The aim of this paper is to introduce some new double difference sequence spaces with the help of the Musielak-Orlicz function ℱ=(Fjk) and four-dimensional bounded-regular (shortly, RH-regular) matrices A=(anmjk). 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 ℬ𝒮, ℬ𝒮(t), 𝒞𝒮p, 𝒞𝒮bp, 𝒞𝒮r, and ℬ𝒱 consisting of all double series whose sequence of partial sums are in the spaces ℳu, ℳu(t), 𝒞p, 𝒞bp, 𝒞r, and ℒu, respectively. Başar and Sever [13] extended the well-known space ℓq from single sequence to double sequences, denoted by ℒq, 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 σ-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 l∞(Δ), c(Δ), and c0(Δ). The notion was further generalized by Et and Çolak [20] by introducing the spaces l∞(Δr), c(Δr), and c0(Δr).
Let w be the space of all complex or real sequences x=(xk) and let r and s be two nonnegative integers. Then for Z=l∞,c,c0, we have the following sequence spaces:
(1)Z(Δsr)={x=(xk)∈w:(Δsrxk)∈Z},
where Δsrx=(Δsrxk)=(Δsr-1xk-Δsr-1xk+1) and Δ0xk=xk for all k∈ℕ, which is equivalent to the following binomial representation:
(2)Δsrxk=∑v=0r(-1)v(rv)xk+sv.
We remark that for s=1 and r=s=1, 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 F:[0,∞)→[0,∞) is continuous, nondecreasing, and convex such that F(0)=0, F(x)>0 for x>0 and F(x)→∞ as x→∞. If convexity of Orlicz function is replaced by F(x+y)≤F(x)+F(y), then this function is called modulus function. Lindenstrauss and Tzafriri [28] used the idea of Orlicz function to define the following sequence space:
(3)ℓF={x=(xk)∈w:∑k=1∞F(|xk|ρ)<∞,ρ>0},
which is known as an Orlicz sequence space. The space ℓF is a Banach space with the norm
(4)∥x∥=inf{ρ>0:∑k=1∞F(|xk|ρ)≤1}.
Also it was shown in [28] that every Orlicz sequence space ℓF contains a subspace isomorphic to ℓp(p≥1). An Orlicz function F can always be represented in the following integral form:
(5)F(x)=∫0xη(t)dt,
where η is known as the kernel of F, is a right differentiable for t≥0,η(0)=0,η(t)>0,η is nondecreasing, and η(t)→∞ as t→∞.
A sequence ℱ=(Fk) of Orlicz functions is said to be a Musielak-Orlicz function (see [29, 30]). A sequence 𝒩=(Nk) is defined by
(6)Nk(v)=sup{|v|u-Fk(u):u≥0},k=1,2,…,
which is called the complementary function of a Musielak-Orlicz function ℱ. For a given Musielak-Orlicz function ℱ, the Musielak-Orlicz sequence space tℱ and its subspace hℱ are defined as follows:
(7)tℱ={x∈w:Iℱ(cx)<∞forsomec>0},hℱ={x∈w:Iℱ(cx)<∞∀c>0},
where Iℱ is a convex modular defined by
(8)Iℱ(x)=∑k=1∞Fk(xk),x=(xk)∈tℱ.
We consider tℱ equipped with the Luxemburg norm
(9)∥x∥=inf{k>0:Iℱ(xk)≤1}
or equipped with the Orlicz norm
(10)∥x∥0=inf{1k(1+Iℱ(kx)):k>0}.
A Musielak-Orlicz function ℱ=(Fk) 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
(11)Fk(2u)≤KFk(u)+ck
holds for all k∈ℕ and u∈ℝ+, whenever Fk(u)≤a.
A double sequence x=(xjk) is said to be bounded if ∥x∥(∞,2)=supj,k|xjk|<∞. We denote by l∞2 the space of all bounded double sequences.
By the convergence of double sequence x=(xjk) we mean the convergence in the Pringsheim sense; that is, a double sequence x=(xjk) is said to converge to the limit L in Pringsheim sense (denoted by, P-limx=L) provided that given ϵ>0 there exists n∈ℕ such that |xjk-L|<ϵ whenever j,k>n (see [31]). We will write more briefly as P-convergent. If, in addition, x∈l∞2, then x is said to be boundedly P-convergent to L. We will denote the space of all bounded convergent double sequences (or boundedly P-convergent) by c∞2.
Let S⊆ℕ×ℕ and let ϵ>0 be given. By χS(x;ϵ), we denote the characteristic function of the set S(x;ϵ)={(j,k)∈ℕ×ℕ:|xjk|≥ϵ}.
Let A=(anmjk) be a four-dimensional infinite matrix of scalers. For all m,n∈ℕ0, where ℕ0:=ℕ∪{0}, the sum
(12)ynm=∑j,k=0,0∞,∞anmjkxjk
is called the A-means of the double sequence (xjk). A double sequence (xjk) is said to be A-summable to the limit L if the A-means exist for all m,n in the sense of Pringsheim’s convergence:
(13)P-limp,q→∞∑j,k=0,0p,qanmjkxjk=ynm,P-limn,m→∞ynm=L.
A four-dimensional matrix A is said to be bounded-regular (or RH-regular) if every bounded P-convergent sequence is A-summable to the same limit and the A-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 A is RH-regular if and only if
P-limn,manmjk=0 for each j and k,
P-limn,m∑j,k=0,0∞,∞|anmjk|=1,
P-limn,m∑j=0∞|anmjk|=0 for each k,
P-limn,m∑k=0∞|anmjk|=0 for each j,
∑j,k=0,0∞,∞|anmjk|<∞ for all n,m∈ℕ0.
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 X over the real field ℝ (the set of real numbers) is said to be a paranormed space if there is a subadditive function g:X→ℝ such that g(θ)=0, g(x)=g(-x), and scalar multiplication is continuous; that is, |αn-α|→0 and g(xn-x)→0 imply g(αnxn-αx|→0 for all α’s in ℝ and all x’s in X, where θ is the zero vector in the linear space X.
The linear spaces l∞(p), c(p), and c0(p) were defined by Maddox [35] (also, see Simons [36]).
Let ℱ=(Fjk) be a Musielak-Orlicz function; that is, ℱ is a sequence of Orlicz functions and let A=(anmjk) be a nonnegative four-dimensional bounded-regular matrix. Then, we define the following:
(14)W02(A,ℱ,u,Δsr,p)={∑j,k=0,0∞,∞x=(xjk):1111111P-limn,m∑j,k=0,0∞,∞anmjk[Fjk(ujk|Δsrxjk|)pjk]=0},W2(A,ℱ,u,Δsr,p)={∑j,k=0,0∞,∞x=(xjk):1111111P-limn,m∑j,k=0,0∞,∞anmjk[Fjk(ujk|Δsrxjk-L|)pjk]1111111=0forsomeL∈ℂ∑j,k=0,0∞,∞},
where p=(pjk) is a double sequence of real numbers such that pjk>0 for j,k, supj,kpjk=H<∞, and u=(ujk) is a double sequence of strictly positive real numbers.
Remark 2.
If we take ℱ(x)=x in W02(A,ℱ,u,Δsr,p) and W2(A,ℱ,u,Δsr,p), then we have the following spaces:
(15)W02(A,u,Δsr,p)={∑j,k=0,0∞,∞x=(xjk):1111111P-limn,m∑j,k=0,0∞,∞anmjk[(ujk|Δsrxjk|)pjk]=0},W2(A,u,Δsr,p)={∑j,k=0,0∞,∞x=(xjk):1111111P-limn,m∑j,k=0,0∞,∞anmjk[(ujk|Δsrxjk-L|)pjk]1111111=0forsomeL∈ℂ∑j,k=0,0∞,∞}.
Remark 3.
Let p=(pjk)=1 for all j,k. Then W02(A,ℱ,u,Δsr,p) and W2(A,ℱ,u,Δsr,p) are reduced to
(16)W02(A,ℱ,u,Δsr)={∑j,k=0,0∞,∞x=(xjk):1111111P-limn,m∑j,k=0,0∞,∞anmjk[Fjk(ujk|Δsrxjk|)]=0},W2(A,ℱ,u,Δsr)={∑j,k=0,0∞,∞x=(xjk):1111111P-limn,m∑j,k=0,0∞,∞anmjk[Fjk(ujk|Δsrxjk-L|)]1111111=0forsomeL∈ℂ∑j,k=0,0∞,∞},
respectively.
Remark 4.
Let u=(ujk)=1 for all j,k. Then, the spaces W02(A,ℱ,u,Δsr,p) and W2(A,ℱ,u,Δsr,p) are reduced to
(17)W02(A,ℱ,Δsr,p)={∑j,k=0,0∞,∞x=(xjk):1111111P-limn,m∑j,k=0,0∞,∞anmjk[Fjk(|Δsrxjk|)pjk]=0},W2(A,ℱ,Δsr,p)={∑j,k=0,0∞,∞x=(xjk):1111111P-limn,m∑j,k=0,0∞,∞anmjk[Fjk(|Δsrxjk-L|)pjk]1111111=0forsomeL∈ℂ∑j,k=0,0∞,∞},
respectively.
Remark 5.
If we take A=(C,1,1) in W02(A,ℱ,u,Δsr,p) and W2(A,ℱ,u,Δsr,p), then we have the following spaces:
(18)W02(ℱ,u,Δsr,p)={∑j,k=0,0∞,∞x=(xjk):1111111P-limn,m∑j,k=0,0m-1,n-1[Fjk(ujk|Δsrxjk|)pjk]=0},W2(ℱ,u,Δsr,p)={∑j,k=0,0m-1,n-1x=(xjk):1111111P-limn,m∑j,k=0,0m-1,n-1[Fjk(ujk|Δsrxjk-L|)pjk]1111111=0forsomeL∈ℂ∑j,k=0,0m-1,n-1}.
Remark 6.
If we take A=(C,1,1) and ℱ(x)=x in W02(A,ℱ,u,Δsr,p) and W2(A,ℱ,u,Δsr,p), then we have the following spaces:
(19)W02(u,Δsr,p)={∑j,k=0,0m-1,n-1x=(xjk):1111111P-limn,m∑j,k=0,0m-1,n-1[(ujk|Δsrxjk|)pjk]=0},W2(u,Δsr,p)={∑j,k=0,0m-1,n-1x=(xjk):1111111P-limn,m∑j,k=0,0m-1,n-1[(ujk|Δsrxjk-L|)pjk]1111111=0forsomeL∈ℂ∑j,k=0,0m-1,n-1}.
Remark 7.
Let pjk=ujk=1 for all j,k. If, in addition, ℱ(x)=F(x) and r=0, then the spaces W02(A,ℱ,u,Δsr,p) and W2(A,ℱ,u,Δsr,p) are reduced to W02(A,F) and W2(A,F) which were introduced and studied by Yurdakadim and Tas [37] as below:
(20)W02(A,F)={x=(xjk):P-limn,m∑j,kanmjkF(|xjk|)=0},W2(A,F)={x=(xjk):P-limn,m∑j,kanmjkF(|xjk-L|)1111111111111,=0forsomeL∈ℂ∑j,k}.
Throughout the paper, we will use the following inequality: let (ajk) and (bjk) be two double sequences. Then
(21)|ajk+bjk|pjk≤K(|ajk|pjk+|bjk|pjk),
where K=max(1,2H-1) and supj,kpjk=H (see [15]). We will also assume throughout this paper that the symbol ℱ will denote the sublinear Musielak-Orlicz function.
3. Main ResultsTheorem 8.
Let ℱ=(Fjk) be a sublinear Musielak-Orlicz function, A=(anmjk) a nonnegative four-dimensional RH-regular matrix, p=(pjk) a bounded sequence of positive real numbers, and u=(ujk) a sequence of strictly positive real numbers. Then W02(A,ℱ,u,Δsr,p) and W2(A,ℱ,u,Δsr,p) are linear spaces over the complex field ℂ.
Proof.
Let x=(xjk),y=(yjk)∈W02(A,ℱ,u,Δsr,p) and α,β∈ℂ. Then there exist integers Mα and Nβ such that |α|≤Mα and |β|≤Nβ.
Since ℱ=(Fjk) is a nondecreasing function, so by inequality (21), we have
(22)∑j,k=0,0∞,∞anmjk[Fjk(ujk|Δsr(αxjk+βyjk)|)pjk]≤∑j,k=0,0∞,∞anmjk[Fjk(ujk|αΔsrxjk+βΔsryjk|)pjk]≤K∑j,k=0,0∞,∞anmjk[FjkMα(ujk|Δsrxjk|)pjk]+K∑j,k=0,0∞,∞anmjk[FjkNβ(ujk|Δsryjk|)pjk]≤KMαH∑j,k=0,0∞,∞anmjk[Fjk(ujk|Δsrxjk|)pjk]+KNβH∑j,k=0,0∞,∞anmjk[Fjk(ujk|Δsryjk|)pjk]⟶0.
Thus αx+βy∈W02(A,ℱ,u,Δsr,p). This proves that W02(A,ℱ,u,Δsr,p) is a linear space. Similarly we can prove that W2(A,ℱ,u,Δsr,p) is also a linear space.
Theorem 9.
Let ℱ=(Fjk) be a sublinear Musielak-Orlicz function, A=(anmjk) a nonnegative four-dimensional RH-regular matrix, p=(pjk) a bounded sequence of positive real numbers, and u=(ujk) a sequence of strictly positive real numbers. Then W02(A,ℱ,u,Δsr,p) and W2(A,ℱ,u,Δsr,p) are paranormed spaces with the paranorm
(23)g(x)=supn,m∑j,k=0,0∞,∞{anmjk[Fjk(ujk|Δsrxjk|)pjk]}1/M,
where 0<pjk≤suppjk=H<∞ and M=max(1,H).
Proof.
We will prove the result for W02(A,ℱ,u,Δsr,p). Let x=(xjk)∈W02(A,ℱ,u,Δsr,p). Then for each x=(xjk)∈W02(A,ℱ,u,Δsr,p), g(x) exists. Also it is clear that g(0)=0,g(-x)=g(x), and g(x+y)≤g(x)+g(y).
We now show that the scalar multiplication is continuous. First observe the following:
(24)g(λx)=supnm∑j,k=0,0∞,∞anmjk[Fjk(ujk|λΔsrxjk|)pjk]g(λx)≤(1+[|λ|])g(x),
where [|λ|] denotes the integer part of |λ|. It is also clear that if x→0 and λ→0 implies g(λx)→0. For fixed λ, if x→0, then g(λx)→0. We need to show that for fixed x,λ→0 implies g(λx)→0. Let x∈W2(A,ℱ,u,Δsr,p). Thus
(25)P-limn,m∑j,k=0,0∞,∞anmjk[Fjk(ujk|Δsrxjk-L|)pjk]=0.
Then, for ϵ>0 there exists N∈ℕ such that
(26)∑j,k=0,0∞,∞anmjk[Fjk(ujk|Δsrxjk-L|)pjk]<ϵ4
for m,n>N. Also, for each m,n with 1≤m,n≤N, since
(27)∑j,k=0,0∞,∞anmjk[Fjk(ujk|Δsrxjk-L|)pjk]<∞,
there exists an integer Mm,n such that
(28)∑j,k>Mm,nanmjk[Fjk(ujk|Δsrxjk-L|)pjk]<ϵ4.
Let M=max1≤(m,n)≤N{Mm,n}. We have for each m,n with 1≤m,n≤N(29)∑j,k>Manmjk[Fjk(ujk|Δsrxjk-L|)pjk]<ϵ4.
Also from (26), for m,n>N, we have
(30)∑j,k>Manmjk[Fjk(ujk|Δsrxjk-L|)pjk]<ϵ4.
Thus M is an integer independent of m,n such that
(31)∑j,k>Manmjk[Fjk(ujk|Δsrxjk-L|)pjk]<ϵ4.
Since |λ|pjk≤max(1,|λ|H), therefore
(32)∑j,k=0,0∞,∞anmjk[Fjk(ujk|λΔsrxjk|)pjk]=∑j,k=0,0∞,∞anmjk[Fjk(ujk|λΔsrxjk-λL+λL|)pjk]≤∑j,k=0,0∞,∞anmjk[Fjk(ujk|λΔsrxjk-λL|)pjk]+∑j,k=0,0∞,∞anmjk[Fjk(ujk|λL|)pjk]≤∑j,k>Manmjk[Fjk(ujk|λΔsrxjk-λL|)pjk]+∑j,k≤Manmjk[Fjk(ujk|λΔsrxjk-λL|)pjk]+∑j≥M,k<Manmjk[Fjk(ujk|λΔsrxjk-λL|)pjk]+∑j<M,k≥Manmjk[Fjk(ujk|λΔsrxjk-λL|)pjk]+∑j,k=0,0∞,∞anmjk[Fjk(ujk|λL|)pjk].
For each m,n and by the continuity of F as λ→0, we have the following:
(33)∑j,k≤Manmjk[Fjk(ujk|λΔsrxjk-λL|)pjk]+∑j,k=0,0∞,∞anmjk[Fjk(ujk|λL|)pjk]⟶0
in Pringsheim’s sense. Now choose δ<1 such that |λ|<δ implies
(34)∑j,k≤Manmjk[Fjk(ujk|λΔsrxjk-λL|)pjk]+∑j,k=0,0∞,∞anmjk[Fjk(ujk|λL|)pjk]<ϵ4.
In the same manner, we have
(35)∑j≥M,k<Manmjk[Fjk(ujk|λΔsrxjk-λL|)pjk]<ϵ4,(36)∑j<M,k≥Manmjk[Fjk(ujk|λΔsrxjk-λL|)pjk]<ϵ4.
It follows from (31), (34), (35), and (36) that
(37)∑j,k=0,0∞,∞anmjk[Fjk(ujk|λΔsrxjk|)pjk]<ϵ∀m,n.
Thus g(λx)→0 as λ→0. Therefore W02(A,ℱ,u,Δsr,p) is a paranormed space. Similarly, we can prove that W2(A,ℱ,u,Δsr,p) is a paranormed space. This completes the proof.
Theorem 10.
Let ℱ=(Fjk) be a sublinear Musielak-Orlicz function, A=(anmjk) a nonnegative four-dimensional RH-regular matrix, p=(pjk) a bounded sequence of positive real numbers, and u=(ujk) a sequence of strictly positive real numbers. Then W02(A,ℱ,u,Δsr,p) and W2(A,ℱ,u,Δsr,p) are complete topological linear spaces.
Proof.
Let (xjkq) be a Cauchy sequence in W02(A,ℱ,u,Δsr,p); that is, g(xq-xt)→0 as q,t→∞. Then, we have
(38)∑j,k=0,0∞,∞anmjk[Fjk(ujk|Δsrxjkq-Δsrxjkt|)pjk]⟶0.
Thus for each fixed j and k as q,t→∞, since A=(anmjk) is nonnegative, we are granted that
(39)Fjk(ujk|Δsrxjkq-Δsrxjkt|)⟶0,
and by continuity of ℱ=(Fjk), (xjkq) is a Cauchy sequence in ℂ for each fixed j and k.
Since ℂ is complete as t→∞, we have xjkq→xjk for each (j,k). Now from (36), we have that, for ϵ>0, there exists a natural number N such that
(40)∑j,k=0,0q,t>N∞,∞anmjk[Fjk(ujk|Δsrxjkq-Δsrxjkt|)pjk]<ϵ∀m,n.
Since for any fixed natural number M, from (38) we have
(41)∑j,k≤Mq,t>N∞,∞anmjk[Fjk(ujk|Δsrxjkq-Δsrxjkt|)pjk]<ϵ∀m,n.
By letting t→∞ in the above expression we obtain
(42)∑j,k≤Mq>N∞,∞anmjk[Fjk(ujk|Δsrxjkq-Δsrxjk|)pjk]<ϵ.
Since M is arbitrary, by letting M→∞ we obtain
(43)∑j,k=0,0∞,∞anmjk[Fjk(ujk|Δsrxjkq-Δsrxjk|)pjk]<ϵ∀m,n.
Thus g(xq-x)→0 as q→∞. This proves that W02(A,ℱ,u,Δsr,p) is a complete topological linear space.
Now we will show that W2(A,ℱ,u,Δsr,p) is a complete topological linear space. For this, since (xq) is also a sequence in W2(A,ℱ,u,Δsr,p) by definition of W2(A,ℱ,u,Δsr,p), for each q, there exists Lq with
(44)∑j,k=0,0∞,∞anmjk[Fjk(ujk|Δsrxjkq-ΔsrLq|)pjk]⟶01111111111111111111111111asm,n⟶∞;
whence from the fact that supnm∑j,k=0,0∞,∞anmjk<∞ and from the definition of Musielak-Orlicz function, we have Fjk|ΔsrLq-ΔsrL|→0 as q→∞ and so Lq converges to L. Thus
(45)∑j,k=0,0∞,∞anmjk[Fjk(ujk|Δsrxjk-L|)pjk]⟶01111111111111111111111asm,n⟶∞.
Hence x∈W2(A,ℱ,u,Δsr,p) and this completes the proof.
Theorem 11.
Let ℱ=(Fjk) be a sublinear Musielak-Orlicz function which satisfies the Δ2-condition. Then W2(A,u,Δsr,p)⊆W2(A,ℱ,u,Δsr,p).
Proof.
Let x=(xk)∈W2(A,u,Δsr,p); that is,
(46)limn,m∑j,kanmjk[(ujk|Δsrxjk-L|)pjk]=0.
Let ϵ>0 and choose δ with 0<δ<1 such that Fjk(t)<ϵ for 0≤t≤δ. Write yjk=(ujk|Δsrxjk-L|) and consider
(47)∑j,kanmjk[Fjk(yjk)pjk]=∑j,k:|yjk|≤δanmjk[Fjk(yjk)pjk]∑j,kanmjk[Fjk(yjk)pjk]=+∑j,k:|yjk|>δanmjk[Fjk(yjk)pjk]∑j,kanmjk[Fjk(yjk)pjk]=ϵ∑j,k:|yjk|≤δanmjk∑j,kanmjk[Fjk(yjk)pjk]=+∑j,k:|yjk|>δanmjk[Fjk(yjk)pjk].
For yjk>δ, we use the fact that yjk<yjk/δ<1+yjk/δ. Hence
(48)Fjk(yjk)<Fjk(1+yjkδ)<Fjk(2)2+12Fjk(2yjkδ).
Since ℱ satisfies the Δ2-condition, we have
(49)Fjk(yjk)<Kyjk2δFjk(2)+Kyjk2δFjk(2)=KyjkδFjk(2),
and hence
(50)∑j,k:|yjk|>δanmjk[Fjk(yjk)pjk]≤KFjkδ(2)∑j,kanmjk[(ujk|Δsrxjk-L|)pjk].
Since A is RH-regular and x∈W2(A,u,Δsr,p), we get x∈W2(A,ℱ,u,Δsr,p).
Theorem 12.
Let ℱ=(Fjk) be a sublinear Musielak-Orlicz function and let A=(anmjk) be a nonnegative four-dimensional RH-regular matrix. Suppose that β=limt→∞(Fjk(t)/t)<∞. Then
(51)W2(A,u,Δsr,p)=W2(A,ℱ,u,Δsr,p).
Proof.
In order to prove that W2(A,u,Δsr,p)=W2(A,ℱ,u,Δsr,p), it is sufficient to show that W2(A,ℱ,u,Δsr,p)⊂W2(A,u,Δsr,p). Now, let β>0. By definition of β, we have Fjk(t)≥βt for all t≥0. Since β>0, we have t≤(1/β)Fjk(t) for all t≥0. Let x=(xjk)∈W2(A,ℱ,u,Δsr,p). Thus, we have
(52)∑j,k=0,0∞,∞anmjk[(ujk|Δsrxjk-L|)pjk]≤1β∑j,k=0,0∞,∞anmjk[Fjk(ujk|Δsrxjk-L|)pjk],
which implies that x=(xjk)∈W2(A,u,Δsr,p). This completes the proof.
Theorem 13.
(i) Let 0<infpjk<pjk≤1. Then
(53)W2(A,ℱ,u,Δsr,p)⊆W2(A,ℱ,u,Δsr).
(ii) Let 1≤pjk≤suppjk<∞. Then
(54)W2(A,ℱ,u,Δsr)⊆W2(A,ℱ,u,Δsr,p).
Proof.
(i) Let x=(xjk)∈W2(A,ℱ,u,Δsr,p). Then since 0<infpjk<pjk≤1, we obtain the following:
(55)∑j,k=0,0∞,∞anmjk[Fjk(ujk|Δsrxjk-L|)]≤∑j,k=0,0∞,∞anmjk[Fjk(ujk|Δsrxjk-L|)pjk].
Thus x=(xjk)∈W2(A,ℱ,u,Δsr).
(ii) Let pjk≥1 for each j and k and suppjk<∞. Let x=(xjk)∈W2(A,ℱ,u,Δsr). Then for each 0<ϵ<1 there exists a positive integer N such that
(56)∑j,k=0,0∞,∞anmjk[Fjk(ujk|Δsrxjk-L|)]≤ϵ<1∀m,n≥N.
This implies that
(57)∑j,k=0,0∞,∞anmjk[Fjk(ujk|Δsrxjk-L|)pjk]≤∑j,k=0,0∞,∞anmjk[Fjk(ujk|Δsrxjk-L|)].
Therefore x=(xjk)∈W2(A,ℱ,u,Δsr,p). This completes the proof.
Lemma 14.
Let ℱ=(Fjk) be a sublinear Musielak-Orlicz function which satisfies the Δ2-condition and let A=(anmjk) be a nonnegative four-dimensional RH-regular matrix. Then W02(A,ℱ,u,Δsr,p)∩l∞2 is an ideal in l∞2.
Proof.
Let x∈W02(A,ℱ,u,Δsr,p)∩l∞2 and y∈l∞2. We need to show that xy∈W02(A,ℱ,u,Δsr,p)∩l∞2. Since y∈l∞2, there exists T1>1 such that ∥y∥<T1. In this case |xjkyjk|<T1|xjk| for all j,k. Since ℱ is nondecreasing and satisfies Δ2-condition, we have
(58)[Fjk(ujk|Δsr(xjkyjk)|)pjk]<[Fjk(ujkT1|Δsrxjk|)pjk][Fjk(ujk|Δsr(xjkyjk)|)pjk]≤T(T1)[Fjk(ujk|Δsrxjk|)pjk],
for all j,k and T>0. Therefore limn,m∑j,kanmjk[Fjk(ujk|Δsr(xjkyjk)|)pjk]=0. Thus xy∈W02(A,ℱ,u,Δsr,p)∩l∞2. This completes the proof.
Lemma 15.
Let G be an ideal in l∞2 and let x=(xjk)∈l∞2. Then x is in the closure of G in l∞2 if and only if χS(x;ϵ)∈G for all ϵ>0.
Proof.
Let x be in the closure of G and let ϵ>0 be given. Suppose that z=(zjk)∈G such that ∥x-z∥<ϵ/2 and observe that S(x;ϵ)⊆S(z;ϵ/2). Define a double sequence y=(yjk)∈l∞2 by
(59)yjk={1zjk,if|zjk|≧ϵ2,0,otherwise.
Clearly yz=χS(z;ϵ/2)∈G. Since S(x;ϵ)⊆S(z;ϵ/2) and χS(x;ϵ)∈l∞2, hence χS(x;ϵ)χS(z;ϵ/2)=χS(x;ϵ)∈G.
Conversely, if x∈l∞2 then ∥x-xχS(x;ϵ)∥<ϵ. It follows that χS(x;ϵ)∈G for all ϵ>0; then x is in the closure of G.
Lemma 16.
If A is a nonnegative four-dimensional RH-regular matrix, then W02(A,u,Δsr,p)∩l∞2 is a closed ideal in l∞2.
Proof.
We have W02(A,ℱ,u,Δsr,p)∩l∞2⊂l∞2 and it is clear that W02(A,ℱ,u,Δsr,p)∩l∞2≠0. For x,y∈W02(A,ℱ,u,Δsr,p)∩l∞2, we get |xjk+yjk|<|xjk|+|yjk|. Now, we have
(60)[Fjk(ujk|Δsr(xjk+yjk)|)pjk]≤[Fjk(ujk|Δsrxjk|+|Δsryjk|)pjk]<12[Fjk(ujk2|Δsrxjk|)pjk]+12[Fjk(ujk2|Δsryjk|)pjk]<12K1[Fjk(ujk|Δsrxjk|)pjk]+12K2[Fjk(ujk|Δsryjk|)pjk]
by the Δ2-condition and the convexity of F. Since
(61)∑j,kanmjk[Fjk(ujk|Δsr(xjk+yjk)|)pjk]≤12K∑j,kanmjk[Fjk(ujk|Δsrxjk|)pjk]+12K∑j,kanmjk[Fjk(ujk|Δsryjk|)pjk],
where K=max{K1,K2}, so x+y,x-y∈W02(A,ℱ,u,Δsr,p)∩l∞2.
Let x∈W02(A,ℱ,u,Δsr,p)∩l∞2 and y∈l∞2. Thus, there exists a positive integer K, so that, for every j,k, we have |xjkyjk|≤K|xjk|. Therefore
(62)[Fjk(ujk|Δsr(xjkyjk)|)pjk]≤[Fjk(ujkK|Δsrxjk|)pjk][Fjk(ujk|Δsr(xjkyjk)|)pjk]≤T[Fjk(ujk|Δsrxjk|)pjk],
and so
(63)∑j,kanmjk[Fjk(ujk|Δsr(xjkyjk)|)pjk]≤T∑j,kanmjk[Fjk(ujk|Δsrxjk|)pjk].
Hence xy∈W02(A,ℱ,u,Δsr,p)∩l∞2. So W02(A,ℱ,u,Δsr,p)∩l∞2 is an ideal in l∞2 for a Musielak-Orlicz function which satisfies the Δ2-condition.
Now, we have to show that W02(A,ℱ,u,Δsr,p)∩l∞2 is closed. Let x∈W02(A,ℱ,u,Δsr,p)∩l∞2¯; there exists xcd=xjkcd∈W02(A,ℱ,u,Δsr,p)∩l∞2 such that xcd→x∈l∞2. For every ϵ>0 there exists N1(ϵ)∈ℕ such that, for all c,d>N1(ϵ), |Δsrxcd-Δsrx|<ϵ. Now, for ϵ>0, we have
(64)∑j,kanmjk[Fjk(ujk|Δsrxjk|)pjk]=∑j,kanmjk[Fjk(ujk|Δsrxjk-Δsrxjkcd+Δsrxjkcd|)pjk]≤∑j,kanmjk[Fjk(ujk|Δsrxjk-Δsrxjkcd|+|Δsrxjkcd|)pjk]≤12∑j,kanmjk[Fjk(ujk2|Δsrxjk-Δsrxjkcd|)pjk]+12∑j,kanmjk[Fjk(ujk2|Δsrxjkcd|)pjk]≤12KFjk(ϵ)∑j,kanmjk+12K∑j,kanmjk[Fjk(ujk|Δsrxjkcd|)pjk].
Since xcd∈W02(A,ℱ,u,Δsr,p)∩l∞2 and A is RH-regular, we get
(65)limn,m∑j,kanmjk[Fjk(ujk|Δsrxjk|)pjk]=0;
so x∈W02(A,ℱ,u,Δsr,p)∩l∞2. This completes the proof.
Theorem 17.
Let x=(xjk) be a bounded sequence, ℱ=(Fjk) a sublinear Musielak-Orlicz function which satisfies the Δ2-condition, and A a nonnegative four-dimensional RH-regular matrix. Then W2(A,ℱ,u,Δsr,p)∩l∞2=W2(A,u,Δsr,p)∩l∞2.
Proof.
Without loss of generality we may take L=0 and establish
(66)W02(A,ℱ,u,Δsr,p)∩l∞2=W02(A,u,Δsr,p)∩l∞2.
Since W02(A,u,Δsr,p)⊆W02(A,ℱ,u,Δsr,p), therefore W02(A,u,Δsr,p)∩l∞2⊆W02(A,ℱ,u,Δsr,p)∩l∞2. We need to show that W02(A,ℱ,u,Δsr,p)∩l∞2⊆W02(A,u,Δsr,p)∩l∞2. Notice that if S⊂ℕ×ℕ, then
(67)∑j,kanmjk[Fjk(χS(j,k))pjk]=Fjk(1)∑j,kanmjk(χS(j,k))pjk,
for all n,m. Observe that χS(j,k)∈W02(A,u,Δsr,p)∩l∞2 whenever x∈W02(A,ℱ,u,Δsr,p)∩l∞2 by Lemmas 14 and 15, so
(68)W02(A,ℱ,u,Δsr,p)∩l∞2⊆W02(A,u,Δsr,p)∩l∞2.
The proof is complete.
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.
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