# Some New Double-Sequence Spaces in 2-Normed Spaces Defined by Ideal Convergence and an Orlicz Function

Authors: Orhan Tuğ; Mutlay Doğan; Abdullah Kurudirek
Journal: ISRN Mathematical Analysis
 (2012)

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## Abstract

We generalize some sequence spaces from single to double, we study some topological properties of these double sequence spaces by using ideal convergence, difference sequence spaces, and an Orlicz function in 2-normed spaces, and we give some results related to these sequence spaces.


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## Body

## 1. Introduction

The concept ofI-convergent was introduced by Das et al. [1] and developed by many scholars. B. Tripathy and B. C. Tripathy [2] extended the concept of I-convergence from single sequence to double sequence. Some difference double-sequence spaces and paranormed double-sequence spaces defined by Orlicz function were introduced by Tripathy and Sarma [3, 4]. And also some new single- and double-sequence spaces defined in 2-normed spaces using ideal convergence and an Orlicz function were introduced by Savaş [5, 6].Recall that an Orlicz Function, which was presented by Karasnoselskii and Rutishky [7] M:[0,∞)→[0,∞) is continuous, convex, nondecreasing function such that M(0)=0 and M(x)>0 for x>0 and M(x)→∞ as x→∞.An Orlicz functionM can be represented in the following integral form: M(x)=∫0xp(t)dt, where p is the known kernel of M, right differential for t≥0,p(0)=0,p(t)>0 for t>0,p is nondecreasing, and p(t)→∞ as t→∞.Ruckle [8] and Maddox [9] presented and discussed that if convexity of Orlicz function M is replaced by M(x+y)≤M(x)+M(y) then this function is called Modulus function.An Orlicz functionM is said to satisfy Δ2-condition for all values of u, if there exists constant K>0, such that M(2u)≤KM(u),(u≥0). The Δ2-condition is equivalent to the inequality M(Lu)≤KLM(u) for all values of u and for L>1 being satisfied [7].The notion of difference double-sequence spaces was introduced by Tripathy and Sarma [3]. These notions are further studied by Tripathy and Sarma [4]. Let (ank) be a double sequence. Then the operator Δ is defined as Δank=an,k-an,k+1-an+1,k+an+1,k+1, for all k,n∈ℕ.Let us recall some well-known concepts that(X,∥·∥) is a normed space and a sequence (xmn)(m,n∈ℕ) of elements of X is called to be statistically convergent to x∈X if the set A(ɛ)={m,n∈ℕ:∥xmn-x∥≥ɛ}has zero natural density for each ɛ>0 [2].A real double-sequencex=(xk,l) is said to be statistically bounded above if there exists a real number D>0 such that δ{k:xk,l>D}=0. A real double-sequence x=(xk,l) is said to be statistically bounded below if there exists a real number E>0 such that δ{k:xk,l<E}=0. If a real double-sequence x=(xk,l) is statistically bounded both above and below, then we say that x=(xk,l) is statistically bounded. It is clear that any bounded double-sequence is also statistically bounded [10].A familyI⊂2Y of subset of a nonempty set Y is said to be an ideal in Y if(i)
∅∈Y,(ii)
A,B∈I imply A∪B∈I,(iii)
A∈I,B⊂A imply B∈I [11, 12].I⊂2Y is called an admissible ideal in Y if and only if it contains {{y}:y∈Y} [13].LetI⊂2ℕ×ℕ be a nontrivial ideal (i.e.,I≠∅ and ℕ×ℕ∉I) in N. The sequence (xmn)(m,n∈ℕ) is said to be I-convergent to x∈X if for each ɛ>0 the set A(ɛ)={m,n∈ℕ:∥xmn-x∥≥ɛ}∈I [1].A real double-sequencex=(xk,l) is said to be I-bounded above if there exists a real number G>0 such that {k:xk,l>G}∈I. A real double-sequence x=(xk,l) is said to be I-bounded below if there exists a real number H>0 such that {k:xk,l<H}∈I. If a real double-sequence x=(xk,l) is I-bounded both above and below, then we say that x=(xk,l) is I-bounded. One can observe easily that any bounded double sequence is I-bounded [14].LetX be a real vector space of dimension d, where 2≤d<∞. A 2-norm on X is a function ∥·,·∥:X×X→ℝ which satisfied the following four conditions:(i)
∥x,y∥=0 if and only if x and y are linear dependent,(ii)
∥x,y∥=∥y,x∥,(iii)
∥αx,y∥=|α|∥x,y∥,α∈R,(iv)
∥x,y+z∥≤∥x,y∥+∥x,z∥, the pair(X,∥·,·∥) is then called a 2-normed space [15, 16].The sequence(xk)k∈ℕ in a 2-normed space (X,∥·,·∥) is said to be convergent to L in X if limk→∞∥xk-L,z∥=0 for every z∈X. In this instance, we write limk→∞∥xk,z∥=∥L,z∥ [17].LetI⊂2ℕ be a nontrivial ideal in ℕ. The sequence (xk)k∈ℕ in a2-normed space X is said to be I-convergent to x, if for every ɛ>0 and z in X the set A(ɛ)={k∈ℕ:∥xk-x,z∥≥ɛ} belongs to I [18].The following inequalities which will be used throughout the paper can be introduced like in Maddox [19]. If 0≤pk,l≤supk,lpk,l=H,D=max(1,2H-1) then |ak,l+bk,l|pk,l≤D{|ak,l|pk,l+|bk,l|pk,l} for all k,l and ak,l,bk,l∈ℂ. And also |a|pk,l≤max(1,2H) for all a∈ℂ.


## 2. Main Results

Letλ=(λn) and μ=(μm) be two nondecreasing sequences of positive real numbers both of which tend to ∞ as n and m approach ∞, respectively. Also let λn+1≤λn+1,λ1=0 and μm+1≤μm+1,μ1=0. Let I be an admissible ideal of ℕ×ℕ, M an Orlicz function, and (X,∥·,·∥) a 2-normed spaces. In addition let p=pk,l be a bounded sequence of positive real numbers. With S′′(2-X), we symbolize the space of all double sequences defined over (X,∥·,·∥). Now we define the following double sequence spaces:(2.1)(WI)2(λ,M,Δ,p,‖⋅,⋅‖)={x∈S′′(2-X):∀ɛ>0{m,n∈N:1λ¯n,m∑k,l∈In,m[M(‖Δxk,l-Lρ,z‖)]pk,l≥ɛ}∈Iforsomeρ>0,L∈Xandeachz∈X{m,n∈N:1λ¯n,m∑k,l∈In,m[M(‖Δxk,l-Lρ,z‖)]pk,l≥ɛ}},(W0I)2(λ,M,Δ,p,‖⋅,⋅‖)={x∈S′′(2-X):∀ɛ>0{m,n∈N:1λ¯n,m∑k,l∈In,m[M(‖Δxk,lρ,z‖)]pk,l≥ɛ}∈Ifor  someρ>0and  eachz∈X{m,n∈N:1λ¯n,m∑k,l∈In,m[M(‖Δxk,l-Lρ,z‖)]pk,l≥ɛ}},(W∞)2(λ,M,Δ,p,‖⋅,⋅‖)={x∈S′′(2-X):∃K>0s.t.sup⁡m,n1λ¯n,m∑k,l∈In,m[M(‖Δxk,lρ,z‖)]pk,l≤Kfor  someρ>0and  eachz∈X{m,n∈N:1λ¯n,m∑k,l∈In,m[M(‖Δxk,l-Lρ,z‖)]pk,l≥ɛ}},(W∞I)2(λ,M,Δ,p,‖⋅,⋅‖)={x∈S′′(2-X):∃K>0s.t.{m,n∈N:1λ¯n,m∑k,l∈In,m[M(‖Δxk,lρ,z‖)]pk,l≥K}∈Ifor  someρ>0and eachz∈X{m,n∈N:1λ¯n,m∑k,l∈In,m[M(‖Δxk,l-Lρ,z‖)]pk,l≥ɛ}},whereIn=[n-λn+1,n],  Im=[m-μm+1,m], and λ¯n,m=λnμm. Throughout this paper we will denote λnμm by λ¯n,m and (k∈In,l∈Im) by (k,l)∈I¯n,m.Theorem 2.1.
(WI)2(λ,M,Δ,p,∥·,·∥),(W0I)2(λ,M,Δ,p,∥·,·∥),(W∞I)2(λ,M,Δ,p,∥·,·∥) are linear spaces.Proof.
Because all the statements can be proved in a similar way, we will prove the assertion for(W0I)2(λ,M,Δ,p,∥·,·∥). Suppose that x,y∈(W0I)2(λ,M,Δ,p,∥·,·∥) and α,β∈ℝ. So
(2.2){m,n∈N:1λ¯n,m∑k,l∈In,m[M(‖Δxk,lρ1,z‖)]pk,l≥ɛfor  someρ1>0}∈I,{m,n∈N:1λ¯n,m∑k,l∈In,m[M(‖Δyk,lρ2,z‖)]pk,l≥ɛfor  someρ2>0}∈I.
Since we study (X,∥·,·∥)  2-normed, Δ is linear and M is an Orlicz function, we have the following inequality:
(2.3)1λ¯n,m∑k,l∈In,m[M(‖Δ(αxk,l+βyk,l)(|α|ρ1+|β|ρ2),z‖)]pk,l≤D1λ¯n,m∑k,l∈In,m[|α|(|α|ρ1+|β|ρ2)M(‖Δxk,lρ1,z‖)]pk,l+D1λ¯n,m∑k,l∈In,m[|β|(|α|ρ1+|β|ρ2)M(‖Δyk,lρ2,z‖)]pk,l≤DF1λ¯n,m∑k,l∈In,m[M(‖Δxk,lρ1,z‖)]pk,l+DF1λ¯n,m∑k,l∈In,m[M(‖Δyk,lρ2,z‖)]pk,l,
where F=max[1,(|α|/(|α|ρ1+|β|ρ2))H,(|β|/(|α|ρ1+|β|ρ2))H].
From the above inequality we get(2.4){m,n∈N:1λ¯n,m∑k,l∈In,m[M(‖Δ(αxk,l+βyk,l)(|α|ρ1+|β|ρ2),z‖)]pk,l≥ɛ}⊆{m,n∈N:DF1λ¯n,m∑k,l∈In,m[M(‖Δxk,lρ1,z‖)]pk,l≥ɛ2}∪{m,n∈N:DF1λ¯n,m∑k,l∈In,m[M(‖Δyk,lρ2,z‖)]pk,l≥ɛ2}
the two set on the right side belonging to I and this completes the proof.Theorem 2.2.
For any fixedm,n∈ℕ,(W∞)2(λ,M,Δ,p,∥·,·∥) is a paranormed space with the paranorm defined by
(2.5)gm,n(x)=inf⁡z∈X1λ¯n,m∑k,l∈In,m‖xk,l,z‖+inf⁡{ρpmn/H:s.t.sup⁡k,l[M(‖Δxk,lρ,z‖)]pk,l≤1ρ>0,∀z∈X}.Proof.
(i)gm,n(0)=0 and (ii) gm,n(-x)=gm,n(x) are easy to prove. So we leave them out. (iii) Let us take x,y∈(W∞)2(λ,M,Δ,p,∥·,·∥). Let
(2.6)A(x)={ρ>0:sup⁡k,l[M(‖Δxk,lρ,z‖)]pk,l≤1,∀z∈X},A(y)={ρ>0:sup⁡k,l[M(‖Δyk,lρ,z‖)]pk,l≤1,∀z∈X}.
Letρ1∈A(x) and ρ2∈A(y). Then if ρ=ρ1+ρ2 then we have
(2.7)M(‖Δ(xk,l+yk,l)ρ,z‖)≤ρ1ρ1+ρ2M(‖Δxk,lρ1,z‖)+ρ2ρ1+ρ2M(‖Δyk,lρ2,z‖).
Thus
(2.8)sup⁡k,lM(‖Δ(xk,l+yk,l)ρ1+ρ2,z‖)≤1,gm,n(x+y)=inf⁡z∈X1λ¯n,m∑k,l∈In,m‖xk,l+yk,l,z‖+inf⁡{(ρ1+ρ2)pmn/H:ρ1∈A(x),ρ2∈A(y)}≤inf⁡z∈X1λ¯n,m∑k,l∈In,m‖xk,l,z‖+inf⁡{(ρ1)pmn/H:ρ1∈A(x)}+inf⁡z∈X1λ¯n,m∑k,l∈In,m‖yk,l,z‖+inf⁡{(ρ2)pmn/H:ρ2∈A(y)}=gm,n(x)+gm,n(y).
(iv) Let σi→σ, where σ,σi∈ℂ and gm,n(xk,li-x)→0 as i→∞. We have to show that gm,n(σixk,li-σx)→0 as i→∞. Let
(2.9)A(xi)={ρi>0:sup⁡k,l[M(‖Δxk,liρi,z‖)]pk,l≤1,∀z∈X},A(xi-x)={ρi′>0:sup⁡k,l[M(‖(Δxk,li-xk,l)ρi′,z‖)]pk,l≤1,∀z∈X}.
If ρi∈A(xi) and ρi′∈A(xi-x) then we observe that
(2.10)M(‖Δ(σixk,li-σxk,l)ρi|σi-σ|+ρi′|σ|,z‖)≤M(‖Δ(σixk,li-σxk,li)ρi|σi-σ|+ρi′|σ|,z‖+‖Δ(σxk,li-σxk,li)ρi|σi-σ|+ρi′|σ|,z‖)≤ρi|σi-σ|ρi|σi-σ|+ρi′|σ|M(‖Δxk,liρi,z‖)+ρi′|σ|ρi|σi-σ|+ρi′|σ|M(‖Δ(xk,li-xk,l)ρi′,z‖).
From above inequality it obviously follows that
(2.11)(M(‖Δ(σixk,li-σxk,l)ρi|σi-σ|+ρi′|σ|,z‖))pk,l≤1
and consequently
(2.12)gm,n(σixi-σx)≤inf⁡z∈X1λ¯n,m∑k,l∈In,m‖σixk,li-σxk,l,z‖+inf⁡{(ρi|σi-σ|+ρi′|σ|)pmn/H:ρi∈A(xi),ρi′∈A(xi-x)}≤|σi-σ|inf⁡z∈X1λ¯n,m∑k,l∈In,m‖xk,li,z‖+|σ|inf⁡z∈X1λ¯n,m∑k,l∈In,m‖xk,li-xk,l,z‖+(|σi-σ|)pmn/Hinf⁡{(ρi)pmn/H:ρi∈A(xi)}+(|σ|)pmn/Hinf⁡{(ρi′)pmn/H:ρi′∈A(xi-x)}≤max⁡{|σi-σ|,(|σi-σ|)pmn/H}gm,n(xi)+max⁡{|σ|,(|σ|)pmn/H}gm,n(xi-x).
Note thatgm,n(xi)<gm,n(x)+gm,n(xi-x) for all i∈ℕ. Hence by our assumption the right-hand side tends to 0 as i→∞ and the results follow. This completes the proof of the theorem.Theorem 2.3.
LetM be an Orlicz function which satisfies the Δ2-condition. Then (W0I)2(λ,M,Δ,p,∥·,·∥)⊂(WI)2(λ,M,Δ,p,∥·,·∥)⊂(W∞I)2(λ,M,Δ,p,∥·,·∥) and the inclusions are strict.Proof.
Because it can be proven in a similar way, we introduce the proof for(WI)2(λ,M,Δ,p,∥·,·∥)⊂(W∞I)2(λ,M,Δ,p,∥·,·∥) only. Let x∈(WI)2(λ,M,Δ,p,∥·,·∥). Then
(2.13)1λ¯n,m∑k,l∈In,m[M(‖Δxk,l2ρ,z‖)]pk,l=1λ¯n,m∑k,l∈In,m[M(‖Δxk,l+L-L2ρ,z‖)]pk,l≤1λ¯n,m∑k,l∈In,m[M(‖Δxk,l-L2ρ,z‖+‖L2ρ,z‖)]pk,l≤DG1λ¯n,m∑k,l∈In,m[M(‖Δxk,l-Lρ,z‖)]pk,l+DG1λ¯n,m∑k,l∈In,m[M(‖Lρ,z‖)],pk,l
where G=max{1,(1/2)H} and we get from Δ2-condition
(2.14)DG1λ¯n,m∑k,l∈In,m([M(‖Δxk,l-Lρ,z‖)]pk,l+[Kρδ‖L,z‖M(2)]H),
so x∈(W∞I)2(λ,M,Δ,p,∥·,·∥) and this completes the proof.Theorem 2.4.
LetX(Δ) stand for (WI)2(λ,M,Δ,p,∥·,·∥),(W0I)2(λ,M,Δ,p,∥·,·∥) or (W∞I)2(λ,M,Δ,p,∥·,·∥). Then the inclusion X(Δ)⊂X(Δ2) is strict.Proof.
Because this can be proven in a similar way, we will give the proof for(W0I)2(λ,M,Δ,p,∥·,·∥) only. Let x∈(W0I)2(λ,M,Δ,p,∥·,·∥). Then given ɛ>0 we have
(2.15){m,n∈N:1λ¯n,m∑k,l∈In,m[M(‖Δxk,lρ,z‖)]pk,l≥ɛfor  someρ>0}∈I.
Since M is nondecreasing and convex it follows that
(2.16)1λ¯n,m∑k,l∈In,m[M(‖Δ2xk,l4ρ,z‖)]pk,l=1λ¯n,m∑k,l∈In,m[M(‖Δxk,l-Δxk+1,l-Δxk,l+1+Δxk+1,l+14ρ,z‖)]pk,l≤1λ¯n,m∑k,l∈In,m[M(‖Δxk,l-Δxk+1,l4ρ,z‖+‖Δxk+1,l+1-Δxk,l+14ρ,z‖)]pk,l≤DG1λ¯n,m∑k,l∈In,m[M(‖Δxk,l-Δxk+1,l2ρ,z‖)]pk,l+DG1λ¯n,m∑k,l∈In,m[M(‖Δxk+1,l+1-Δxk,l+12ρ,z‖)]pk,l,
where G=max{1,(1/2)H}. Similarly from the convexity and nondecreasing properties of M(2.17)DG1λ¯n,m∑k,l∈In,m([M(‖Δxk,l-Δxk+1,l2ρ,z‖)]pk,l+[M(‖Δxk+1,l+1-Δxk,l+12ρ,z‖)]pk,l)≤DG1λ¯n,m∑k,l∈In,m[M(‖Δxk,l2ρ,z‖+‖Δxk+1,l2ρ,z‖)]pk,l+DG1λ¯n,m∑k,l∈In,m[M(‖Δxk+1,l+12ρ,z‖+‖Δxk,l+12ρ,z‖)]pk,l≤D2G21λ¯n,m∑k,l∈In,m([M(‖Δxk,lρ,z‖)]pk,l+[M(‖Δxk+1,lρ,z‖)]pk,l)+D2G21λ¯n,m∑k,l∈In,m([M(‖Δxk,l+1ρ,z‖)]pk,l+[M(‖Δxk+1,l+1ρ,z‖)]pk,l),{m,n∈N:1λ¯n,m∑k,l∈In,m[M(‖Δ2xk,lρ,z‖)]pk,l≥ɛ}⊆{m,n∈N:D2G21λ¯n,m∑k,l∈In,m[M(‖Δxk,lρ,z‖)]pk,l≥ɛ4}∪{m,n∈N:D2G21λ¯n,m∑k,l∈In,m[M(‖Δxk+1,lρ,z‖)]pk,l≥ɛ4}∪{m,n∈N:D2G21λ¯n,m∑k,l∈In,m[M(‖Δxk,l+1ρ,z‖)]pk,l≥ɛ4}∪{m,n∈N:D2G21λ¯n,m∑k,l∈In,m[M(‖Δxk+1,l+1ρ,z‖)]pk,l≥ɛ4}.
Since the right-hand side belongs toI so does the left hand side.Theorem 2.5.
LetM,M1,M2 be Orlicz function. Then one has(i)
(W0I)2(λ,M1,Δ,p,∥·,·∥)⊆(W0I)2(λ,MoM1,Δ,p,∥·,·∥) provided (pk,l) is such that H0=infpk,l>0,(ii)
(W0I)2(λ,M1,Δ,p,∥·,·∥)∩(W0I)2(λ,M2,Δ,p,∥·,·∥)⊆(W0I)2(λ,M1+M2,Δ,p,∥·,·∥).Proof.
(i) For givenɛ>0, we can choose ɛ0>0 such that max{ɛ0H,ɛ0H0}<ɛ. Because of the continuity of M, we can choose 0<δ<1 such that 0<t<δ⇒M(t)<ɛ0. Let x∈(W0I)2(λ,M1,Δ,p,∥·,·∥). We can write from the definition
(2.18)A(δ)={m,n∈N:1λ¯n,m∑k,l∈In,m[M1(‖Δxk,lρ,z‖)]pk,l≥δH}∈I.
Hence ifm,n∉A(δ), then
(2.19)1λ¯n,m∑k,l∈In,m[M1(‖Δxk,lρ,z‖)]pk,l<δH,
that is,
(2.20)∑k,l∈In,m[M1(‖Δxk,lρ,z‖)]pk,l<λ¯n,mδH,
that is,
(2.21)[M1(‖Δxk,lρ,z‖)]pk,l<δH,∀k,l∈In,m,
that is,
(2.22)M1(‖Δxk,lρ,z‖)<δ,∀k,l∈In,m.
From this inequality using continuity of M we must have
(2.23)M(M1(‖Δxk,lρ,z‖))<ɛ0,∀k,l∈In,m,
which consequently implies that
(2.24)∑k,l∈In,m[M(M1(‖Δxk,lρ,z‖))]pk,l<λ¯n,mmax⁡{ɛ0H,ɛ0H0}<λ¯n,mɛ,
that is,
(2.25)1λ¯n,m∑k,l∈In,m[M(M1(‖Δxk,lρ,z‖))]pk,l<ɛ.
This show that
(2.26){m,n∈N:1λ¯n,m∑k,l∈In,m[M(M1(‖Δxk,lρ,z‖))]pk,l≥ɛ}⊂A(δ),
and from the definition ideal the left side belongs to I. This proves the result.
(ii)  Let(xk,l)∈(W0I)2(λ,M1,Δ,p,∥·,·∥)∩(W0I)2(λ,M2,Δ,p,∥·,·∥). Then the fact
(2.27)[(M1+M2)(‖Δxk,lρ,z‖)]pk,l≤D([M1(‖Δxk,lρ,z‖)]pk,l+[M2(‖Δxk,lρ,z‖)]pk,l)
give us the result. Thus (xk,l)∈(W0I)2(λ,M1+M2,Δ,p,∥·,·∥), hence the proof.Definition 2.6.
LetX be sequence space. Then X is called solid if (αkxk)∈X whenever (xk)∈X for all sequence (αk) of scalar with |αk|≤1 for all k∈ℕ.Theorem 2.7.
The sequence spaces(W0I)2(λ,M,Δ,p,∥·,·∥),(W∞I)2(λ,M,Δ,p,∥·,·∥) are solid.Proof.
We give the proof for(W0I)2(λ,M,Δ,p,∥·,·∥) only. Let (xk,l)∈(W0I)2(λ,M,Δ,p,∥·,·∥) and (αk,l) be a sequence of scalars such that |αk,l|≤1 for all k,l∈ℕ. Then we have
(2.28){m,n∈N:1λ¯n,m∑k,l∈In,m[M(‖Δ(αk,lxk,l)ρ,z‖)]pk,l≥ɛ}⊆{m,n∈N:Cλ¯n,m∑k,l∈In,m[M(‖Δxk,lρ,z‖)]pk,l≥ɛ}∈I,
where C=max{1,(|αk,l|)H}. Hence (αk,lxk,l)∈(W0I)2(λ,M,Δ,p,∥·,·∥) for all sequences of scalars (αk,l) with |αk,l|≤1 for all k,l∈ℕ whenever (xk,l)∈(W0I)2(λ,M,Δ,p,∥·,·∥).



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