# On Paranormed Ideal Convergent Generalized Difference Strongly Summable Sequence Spaces Defined overn-Normed Spaces

Authors: Bipan Hazarika
Journal: ISRN Mathematical Analysis
 (2011)

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## Abstract

We introduce a new class of ideal convergent (shortlyI-convergent) sequence spaces using an Orlicz function and difference operator of order s≥1 defined over the n-normed spaces. We investigate these spaces for some linear topological structures. These investigations will enhance the acceptability of the notion of n-norm by giving a way to construct different sequence spaces with elements in n-normed spaces. We also give some relations related to these sequence spaces.


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## Body

## 1. Introduction

Throughout the paperw, ℓ∞, c, c0, and ℓp  denote the classes of all, bounded, convergent, null, and p-absolutely summable sequences of complex numbers.The notion of statistical convergence is a very useful functional tool for studying the convergence problems of numerical sequences/matrices (double sequences) through the concept of density. It was first introduced by Fast [1], and Schoenberg [2], independently for the real sequences. Later on, it was further investigated from sequence point of view and linked with the summability theory by Fridy [3] and many others. The idea is based on the notion of natural density of subsets of ℕ, the set of positive integers, which is defined as follows: the natural density of a subset E of ℕ is denoted by δ(E) and is defined by(1.1)δ(E)=lim⁡n→∞⁡1n|{k∈E:k≤n}|,
where the vertical bar denotes the cardinality of the respective set.The notion ofI-convergence (I denotes the ideal of subsets of ℕ), which is a generalization of statistical convergence, was introduced by Kastyrko et al. [4] and further studied by many other authors.The notion of difference sequence space was introduced by Kizmaz [5], who studied the difference sequence spaces ℓ∞(Δ), c(Δ), and c0(Δ). The notion was further generalized by Et and Çolak [6] by introducing the sequence spaces ℓ∞(Δs), c(Δs), and c0(Δs). Another type of generalization of the difference sequence spaces is due to Tripathy and Esi [7], who studied the spaces ℓ∞(Δm), c(Δm), and c0(Δm).  Tripathy et al. [8] generalized the above notion and unified these as follows.For nonnegative integersm, s, the generalized difference sequence spaces are defined as follows: for a given sequence space Z, we have (1.2)Z(Δms)={x=(xk)∈w:(Δmsxk)∈Z},
where Δmsx= (Δmsxk)=(Δms-1xk-Δms-1xk+m), Δm0xk=xk,for all k∈ℕ, the difference operator is equivalent to the following binomial representation: (1.3)Δmsxk=∑ν=0s(-1)ν(sν)xk+mν.
Taking s=1, we get the spaces ℓ∞(Δm), c(Δm), and c0(Δm), introduced and studied by Tripathy and Et [7]. Taking m=1, we get the spaces ℓ∞(Δs), c(Δs), and c0(Δs) studied by Et and Çolak [6]. Taking s=m=1, we get the spaces ℓ∞(Δ), c(Δ), and c0(Δ) introduced and studied by Kizmaz [5].Letm, s be nonnegative integers, then for a given sequence space Z, we introduced(1.4)Z(Δm(s))={x=(xk)∈w:(Δm(s)xk)∈Z},
where Δm(s)x=(Δm(s)xk)=(Δm(s-1)xk-Δm(s-1)xk-m),Δm(0)xk=xk, for all  k∈ℕ, the difference operator is equivalent to the following binomial representation: (1.5)Δm(s)xk=∑ν=0s(-1)ν(sν)xk-mν,
where xk=0, for k<0.The concept of 2-normed space was initially introduced by Gähler [9], in the mid of 1960s, while that of n-normed spaces can be found in Misiak [10]. Since then, many others authors have used this concept and obtained various results, see, for instance, Gunawan [11] and Gunawan and Mashadi [12, 13]. Recently, a lot of activities have started to study summability, sequence spaces, and related topics in these spaces (see [14, 15]).The notion of ideal-convergence in 2-normed spaces was initially introduced by Gürdal [16]. Later on, it was extended to n-normed spaces by Gürdal and Sahnier [17].An Orlicz function is a functionM:[0,∞)→[0,∞), which is continuous, nondecreasing and convex with M(0)=0,M(0)>0,as x>0 and  M(x)→∞,as x→∞.An Orlicz functionM can always be represented in the following integral form:(1.6)M(x)=∫0xp(t)dt,
where p is the known kernel of M, right differentiable for t≥0, p(0)=0, p(t)>0, for t>0 and p(t)→∞, as t→∞.If convexity of Orlicz functionM is replaced by M(x+y)≤M(x)+M(y), then this function is called Modulus function, which was presented and discussed by Maddox [18].Lindenstrauss and Tzafriri [19] used the idea of Orlicz function to construct the sequence space(1.7)lM={(xk)∈w:∑k=1∞M(|xk|ρ)<∞,for someρ>0}.
The space ℓM with the norm(1.8)‖x‖=inf⁡{ρ>0:∑k=1∞M(|xk|ρ)≤1}
becomes a Banach space which is called an Orlicz sequence space.The spaceℓM  is closely related to the space ℓp which is an Orlicz sequence space with M(t)=|t|p, for 1≤p<∞.Subsequently, Orlicz function was used to define sequence spaces by Parashar and Choudhary [20] and others (see [10, 15, 21–23]).Remark 1.1.
It is well known ifM is a convex function and M(0) = 0, then
(1.9)M(λx)≤λM(x),∀λwith0<λ<1.
In this paper, we introduce some new ideal convergent sequences over n-normed spaces by using Orlicz functions and generalized difference sequence spaces.


## 2. Definitions and Preliminaries

Letn be a nonnegative integer and X a real vector space of dimension d≥n (d may be infinite). A real-valued function ∥·,…,·∥ on Xn satisfying the following conditions:(1)
∥x1,x2,…,xn∥=0 if and only if x1,x2,…,xn are linearly dependent,(2)
∥x1,x2,…,xn∥ is invariant under permutation,(3)
∥αx1,x2,…,xn∥=|α|∥x1,x2,…,xn∥, for any α∈ℝ,(4)
∥x+x′,x2,…,xn∥≤∥x,x2,…,xn∥+∥x′,x2,…,xn∥, is called an n-norm on X and the pair (X,∥.,…,.∥) is called an n-normed space.A trivial example of ann-normed space is X=ℝn, equipped with the Euclidean n-norm ∥x1,x2,…,xn∥E= the volume of the n-dimensional parallelopiped spanned by the vectors x1,x2,…,xn, which may be given explicitly by the formula(2.1)‖x1,x2,…,xn‖E=|det⁡(xij)|=abs(|〈x1,x1〉⋯〈x1,xn〉⋅⋯⋅⋅⋯⋅〈xn,x1〉⋯〈xn,xn〉|),
where xi=(xi1,xi2,…,xin)∈ℝn, for each i=1,2,…,n.Let(X,∥.,…,.∥) be an n-normed space of dimension d≥n≥2 and {a1,a2,…,an} a linearly independent set in X. Then, the function ∥·,…,·∥∞onXn-1 is defined by
(2.2)‖x1,x2,…,xn‖∞=max⁡1≤i≤n{‖x1,x2,…,xn-1,ai‖}
defines as (n-1)-norm on X with respect to {a1,a2,…,an}, and this is known as the derived (n-1)-norm (see [12] for details).The standardn-norm on X, a real inner product space of dimension d≥n, is as follows:(2.3)‖x1,x2,…,xn‖S=|〈x1,x1〉⋯〈x1,xn〉⋅⋯⋅⋅⋯⋅〈xn,x1〉⋯〈xn,xn〉|1/2,
where 〈·,·〉 denotes the inner product on X. If we take X=ℝn, then this n-norm is exactly the same as the Euclidean n-norm ∥x1,x2,…,xn∥E  mentioned earlier.Forn=1, this n-norm is the usual norm ∥x1∥=〈x1,x1〉.Definition 2.1.
A sequence (xk) in an n-normed space (X,∥·,…,·∥) is said to converge to some l∈X with respect the n-norm if for each ε>0, there exists an positive integer n0 such that
(2.4)‖xk-l,z1,z2,…,zn-1‖<ε,∀k≥n0for everyz1,z2,…,zn-1∈X.Definition 2.2.
A sequence (xk) in an n-normed space (X,∥.,…,.∥) is said to be Cauchy with respect the n-norm if for each ε>0, there exists an positive integer n0=n0(ε) such that
(2.5)‖xk-xm,z1,z2,…,zn-1‖<ε,∀k,m≥n0for everyz1,z2,…,zn-1∈X.
If every Cauchy sequence in X converges to some l∈X, then X is said to be complete with respect to the n-norm. Any complete n-normed space is said to be n-Banach space.Definition 2.3.
LetS be a nonempty set. A nonempty family of sets I⊆P(S) (power set of S) is called an ideal in S if (i) for each A,B∈I, we have A∪B∈I; (ii) for each A∈I and B⊆A, we have B∈I.Definition 2.4.
LetS be a nonempty set. A family F⊆P(S) (power set of S) is called a filter on S if (i) ∅∉F; (ii) for each A, B∈F, we have A∩B∈F; (iii) for each A∈F and B⊇A, we have B∈F.An idealI is called nontrivial if I≠∅ and S∉I. It is clear that I⊆P(S) is a nontrivial ideal if and only if the class F=F(I)={S-A:A∈I} is a filter on S. The filter F(I) is called the filter associated with the ideal I.A nontrivial idealI⊆P(S) is called an admissible ideal in S if and only if it contains all singletons, that is, if it contains {{x}:x∈S}.Definition 2.5.
A sequence (xk) in a normed space (X,||·||) is said to be statistically convergent to x0∈X if for each ε>0, the set E(ε)={k∈ℕ:∥xk-x0∥≥ε} has natural density zero.Definition 2.6.
A sequence (xk) in a normed space (X,||·||) is said to be I-convergent to x0∈X if for each ε>0, the set
(2.6)E(ε)={k∈N:‖xk-x0‖≥ε}belongs toI.Definition 2.7.
A sequence (xk) in a normed space (X,||·||) is said to be I-bounded if there exists M>0 such that the set {k∈ℕ:∥xk∥>M} belongs to I.Definition 2.8.
A sequence (xk) in a normed space (X,∥·∥) is said to be I-Cauchy if for each ε>0, there exists an positive integer m=m(ε) such that the set
(2.7)E(ε)={k∈N:‖xk-xm‖≥ε}belongs toI.Definition 2.9.
A sequence (xk) in an n-normed space (X,∥·,…,·∥) is said to be I-convergent to x0∈X with respect the n-norm, if for each ε>0, the set
(2.8)E(ε)={k∈N:‖xk-x0,z1,z2,…,zn-1‖≥ε,for everyz1,z2,…,zn-1∈X}belongs toI.Definition 2.10.
A sequence (xk) in an n-normed space (X,∥·,…,·∥) is said to be I-Cauchy with respect to the n-norm if for each ε>0, there exists a positive integer m=m(ε) such that the set
(2.9)E(ε)={k∈N:‖xk-xm,z1,z2,…,zn-1‖≥ε,for everyz1,z2,…,zn-1∈X}belongs toI.Definition 2.11.
A sequence spaceE is said to be normal (or solid) if (αkxk)∈E,whenever (xk)∈E and for all sequence (αk) of scalars with |αk|≤1, for all k∈ℕ.LetK={k1<k2<⋯}⊆ℕ and E a sequence space.A K-step space of E is a sequence space λKE={(xkn)∈w:(kn)∈E}.A canonical preimage of a sequence{xkn}∈λKE is a sequence {yk}∈w defined as
(2.10)yk={xk,ifk∈K,0,otherwise.
A canonical preimage of a step space λKE is a set of canonical preimages of all elements inλKE; that is,  y is in canonical preimage of λKE if and only if y is canonical preimage of some x∈λKE.Definition 2.12.
A sequence spaceE is said to be monotoneif it contains the canonical preimages of all its step spaces.Example 2.13.
If we takeI=If={A⊆ℕ:Aisafinitesubset}. Then, If is a nontrivial admissible ideal of  ℕ  and the corresponding convergence coincide with usual convergence of sequences in n-normed spaces.Example 2.14.
If we takeI=Iδ={A⊆ℕ:δ(A)=0}, where δ(A) denote the asymptotic density of the set A then Iδ is a nontrivial admissible ideal of ℕ and the corresponding convergence coincide with statistical convergence in n-normed spaces.Now, we state the following lemmas (see [12] for details).Lemma 2.15.
Every n-normed space is an(n-r)-normed space for all  r=1,2,…,n-1. In particular, every n-normed space is a normed space.Lemma 2.16.
A standard n-normed space is complete if and only if it is complete with respect to the usual norm∥·∥=〈,〉.Lemma 2.17.
On a standard n-normed spaceX, the derived (n-1)-norm ∥·,…,·∥∞ defined with respect to the orthogonal set {e1,e2,…,en} is equivalent to the standard (n-1)-norm ∥·,…,·∥S. To be precise, one has
(2.11)‖x1,x2,…,xn-1‖∞≤‖x1,x2,…,xn-1‖S≤n‖x1,x2,…,xn-1‖∞,
for all x1,x2,…,xn-1∈X, where∥x1,x2,…,xn-1∥∞=max1≤i≤n{∥x1,x2,…,xn-1,ei∥S}.Lemma 2.18 (Kamthan and Gupta [24]).
Every normal space is monotone.


## 3. Main Results

In this section, we define some new ideal convergent sequence spaces and investigate their linear topological structures. Also, we find out some relations related to these sequence spaces.LetI be an admissible ideal of ℕ, M an Orlicz function, and (X,∥·,…,·∥) an n-normed space. Further, let p=(pk) be any bounded sequence of positive real numbers, and let w(n-X), denote the spaces of all X-valued sequences. For each ε>0, we define the following sequence spaces:(3.1)WI(Δms,M,p,‖⋅,…,⋅‖)={(xk)∈w(n-X):{n∈N:1n∑k=1n[M(‖Δmsxk-lρ,z1z2,…,zn-1‖)]pk≥ε}∈I,forsomeρ>0,l∈Xand∀z1,z2…zn-1∈X∃K>0s.t.{n∈N:1n∑k=1n[M(‖Δm(s)xkρ,z1z2,…,zn-1‖)]pk≥K}∈I},(3.2)WI(Δms,M,p,‖⋅,…,⋅‖)={(xk)∈w(n-X):{n∈N:1n∑k=1n[M(‖Δmsxkρ,z1z2,…,zn-1‖)]pk≥ε}∈I,forsomeρ>0and∀z1,z2…zn-1∈X∑k=1n[M(‖Δmsxk-lρ,z1z2,…,zn-1‖)]pk≥ε},(3.3)WI(Δms,M,p,‖⋅,…,⋅‖)={(xk)∈w(n-X):sup⁡n∈N1n∑k=1n[M(‖Δmsxkρ,z1z2,…,zn-1‖)]pk<∞,forsomeρ>0and∀z1,z2…zn-1∈X∑k=1n[M(‖Δmsxk-lρ,z1z2,…,zn-1‖)]pk≥ε},(3.4)WI(Δms,M,p,‖⋅,…,⋅‖)={(xk)∈w(n-X):∃K>0s.t.{n∈N:1n∑k=1n[M(‖Δmsxkρ,z1z2,…,zn-1‖)]pk≥K}∈I,forsomeρ>0and∀z1,z2…zn-1∈X∑k=1n[M(‖Δmsxk-lρ,z1z2,…,zn-1‖)]pk≥ε}.The following well-known inequality will be used throughout the article. Letp=(pk) be any sequence of positive real numbers with 0≤pk≤supkpk=G,D=max{1,2G-1}, then  |ak+bk|pk≤D(|ak|pk+|bk|pk), for all k∈ℕ and ak,bk∈ℂ. Also, |a|pk≤max{1,|a|G},  for all a∈ℂ.Theorem 3.1.
The spacesWI(Δms,M,p,∥·,…,·∥), W0I(Δms,M,p,||·,…,·||)  and  W∞I(Δms,M,p, ∥·,…,·∥)   are linear.Proof.
We will prove the result for the spaceW0I(Δms,M,p,∥·,…,·∥) only and the others can be proved similarly.
Letx=(xk) and y=(yk) be any two elements of the space W0I(Δms,M,p,∥·,…,·∥). Then, there exist ρ1>0 and ρ2>0 such that the sets
(3.5)Aε/2(x)={n∈N:1n∑k=1n[M(‖Δmsxkρ1,z1,z2,....,zn-1‖)]pk≥ε2,for everyz1,z2,…,zn-1∈X∑k=1n[M(‖Δmsxkρ1,z1,z2,....,zn-1‖)]pk≥ε2,},Bε/2(y)={n∈N:1n∑k=1n[M(‖Δmsykρ2,z1,z2,…,zn-1‖)]pk≥ε2,for everyz1,z2,…,zn-1∈X∑k=1n[M(‖Δmsxkρ1,z1,z2,....,zn-1‖)]pk≥ε2,}
belong to I.
Letα,β∈ℝ be two scalars. Since ∥·,…,·∥ is an n-norm, the operator Δs is linear and the Orlicz function M is continuous, the following inequality holds:
(3.6)1n∑k=1n[M(‖Δms(αxk+βyk)(|α|ρ1+|β|ρ2),z1,z2,…,zn-1‖)]pk≤D1n∑k=1n[|α|(|α|ρ1+|β|ρ2)M(‖Δmsxkρ1,z1,z2,…,zn-1‖)]pk+D1n∑k=1n[|β|(|α|ρ1+|β|ρ2)M(‖Δmsykρ2,z1,z2,…,zn-1‖)]pk≤DK1n∑k=1n[M(‖Δmsxkρ1,z1,z2,…,zn-1‖)]pk+DK1n∑k=1n[M(‖Δmsykρ2,z1,z2,…,zn-1‖)]pk,
where K=max{1,(|α|/(|α|ρ1+|β|ρ2)),(|β|/(|α|ρ1+|β|ρ2))}.
From the above relation, we get(3.7){n∈N:1n∑k=1n[M(‖Δms(αxk+βyk)(|α|ρ1+|β|ρ2),z1,z2,…,zn-1‖)]pk≥ε}⊆{n∈N:DK1n∑k=1n[M(‖Δmsxkρ1,z1,z2,…,zn-1‖)]pk≥ε2}∪{n∈N:DK1n∑k=1n[M(‖Δmsykρ2,z1,z2,…,zn-1‖)]pk≥ε2}.
Since both the sets on the right hand side of the relation (3.7) belong to I, this completes the proof.Note 1.
It is easy to verify that the spaceW∞(Δms,M,p,||·,…,·||) is a linear space.Theorem 3.2.
The spaceWI(Δms,M,p,||·,…,·||),  W0I(Δms,M,p,||·,…,·||),  W∞I(Δms,M,p,  ||·,…,·||),   and  W∞(Δms,M,p,  ||·,…,·||)are paranormed spaces  (not totally paranormed ) with respect to the paranorm  gΔ defined by
(3.8)gΔ(x)=∑k=1ms‖xk,z1,z2,…,zn-1‖+inf⁡{ρpk/H:sup⁡kM(‖Δmsxkρ,z1,z2,…,zn-1‖)≤1,for someρ>0and∀z1,z2,…,zn-1∈X∑k=1ms‖xk,z1,z2,…,zn-1‖(‖Δmsxkρ,z1,z2,…,zn-1‖)}
where H=max{1,suppk}.Proof.
Clearly,gΔ(-x)=gΔ(x) and gΔ(θ)=0. Let x=(xk) and y=(yk)be any two elements of the space Z(Δms,M,p,||·,…,·||), where Z=WI,W0I,W∞IandW∞.
Then,forρ>0, we set
(3.9)A1={ρ>0:sup⁡kM(‖Δmsxkρ,z1,z2,…,zn-1‖)≤1,∀z1,z2,…,zn-1∈X},A2={ρ>0:sup⁡kM(‖Δmsykρ,z1,z2,…,zn-1‖)≤1,∀z1,z2,…,zn-1∈X}.
Let ρ1∈A1 and ρ2∈A2. If ρ=ρ1+ρ2, then we have
(3.10)M(‖Δms(xk+yk)ρ,z1,z2,…,zn-1‖)≤ρ1ρ1+ρ2M(‖Δmsxkρ1,z1,z2,…,zn-1‖)+ρ2ρ1+ρ2M(‖Δmsykρ2,z1,z2,…,zn-1‖).
Thus, we have
(3.11)sup⁡k[M(‖Δms(xk+yk)ρ,z1,z2,…,zn-1‖)]pk≤1,gΔ(x+y)=∑k=1ms‖xk+yk,z1,z2,…,zn-1‖+inf⁡{(ρ1+ρ2)pk/H:ρ1∈A1,ρ2∈A2}≤∑k=1ms‖xk,z1,z2,…,zn-1‖+inf⁡{(ρ1)pk/H:ρ1∈A1}+∑k=1ms‖yk,z1,z2,…,zn-1‖inf⁡{(ρ2)pk/H:ρ2∈A2}=gΔ(x)+gΔ(y).
Let λs→λ, where λs,λ∈ℂ, and let gΔ(xs-x)→0 as s→∞. We have to show that gΔ(λsxs-λx)→0 as s→∞.
We set(3.12)A3={ρs>0:sup⁡k[M(‖Δmsxksρs,z1,z2,…,zn-1‖)]pk≤1,∀z1,z2,…,zn-1∈X},A4={ρs′>0:sup⁡k[M(‖Δms(xks-xk)ρs′,z1,z2,…,zn-1‖)]pk≤1,∀z1,z2,…,zn-1∈X}.
If ρs∈A3 and  ρs′∈A4, we observe by the continuity of the Orlicz function M that
(3.13)M(‖Δms(λsxks-λxk)(|λs-λ|ρs+|λ|ρs′),z1,z2,…,zn-1‖)(3.14)≤M(‖Δms(λsxks-λxks)(|λs-λ|ρs+|λ|ρs′),z1,z2,…,zn-1‖)+M(‖Δms(λxks-λxk)(|λs-λ|ρs+|λ|ρs′),z1,z2,…,zn-1‖)(3.15)≤|λs-λ|ρs|λs-λ|ρs+|λ|ρs′M(‖Δms(xks)ρs,z1,z2,…,zn-1‖)+|λ|ρs′|λs-λ|ρs+|λ|ρs′M(‖Δms(xks-xk)ρs′,z1,z2,…,zn-1‖).
From the above inequality, it follows that
(3.16)sup⁡k[M(‖Δms(λsxks-λxk)(|λs-λ|ρs+|λ|ρs′),z1,z2,…,zn-1‖)]pk≤1,
and consequently
(3.17)gΔ(λsxs-λx)=∑k=1ms‖λsxks-λxk,z1,z2,…,zn-1‖+inf⁡{(|λs-λ|ρs+|λ|ρs′)pk/H:ρs∈A3,ρs′∈A4}≤|λs-λ|∑k=1ms‖xks,z1,z2,…,zn-1‖+|λs-λ|pk/Hinf⁡{(ρs)pk/H:ρs∈A3}+|λ|∑k=1ms‖xks-xk,z1,z2,…,zn-1‖+|λ|pk/Hinf⁡{(ρs′)pk/H:ρs′∈A4}≤max⁡{|λs-λ|,|λs-λ|pk/H}gΔ(xs)+max⁡{|λ|,|λ|pk/H}gΔ(xs-x).
Note that gΔ(xs)≤gΔ(x)+gΔ(xs-x), for all s∈ℕ.
Hence, by our assumption, the right hand side (3.17) tends to 0 as  s→∞, and the result follows. This completes the proof of the theorem.Theorem 3.3.
LetM,M1, and  M2  be Orlicz functions. Then, the following hold:(a)
W0I(Δms,M,p,∥·,…,·∥)⊆W0I(Δms,M.M1,p,∥·,…,·∥), provided  (pk) be such that  G0=infpk>0,(b)
W0I(Δms,M1,p,∥·,…,·∥)∩W0I(Δms,M2,p,∥·,…,·∥)⊆W0I(Δms,M1+M2,p,∥·,…,·∥).Proof.
(a) Letε>0 be given. Choose ε1>0 such that max{ε1G,ε1G0}<ε.
Using the continuity of the Orlicz functionM, choose 0<δ<1 such that 0<t<δ implies that M(t)<ε1.
Letx=(xk) be any element in W0I(Δms,M1,p,∥·,…,·∥). Put
(3.18)Aδ={n∈N:1n∑k=1n[M1(‖Δmsxkρ,z1,z2,…,zn-1‖)]pk≥δG}.
Then, by definition of ideal, we have the set Aδ∈I.
Ifn∉Aδ, then we have
(3.19)1n∑k=1n[M1(‖Δmsxkρ,z1,z2,…,zn-1‖)]pk<δG⟹∑k=1n[M1(‖Δmsxkρ,z1,z2,…,zn-1‖)]pk<nδG⟹[M1(‖Δmsxkρ,z1,z2,…,zn-1‖)]pk<δG,∀k=1,2,…,n⟹M1(‖Δmsxkρ,z1,z2,…,zn-1‖)<δ,∀k=1,2,…,n.
Using the continuity of the Orlicz function M, then from the relation (3.19), we have
(3.20)∑k=1nM(M1(‖Δmsxkρ,z1,z2,…,zn-1‖))<ε1,∀k=1,2,…,n.
Consequently, we get
(3.21)∑k=1n[M(M1(‖Δmsxkρ,z1,z2,…,zn-1‖))]pk<nmax⁡{ε1G,ε1G0}<nε⟹1n∑k=1n[M(M1(‖Δmsxkρ,z1,z2,…,zn-1‖))]pk<ε.
This shows that
(3.22){n∈N:1n∑k=1n[M(M1(‖Δmsxkρ,z1,z2,…,zn-1‖))]pk≥ε,}⊆Aδ∈I.
This proves the result.
(b) Letx=(xk)∈W0I(Δms,M1,p,∥·,…,·∥)∩W0I(Δms,M2,p,∥·,…,·∥). Then, by the following inequality, the result follows:
(3.23)1n∑k=1n[(M1+M2)(‖Δmsxkρ,z1,z2,…,zn-1‖)]pk≤D1n∑k=1n[M1(‖Δmsxkρ,z1,z2,…,zn-1‖)]pk+D1n∑k=1n[M2(‖Δmsxkρ,z1,z2,…,zn-1‖)]pk.Theorem 3.4.
The inclusionsZ(Δms-1,M,p,∥·,…,·∥)⊆Z(Δms,M,p,∥·,…,·∥)   are strict for  s≥1. In general, Z(Δmj,M,p,∥·,…,·∥)⊆Z(Δms,M,p,∥·,…,·∥), for  j=0,1,2,…,s-1,  and the inclusion is strict, where  Z=WI,W0I,andW∞I.Proof.
We will give the proof forW0I(Δms,M,p,∥·,…,·∥) only. The others can be proved by similar arguments.
Letx=(xk) be any element in the space W0I(Δms,M,p,∥·,…,·∥). Let ε>0 be given. Then, there exists ρ > 0 such that the set
(3.24){n∈N:1n∑k=1n[M(‖Δmsxkρ,z1z2,…,zn-1‖)]pk≥ε}∈I.
Since M is nondecreasing and convex, it follows that
(3.25)1n∑k=1n[M(‖Δmsxk2ρ,z1z2,…,zn-1‖)]pk=1n∑k=1n[M(‖Δms-1xk+1-Δms-1xk2ρ,z1z2,…,zn-1‖)]pk≤Dn∑k=1n[12M(‖Δms-1xk+1ρ,z1z2,…,zn-1‖)]pk+Dn∑k=1n[12M(‖Δms-1xkρ,z1z2,…,zn-1‖)]pk≤DHn∑k=1n([M(‖Δms-1xk+1ρ,z1z2,…,zn-1‖)]pk+[M(‖Δms-1xkρ,z1z2,…,zn-1‖)]pk),
where H=max{1,(1/2)G}.
Thus, we have(3.26){n∈N:1n∑k=1n[M(‖Δmsxk2ρ,z1z2,…,zn-1‖)]pk≥ε}⊆{n∈N:DHn∑k=1n[M(‖Δms-1xk+1ρ,z1z2,…,zn-1‖)]pk≥ε2}∪{n∈N:DHn∑k=1n[M(‖Δms-1xkρ,z1z2,…,zn-1‖)]pk≥ε2}.
Since both the sets in the right side of the relation (3.26) belongs to I, we get the set
(3.27){n∈N:1n∑k=1n[M(‖Δmsxk2ρ,z1z2,…,zn-1‖)]pk≥ε}∈I.
Let M(x)=x2, for all x∈[0,∞), pk=1, for all k∈ℕ. Consider a sequence x=(xk)=((k+1)r). Then, x belongs to W0I(Δms-1,M,p,∥·,…,·∥) but does not belong to W0I(Δms,M,p,∥·,…,·∥). This shows that the inclusion W0I(Δms-1,M,p,∥·,…,·∥)⊂W0I(Δms,M,p,∥·,…,·∥) is strict.Theorem 3.5.
Let0<pk≤qk  and (qk/pk)  be bounded, then
(3.28)WI(Δms,M,q,‖⋅,…,⋅‖)⊆WI(Δms,M,p,‖⋅,…,⋅‖).Proof.
Letx=(xk)∈WI(Δms,M,q,∥·,…,·∥). We put
(3.29)yk=(M(‖Δmsxk-lρ,z1,z2,…,zn-1‖))qkandβk=pkqk,∀k∈N.Then,0<βk≤1.
For all k∈ℕ. Let β be such that 0<β<βk, for all k∈ℕ.
Define the sequences(ak) and (bk) as follows:  for yk≥1, let ak=yk and bk=0; for yk<1, let ak=0 and bk=yk.
Then, clearly, for allk∈ℕ, we have yk=ak+bk,ykβk=akβk+bkβk;akβk≤ak≤yk and bkβk≤bkβ. Therefore, we have
(3.30)1n∑k=1nykβk≤1n∑k=1nyk≤[1n∑k=1nbk]β.
Hence, (xk)∈WI(Δms,M,p,∥·,…,·∥).Theorem 3.6.
For any two sequencesp=(pk) and  q=(qk)   of positive real numbers and for any two n-norms ∥·,…,·∥1  and   ∥·,…,·∥2  on X, the following holds:
(3.31)Z(Δms,M,p,‖⋅,…,⋅‖1)∩Z(Δms,M,p,‖⋅,…,⋅‖2)≠∅,
where Z=WI,W0I,W∞I,andW∞.Proof.
Proof of the theorem is obvious, because the zero element belongs to each of the sequence spaces involved in the intersection.Theorem 3.7.
The sequence spacesZ(Δms,M,p,∥·,…,·∥) are normal as well as monotone, where  Z=W0I,W∞I.Proof.
We will give the proof forW0I(Δms,M,p,∥·,…,·∥) only. Let x=(xk) be any element in W0I(Δms,M,p,||·,…,·||), and let (αk)  be a sequence of scalars such that |αk|≤1, for all k∈ℕ. Then, we have
(3.32){n∈N:1n∑k=1n[M(‖Δs(αkxk)ρ,z1z2,…,zn-1‖)]pk≥ε}⊆{n∈N:En∑k=1n[M(‖Δsxkρ,z1z2,…,zn-1‖)]pk≥ε},
where E=max{1,|αk|G}.
Hence,(αkxk)∈W0I(Δms,M,p,||·,…,·||), for all sequence (αk) of scalars with |αk|≤1, for all k∈ℕ, whenever (xk)∈W0I(Δms,M,p,||·,…,·||).
By Lemma2.18, we have the space W0I(Δms,M,p,||·,…,·||) is monotone.Remark 3.8.
If we replace the difference operatorΔms by Δm(s), then for each ε>0, we get the following sequence spaces:
(3.33)WI(Δm(s),M,p,‖⋅,…,⋅‖)={(xk)∈w(n-X):{n∈N:1n∑k=1n[M(‖Δm(s)xk-lρ,z1,z2…zn-1‖)]pk≥ε}∈I,forsomeρ>0,l∈Xand∀z1,z2…zn-1∈X∃K>0s.t.{n∈N:1n∑k=1n[M(‖Δm(s)xkρ,z1z2,…,zn-1‖)]pk≥K}∈I},(3.34)W0I(Δm(s),M,p,‖⋅,…,⋅‖)={(xk)∈w(n-X):{n∈N:1n∑k=1n[M(‖Δm(s)xkρ,z1,z2…zn-1‖)]pk≥ε}∈I,for someρ>0,l∈Xand∀z1,z2…zn-1∈X∃K>0s.t.{n∈N:1n∑k=1n[M(‖Δm(s)xkρ,z1z2,…,zn-1‖)]pk≥K}∈I},(3.35)W∞(Δm(s),M,p,‖⋅,…,⋅‖)={(xk)∈w(n-X):sup⁡n∈N1n∑k=1n[M(‖Δm(s)xkρ,z1z2,…,zn-1‖)]pk<∞,forsomeρ>0,l∈Xand∀z1,z2…zn-1∈X∃K>0s.t.{n∈N:1n∑k=1n[M(‖Δm(s)xkρ,z1z2,…,zn-1‖)]pk≥K}∈I},(3.36)W∞I(Δm(s),M,p,‖⋅,…,⋅‖)={(xk)∈w(n-X):∃K>0s.t.{n∈N:1n∑k=1n[M(‖Δm(s)xkρ,z1z2,…,zn-1‖)]pk≥K}∈I,forsomeρ>0,l∈Xand∀z1,z2…zn-1∈X∃K>0s.t.{n∈N:1n∑k=1n[M(‖Δm(s)xkρ,z1z2,…,zn-1‖)]pk≥K}∈I},Note 2.
Fors=0, we write the above spaces as Z(M,p,∥·,…,·∥), where Z=WI, W0I, W∞,  and W∞I.
It is clear from definitions that(3.37)W0I(Δm(s),M,p,‖⋅,…,⋅‖)⊂WI(Δm(s),M,p,‖⋅,…,⋅‖)⊂W∞(Δm(s),M,p,‖⋅,…,⋅‖).Corollary 3.9.
The sequence spacesZ(Δm(s),M,p,∥·,…,·∥) are paranormed spaces  (not totally paranormed ) with respect to the paranormed  hΔ  defined by
(3.38)hΔ(x)=inf⁡{ρpk/H:sup⁡kM(‖Δm(s)xkρ,z1,z2,…,zn-1‖)≤1,for someρ>0and∀z1,z2,…,zn-1∈X(‖Δm(s)xkρ,z1,z2,…,zn-1‖)},
where H=max{1,suppk}and Z=WI, W0I, W∞, and W∞I.Remark 3.10.
It is obvious that(xk)∈Z(Δms,M,p,∥·,…,·∥) if and only if Z(Δm(s),M,p,∥·,…,·∥), for Z=WI, W0I, W∞I  andW∞.
Also it is clear that the paranormgΔ and hΔ are equivalent.We state the following Theorem in view of the Lemma2.17.Theorem 3.11.
Let X be a standard n-normed space and{e1,e2,…,en} an orthogonal set in X. Then, the following hold:(a)
WI(Δms,M,p,∥·,…,·∥∞)=WI(Δms,M,p,∥·,…,·∥(n-1)),(b)
W0I(Δms,M,p,∥·,…,·∥∞)=W0I(Δms,M,p,∥·,…,·∥(n-1)),(c)
W∞(Δms,M,p,∥·,…,·∥∞)=W∞(Δms,M,p,∥·,…,·∥(n-1)),(d)
W∞I(Δms,M,p,∥·,…,·∥∞)=W∞I(Δms,M,p,∥·,…,·∥(n-1)),   where  ∥·,…,·∥∞ is the derived (n-1)-norm defined with respect to the  set  {e1,e2,…,en} and  ∥·,…,·∥(n-1) is the standard  (n-1)-norm on X.Remark 3.12.
Theorem3.11 also holds if we replace the difference operator Δms  by the difference operator Δm(s).Theorem 3.13.
The spacesZ(Δm(s),M,p,∥·,…,·∥) and Z(M,p,∥·,…,·∥) are equivalent as topological spaces, where  Z=WI,W0I,W∞I, and W∞.Proof.
Consider the mappingT:Z(Δm(s),M,p,∥·,…,·∥)→Z(M,p,∥·,…,·∥) defined by
(3.39)Tx=(Δ(s)xk),for eachx=(xk)∈Z(Δm(s),M,p,‖⋅,…,⋅‖).
Then, clearly,T is a linear homeomorphism and the proof follows.



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*Source: 317423-2011-06-16.xml*