielak-Orlicz function, p=(pij) a bounded double sequence of positive real numbers, and u=(uij) a double sequence of strictly positive real numbers. Then the following hold.
If 0 < infpij < pij ≤ 1, then w-∞2(M,p,u,Δr,A)⊂w-2∞(M,u,Δr,A).
If 1 ≤ pij ≤ suppij < ∞, then w-∞2(M,u,Δr,A)⊂w-2∞(M,p,u,Δr,A).
Proof.
(i) Let x-=(x-ij)∈w-2∞(M,p,u,Δr,A). Since 0<infpij≤1, we obtain the following:
(24)supmn1mn∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,0-)ρ)]≤supmn1mn∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,0-)ρ)]pij<∞,
and hence x-=(x-ij)∈w-2∞(M,u,Δr,A).
(ii) Let pij≥1 for each i and j and suppij<∞. Let x-=(x-ij)∈w-2∞(M,u,Δr,A). Then for each 0<ϵ<1 there exists a positive integer N such that
(25)supmn1mn∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,0-)ρ)]≤ϵ<1∀n,m≥N.
This implies that
(26)supmn1mn∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,0-)ρ)]pij≤supmn1mn∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,0-)ρ)]<∞.
Therefore, x-=(x-ij)∈w-2∞(M,p,u,Δr,A). This completes the proof.
Theorem 12.
Let 0<pij≤qij for all i,j∈N and (qij/pij) be bounded. Then we have w-2∞(M,q,u,Δr,A)⊂w-2∞(M,p,u,Δr,A).
Proof.
Let x-=(x-ij)∈w-2∞(M,q,u,Δr,A). Then
(27)supmn1mn∑i,j=1,1m,namnij[Mij(uijd(Δrxij,0-)ρ)]qij<∞,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiforsomeρ>0.
Let sij=supmn(1/mn)∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,0-)/ρ)]qij and λij=pij/qij. Since pij≤qij, we have 0≤λij≤1. Take 0<λ<λij.
Define
(28)uij={sijifsij≥10ifsij<1,vij={0ifsij≥1sijifsij<1,sij=uij+vij,sijλij=uijλij+vijλij. It follows that uijλij≤uij≤sij,vijλij≤vijλ. since sijλij=uijλij+vijλij, then sijλij≤sij+vijλ(29)supmn1mn∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,0-)ρ)qij]λij≤supmn1mn∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,0-)ρ)]qij⟹supmn1mn∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,0-)ρ)qij]pij/qij≤supmn1mn∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,0-)ρ)]qij⟹supmn1mn∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,0-)ρ)]pij≤supmn1mn∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,0-)ρ)]qij,
but
(30)supmn1mn∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,0-)ρ)]qij<∞forsomeρ>0.
Therefore,
(31)supmn1mn∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,0-)ρ)]pij<∞forsomeρ>0.
Hence x-=(x-ij)∈W-2∞(M,p,u,Δr,A). Thus, we get W-2∞(M,q,u,Δr,A)⊂w-2∞(M,p,u,Δr,A).
Theorem 13.
Let M'=(Mij′) and M''=(Mij′′) be two Musielak-Orlicz functions,
(32)w-2∞(M′,p,u,Δr,A)∩w-2∞(M′′,p,u,Δr,A)⊂w-2∞(M′+M′′,p,u,Δr,A).
Proof.
Let x-=(x-ij)∈w-2∞(M',p,u,Δr,A)∩w-2∞(M'',p,u,Δr,A). Then
(33)supmn1mn∑i,j=1,1m,namnij[Mij′(uijd(Δrx-ij,0-)ρ1)]pij<∞,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiforsomeρ1>0,supmn1mn∑i,j=1,1m,namnij[Mij′′(uijd(Δrx-ij,0-)ρ2)]pij<∞,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiforsomeρ2>0.
Let ρ=max{ρ1,ρ2}. The result follows from the inequality
(34)supmn1mn∑i,j=1,1m,namnij[(Mij′+Mij′′)(uijd(Δrx-ij,0-)ρ)]pij=supmn1mn∑i,j=1,1m,namnij[Mij′(uijd(Δrx-ij,0-)ρ1)]pij+supmn1mn∑i,j=1,1m,namnij[Mij′′(uijd(Δrx-ij,0-)ρ2)]pij≤Ksupmn1mn∑i,j=1,1m,namnij[Mij′(uijd(Δrx-ij,0-)ρ1)]pij+Ksupmn1mn∑i,j=1,1m,namnij[Mij′′(uijd(Δrx-ij,0-)ρ2)]pij<∞.
Thus, supmn(1/mn)∑i,j=1,1m,namnij[(Mij′+Mij′′)(uijd(Δry-ij,0-)/ρ)]pij<∞. Therefore, x-=(x-ij)∈w-2∞(M'+M'',p,u,Δr,A).
Theorem 14.
Let M=(Mij) be a Musielak-Orlicz function and let A=(anmij) be a nonnegative four-dimensional regular summability method. Suppose that β=limt→∞(Mij(t)/t)<∞. Then w-2(p,u,Δr,A)=w-2(M,p,u,Δr,A).
Proof.
In order to prove that W-2(p,u,Δr,A)=w-2(M,p,u,Δr,A), it is sufficient to show that W-2(M,p,u,Δr,A)⊂w-2(p,u,Δr,A). Now, let β>0. By definition of β, we have Mij(t)≥βt for all t≥0. Since β>0, we have t≤(1/β)Mij(t) for all t≥0. Let x-=(x-ij)∈w-2(M,p,u,Δr,A). Thus, we have
(35)supmn1mn∑i,j=1,1m,namnij[(uijd(Δrx-ij,x-0)ρ)]pij≤1βsupmn1mn∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,x-0)ρ)]pij<∞
which implies that x-=(x-ij)∈w-2(p,u,Δr,A). This completes the proof.
Theorem 15.
Let 0<h=
inf
pij≤pij≤
sup
pij=H<∞. Then for a Musielak-Orlicz function M=(Mij) which satisfies the Δ2-condition, we have w-2(p,u,Δr,A)=w-2(M,p,u,Δr,A).
Proof.
Let x-=(x-ij)∈w-2(p,u,Δr,A); that is,
(36)1mn∑i,j=1,1m,namnij[(uijd(Δrx-ij,x-0)ρ)]pij=0,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiforsomeρ>0.
Let ϵ>0 and choose δ with 0<δ<1 such that Mij(t)<ϵ for 0≤t≤δ. Then
(37)1mn∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,x0¯)ρ)]pij=1mn∑i,j=1,1d(Δrx-ij,x-0)≤δm,namnij[Mij(uijd(Δrx-ij,x-0)ρ)]pij+1mn∑i,j=1,1d(Δrx-ij,x-0)>δm,namnij[Mij(uijd(Δrx-ij,x-0)ρ)]pij=∑1+∑2,
where
(38)∑1=1mn∑i,j=1,1d(Δrx-ij,x-0)≤δm,namnij[Mij(uijd(Δrx-ij,x-0)ρ)]pij<max(ϵ,ϵH)
by using continuity of (Mij). For the second summation, we will make the following procedure. Thus we have
(39)d(Δrxij,x-0)ρ<1+d(Δrx-ij,x-0)/ρδ.
Since M=(Mij) is nondecreasing and convex, so we have
(40)amnij[Mij(uijd(Δrx-ij,x-0)ρ)]<amnij[Mij{1+uijd(Δrx-ij,x-0)/ρδ}]≤12amnij[(uij)Mij(2)]+12amnij[Mij{2uijd(Δrx-ij,x-0)/ρδ}].
Again, since M=(Mij) satisfies the Δ2-condition, it follows that
(41)amnij[Mij(uijd(Δrx-ij,x-0)ρ)]≤12K{d(Δrx-ij,x-0)/ρδ}amnij[(uij)Mij(2)]+12K{d(Δrx-ij,x-0)/ρδ}amnij[(uij)Mij(2)]=K{d(Δrx-ij,x-0)/ρδ}amnij[(uij)Mij(2)].
Thus, it follows that
(42)∑2=max{1,[Kamnij[(uij)Mij(2)]δ]H}×1mn∑i,j=1,1m,n[d(Δrx-ij,x-0)ρ]pij.
Taking the limit as ϵ→0 and m,n→∞, it follows that x-=(x-ij)∈w-2(M,p,u,Δr,A).
Theorem 16.
Suppose that M=(Mij) is a Musielak-Orlicz function, p=(pij) a bounded double sequence of positive real numbers, and u=(uij) a double sequence of strictly positive real numbers. If
sup
i,j(Mij(x))pij<∞ for all fixed x>0, then
(43)w-2(M,p,u,Δr,A)⊂w-2∞(M,p,u,Δr,A).
Proof.
Let x-=(x-ij)∈w-2(M,p,u,Δr,A). Then there exists a positive number ρ1>0 such that
(44)1mn∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,x-0)ρ1)]pij=0,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiforsomeρ1>0.
Define ρ=2ρ1. Since M=(Mij) is nondecreasing and convex, for each i,j, so by using (20), we have
(45)supmn1mn∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,0-)ρ)]pij≤supmn1mn∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,x-0)+d(x-0,0-)ρ)]pij≤K{supmn1mn∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,x-0)ρ1)]pij+supmn1mn∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,0-)ρ1)]pij}<∞.
Thus x-=(x-ij)∈w-2∞(M,p,u,Δr,A). This completes the proof of the theorem.
Theorem 17.
The double sequence space w-∞2(M,p,u,Δr,A) is solid.
Proof.
Suppose x-=(x-ij)∈w-2∞(M,p,u,Δr,A)(46)supmn1mn∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,0-)ρ)]pij<∞,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiforsomeρ>0.
Let (αij) be a double sequence of scalars such that |αij|≤1 for all i,j∈N. Then we get
(47)supmn1mn∑i,j=1,1m,namnij[Mij(uijd(Δrαijx-ij,0-)ρ)]pij≤supmn1mn∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,0-)ρ)]pij<∞.
This completes the proof.
Theorem 18.
The double sequence space w-∞2(M,p,u,Δr,A) 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.
DwyerP. S.1951New York, NY, USAWileyMooreR. E.1959Lockheed Missiles and Space CompanyMooreR. E.YangC. T.Interval analysis I1962LMSD-285875Lockheed Missiles and Space Company285875DwyerP. S.Errors of matrix computation, simultaneous equations and eigenvalues, National Bureu of Standarts1953294958MarkovS.Quasilinear spaces and their relation to vector spaces200521121MooreR. E.YangC. T.1958Tokyo, JapanGaukutsu Bunken Fukeyu-kaiRAAG Memories IIChiaoK. P.Fundamental properties of interval vector max-norm2002182219233MR1955480ŞengönülM.EryilmazA.On the sequence spaces of interval numbers201083503510MR2763672EsiA.Strongly almost λ-convergence and statistically almost λ-convergence of interval numbers201172117122EsiA.Lacunary sequence spaces of interval numbers2012102445451MR3001866EsiA.HazarikaB.On interval valued generalized difference classes defined by Orlicz function2013114853EsiA.BrahaN.On asymptotically λ-statistical equivalent sequences of interval numbers2013353515520EsiA.EsiA.Asymptotically lacunary statistically equivalent sequences of interval numbers2013114348EsiA.λ-sequence spaces of interval numbers2014831099110210.12785/amis/080320MR3157606EsiA.Double sequences of interval numbers defined by Orlicz functions2013171576410.12697/ACUTM.2013.17.04MR3062982ZBL06225564EsiA.YaseminS.Some spaces of sequences of interval numbers defined by a modulus function2014211116EsiA.Double lacunary sequence spaces of double sequence of interval numbers201231329930810.4067/S0716-09172012000300008MR2995556ZBL1263.46005KizmazH.On certain sequence spaces198124216917610.4153/CMB-1981-027-5MR619442EtM.ÇolakR.On some generalized difference sequence spaces1995214377386MR1362304EtM.Generalized Cesàro difference sequence spaces of non-absolute type involving lacunary sequences2013219179372937610.1016/j.amc.2013.03.039MR3047834EtM.Spaces of Cesàro difference sequences of order r defined by a modulus function in a locally convex space2006104865879MR2229627EtM.AltinY.ChoudharyB.TripathyB. C.On some classes of sequences defined by sequences of Orlicz functions20069233534210.7153/mia-09-33MR2225019KarakayaV.SavasE.PolatH.Some paranormed Euler sequence spaces of difference sequences of order m201363484986210.2478/s12175-013-0139-9MR3092811MohiuddineS. A.AlotaibiA.Some spaces of double sequences obtained through invariant mean and related concepts2013201311507950MR304500210.1155/2013/507950MohiuddineS. A.RajK.AlotaibiA.Some paranormed double difference sequence spaces for Orlicz functions and bounded-regular matrices201420141041906410.1155/2014/419064MR3182279MursaleenM.MohiuddineS. A.Some new double sequence spaces of invariant means201045113915310.3336/gm.45.1.11MR2646443MursaleenM.SharmaS. K.KiliçmanA.Sequence spaces defined by Musielak-Orlicz function over n-normed spaces2013201310364743MR312408610.1155/2013/364743TripathyB. C.Generalized difference paranormed statistically convergent sequences defined by Orlicz function in a locally convex space2004304431446MR2106062ZBL1066.40005RajK.SharmaS. K.Some difference sequence spaces defined by Musielak-Orlicz functions2013243343RajK.SharmaS. K.Some multiplier double sequence spaces2012373391405MR3027229ParasharS. D.ChoudharyB.Sequence spaces defined by Orlicz functions1994254419428MR1272814ZBL0802.46020SimonsS.The sequence spaces l(pv) and m(pv)1965153422436MR0176325EtM.AlotaibiA.MohiuddineS. A.On (∆m,I)-statistical convergence of order α20142014553541910.1155/2014/535419LindenstraussJ.TzafririL.On Orlicz sequence spaces19711037939010.1007/BF02771656MR0313780ZBL0227.46042MaligrandaL.19895Warsaw, PolandPolish Academy of ScienceMR2264389MusielakJ.19831034Berlin, GermanySpringerLecture Notes in MathematicsMR724434HamiltonH. J.Transformations of multiple sequences193621296010.1215/S0012-7094-36-00204-1MR1545904RobisonG. M.Divergent double sequences and series1926281507310.2307/1989172MR1501332EsiA.HazarikaB.Some double sequence spaces of interval numbers defined by Orlicz functions201410.1016/j.joems.2013.12.004