# On Some Classes of Double Difference Sequences of Interval Numbers

Authors: S. A. Mohiuddine; Kuldip Raj; Abdullah Alotaibi
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

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


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## Body

## 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 numbersx 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 sequencex-=(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 lim⁡kx-k=x-0. Thus, lim⁡kx-k=x-0⇔lim⁡kxkl=x0l and lim⁡kxkr=x0r.Definition 2.
A transformationf 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 sequencex-=(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-lim⁡x-ij=x-0. The interval number x-0 is called the Pringsheim limit of x-=(x-ij).
More exactly, we say that a double sequencex-=(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 sequencex-=(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.
LetA=(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 functionM:[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 sequenceM=(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.
LetX be a linear metric space. A function p: X→R is called paranorm, if (1)
p(x)≥0 for all x∈X;(2)
p(-x)=p(x) for all x∈X;(3)
p(x+y)≤p(x)+p(y) for all x,y∈X;(4)
(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 paranormp 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.LetM=(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-lim⁡mn1mn×∑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-lim⁡mn1mn×∑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):sup⁡mn1mn×∑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.(i)
IfM=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).(ii)
Ifp = (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).(iii)
Ifu = (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).(iv)
IfA=(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).(v)
LetA = (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. If0≤pij≤sup⁡pij=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 spacesw-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 Results

Theorem 10.
If0<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.
Letx¯=(x-ij)∈w-2∞(M,p,u,Δr,A). Then there exists ρ>0 such that
(21)sup⁡mn1mn∑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)sup⁡mn⁡1mn∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,0-)ρ)]qij≤sup⁡mn1mn∑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 thatM=(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.(i)
If0 < inf⁡pij < pij ≤ 1, then w-∞2(M,p,u,Δr,A)⊂w-2∞(M,u,Δr,A).(ii)
If1 ≤ pij ≤ sup⁡pij < ∞, then w-∞2(M,u,Δr,A)⊂w-2∞(M,p,u,Δr,A).Proof.
(i) Letx-=(x-ij)∈w-2∞(M,p,u,Δr,A). Since 0<inf⁡pij≤1, we obtain the following:
(24)sup⁡mn⁡1mn∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,0-)ρ)]≤sup⁡mn1mn∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,0-)ρ)]pij<∞,
and hence x-=(x-ij)∈w-2∞(M,u,Δr,A).
(ii) Letpij≥1 for each i and j and sup⁡pij<∞. 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)sup⁡mn1mn∑i,j=1,1m,n‍amnij[Mij(uijd(Δrx-ij,0-)ρ)]≤ϵ<1∀n,m≥N.
This implies that
(26)sup⁡mn⁡1mn∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,0-)ρ)]pij≤sup⁡mn1mn∑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.
Let0<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.
Letx-=(x-ij)∈w-2∞(M,q,u,Δr,A). Then
(27)sup⁡mn1mn∑i,j=1,1m,namnij[Mij(uijd(Δrxij,0-)ρ)]qij<∞,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiforsomeρ>0.
Let sij=sup⁡mn(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)sup⁡mn1mn∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,0-)ρ)qij]λij≤sup⁡mn1mn∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,0-)ρ)]qij⟹sup⁡mn1mn∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,0-)ρ)qij]pij/qij≤sup⁡mn1mn∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,0-)ρ)]qij⟹sup⁡mn1mn∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,0-)ρ)]pij≤sup⁡mn1mn∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,0-)ρ)]qij,
but
(30)sup⁡mn1mn∑i,j=1,1m,n‍amnij[Mij(uijd(Δrx-ij,0-)ρ)]qij<∞forsomeρ>0.
Therefore,
(31)sup⁡mn1mn∑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.
LetM'=(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.
Letx-=(x-ij)∈w-2∞(M',p,u,Δr,A)∩w-2∞(M'',p,u,Δr,A). Then
(33)sup⁡mn1mn∑i,j=1,1m,namnij[Mij′(uijd(Δrx-ij,0-)ρ1)]pij<∞,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiforsomeρ1>0,sup⁡mn1mn∑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)sup⁡mn⁡1mn∑i,j=1,1m,namnij[(Mij′+Mij′′)(uijd(Δrx-ij,0-)ρ)]pij=sup⁡mn1mn∑i,j=1,1m,namnij[Mij′(uijd(Δrx-ij,0-)ρ1)]pij+sup⁡mn1mn∑i,j=1,1m,namnij[Mij′′(uijd(Δrx-ij,0-)ρ2)]pij≤Ksup⁡mn1mn∑i,j=1,1m,namnij[Mij′(uijd(Δrx-ij,0-)ρ1)]pij+Ksup⁡mn1mn∑i,j=1,1m,namnij[Mij′′(uijd(Δrx-ij,0-)ρ2)]pij<∞.
Thus, sup⁡mn(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.
LetM=(Mij) be a Musielak-Orlicz function and let A=(anmij) be a nonnegative four-dimensional regular summability method. Suppose that β=lim⁡t→∞(Mij(t)/t)<∞. Then w-2(p,u,Δr,A)=w-2(M,p,u,Δr,A).Proof.
In order to prove thatW-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)sup⁡mn1mn∑i,j=1,1m,namnij[(uijd(Δrx-ij,x-0)ρ)]pij≤1βsup⁡mn1mn∑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.
Let0<h=infpij≤pij≤suppij=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.
Letx-=(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 thatM=(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 supi,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.
Letx-=(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)sup⁡mn⁡1mn∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,0-)ρ)]pij≤sup⁡mn1mn∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,x-0)+d(x-0,0-)ρ)]pij≤K{sup⁡mn1mn∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,x-0)ρ1)]pij+sup⁡mn1mn∑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 spacew-∞2(M,p,u,Δr,A) is solid.Proof.
Supposex-=(x-ij)∈w-2∞(M,p,u,Δr,A)(46)sup⁡mn1mn∑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)sup⁡mn⁡1mn∑i,j=1,1m,namnij[Mij(uijd(Δrαijx-ij,0-)ρ)]pij≤sup⁡mn1mn∑i,j=1,1m,namnij[Mij(uijd(Δrx-ij,0-)ρ)]pij<∞.
This completes the proof.Theorem 18.
The double sequence spacew-∞2(M,p,u,Δr,A) is monotone.Proof.
The proof is trivial so we omit it.



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*Source: 516956-2014-07-24.xml*