# Some Almost Lacunary Double Sequence Spaces Defined by Orlicz Functions in 2-Normed Spaces

Authors: Orhan Tuğ
Journal: ISRN Mathematical Analysis
 (2012)

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## Abstract

Das and Patel (1989) introduced two new sequence spaces which are called lacunary almost convergent and lacunary strongly almost convergent sequence spaces. Móricz and Rhoades (1988) defined and studied almostP-convergent double sequences spaces. Savaş and Patterson (2005) introduce the almost lacunary strong P-convergent double sequence spaces by using Orlicz functions and examined some properties of these sequences spaces. In this paper, some almost lacunary double sequences spaces are given by using 2-normed spaces.


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## Body

## 1. Introduction

By the convergence of double sequence which is known the convergence on the Pringsheim sense, that is, a double sequencex=(xk,l) has Pringsheim limit L denoted by P-limx=L, provided given ϵ>0 there exists N∈ℕ such that |xk,l-L|<ϵ whenever k,l>N. We write briefly as P-convergent [1–3].Freedman et al. [4] presented a definition for lacunary refinement as follow: p={k¯r} is called a lacunary refinement of the lacunary sequence θ={kr} if {kr}⊆{k¯r} and studied many scholar [4–6]. Savaş and Patterson [7] gave some properties and theorem and also defined Sθr,s-P-convergence.By a lacunaryθ=(kr); r=0,1,2,… where k0=0, we shall mean an increasing sequence of non-negative integers with kr-kr-1→∞ as r→∞. The intervals determined by θ denoted by Ir=(kr-1,kr] and hr=kr-kr-1. The ratio kr/kr-1 is denoted by qr [4].An Orlicz Function, which was presented by Krasnoselskii and Rutisky [8], M:[0,∞)→[0,∞) is continuous, convex, non-decreasing 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 non-decreasing and p(t)→∞ as t→∞.Ruckle [9] and Maddox [10] described that if convexity of Orlicz function M is replaced by M(x+y)≤M(x)+M(y) then this function is called Modulus function.(X,∥·∥) be 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(ε)={m,n∈ℕ:∥xmn-x∥≥ε} has natural density zero for each ε>0 [11].LetX be a real vector space of dimension d, where 2≤d<∞. A 2-norm on X is a function ∥·,·∥:XxX→ℝ which satisfy 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 spaces [12, 13].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 case, we write limk→∞∥xk,z∥=∥L,z∥ [14].


## 2. Notations and Known Results

AlmostP-convergent sequences have been defined by Móricz and Rhoades [15] as follow:Definition 2.1.
A double sequencex=(xk,l) of real numbers is called almost P-convergent to a limit L if
(2.1)P-lim⁡p,q→∞sup⁡m,n≥01pq∑k=mm+p-1∑l=nn+q-1|xk,l-L|=0.
That is, the average(xk,l) take over any rectangle
(2.2){(k,l):m≤k≤m+p-1,n≤l≤n+q-1},
tends to L as both p and q tend to ∞, and this P-convergence is uniform in m and n. The set of sequence which satisfy this property was denoted as [ĉ2] by Savaş and Paterson [16].
We can define the set of almostP-convergent double sequence in (X,∥·,·∥) similar to above definition as follow:
(2.3)[ĉ2,‖⋅,⋅‖]={x=(xk,l):P-lim⁡p,q⟶∞sup⁡m,n≥01pq∑k=mm+p-1∑l=nn+q-1‖xk,l-L,z‖=0foreveryz∈X}.Definition 2.2.
LetM be an Orlicz function, p=(pk,l) be any factorable double sequence of strictly positive reel numbers and S′′(2-X) denote all double sequence in (X,∥·,·∥)  2-normed space we can define the following double sequence space
(2.4)[ĉ2,M,p,‖⋅,⋅‖]={x=(xk,l):P-lim⁡p,q1pq∑k,l=1,1p,q[M(‖xk+m,l+n-Lρ,z‖)]pk,l=0uniformlyinmandn,for someρ>0andL,andeveryz∈X}.
If we chooseM(x)=x and (pk,l)=1 for all k and l, then [ĉ2,M,p,∥·,·∥]=[ĉ2,∥·,·∥] which was defined above.Definition 2.3.
The double sequnceθr,s={kr,ls} is called double lacunary if there exist two increasing sequences of integers such that
(2.5)k0=0,hr=kr-kr-1⟶∞asr⟶∞,l0=0,hs¯=ls-ls-1⟶∞ass⟶∞.
Letkr,s=krls, hr,s=hrhs¯ and θr,s is defined by
(2.6)Ir,s={(k,l):kr-1<k≤krandls-1<l≤ls},qr=krkr-1,qs¯=lsls-1,qr,s=qrqs¯.Definition 2.4.
LetM be an Orlicz function, S′′(2-X) denote all double sequence in (X,∥·,·∥) 2-normed space, and p=(pk,l) be any factorable double sequence of strictly positive reel numbers, now we can define the following sequence spaces in (X,∥·,·∥) 2-normed space as follows:
(2.7)[ACθr,s,‖⋅,⋅‖]={x=(xk,l):P-lim⁡r,s1hr,s∑k,l∈Ir,s‖xk+m,l+n-L,z‖=0uniformlyinmandn,forsomeLandeveryz∈X}[ACθr,s,‖⋅,⋅‖]0={x=(xk,l):P-lim⁡r,s1hr,s∑k,l∈Ir,s‖xk+m,l+n,z‖=0uniformlyinmandnandeveryz∈X}[ACθr,s,M,p,‖⋅,⋅‖]={x=(xk,l):P-lim⁡r,s1hr,s∑k,l∈Ir,s[M(‖xk+m,l+n-Lρ,z‖)]pk,l=0uniformlyinmandn,forsomeρ>0andL,andeveryz∈X}[ACθr,s,M,p,‖⋅,⋅‖]0={x=(xk,l):P-lim⁡r,s1hr,s∑k,l∈Ir,s[M(‖xk+m,l+nρ,z‖)]pk,l=0uniformlyinmandn,forsomeρ>0andeveryz∈X}.
When(pk,l)=1 for all k and l, we shall denote [ACθr,s,M,p,∥·,·∥] and [ACθr,s,M,p,∥·,·∥]0 as [ACθr,s,M,∥·,·∥] and [ACθr,s,M,∥·,·∥]0. That is,
(2.8)[ACθr,sM,‖⋅,⋅‖]={x=(xk,l):P-lim⁡r,s1hr,s∑k,l∈Ir,s[M(‖xk+m,l+n-Lρ,z‖)]=0uniformlyinmandn,forsomeρ>0andL,andeveryz∈X},[ACθr,sM,‖⋅,⋅‖]0={x=(xk,l):P-lim⁡r,s1hr,s∑k,l∈Ir,s[M(‖xk+m,l+nρ,z‖)]=0uniformlyinmandn,for someρ>0andL,andeveryz∈X}.
Ifx∈[ACθr,sM,∥·,·∥] we shall say that x is almost lacunary strongly P-convergent with respect to the Orlicz function M in 2-normed space. In addition if M(x)=x and (pk,l)=1 for all k and l, then [ACθr,s,M,p,∥·,·∥]=[ACθr,s,∥·,·∥] and [ACθr,s,M,p,∥·,·∥]0=[ACθr,s,∥·,·∥]0 which are defined above. Also note that if (pk,l)=1 for all k and l, then [ACθr,s,M,p,∥·,·∥]=[ACθr,s,M,∥·,·∥] and [ACθr,s,M,p,∥·,·∥]0=[ACθr,s,M,∥·,·∥]0 which are defined above.
Let us generalized almostP-convergent double sequence to Orlicz function in 2-normed spaces.


## 3. Main Results

Theorem 3.1.
For any Orlicz functionM and a bounded factorable positive double sequence pk,l, [ACθr,s,M,p,∥·,·∥] and [ACθr,s,M,p,∥·,·∥]0 are linear spaces.Proof.
Suppose thatx, y∈[ACθr,s,M,p,∥·,·∥]0 and α, β∈ℝ. So we have
(3.1)Ar,s1={x=(xk,l):P-lim⁡r,s1hr,s∑k,l∈Ir,s[M(‖xk+m,l+nρ1,z‖)]pk,l=0uniformlyinmandn,forsomeρ1>0andeveryz∈X},Ar,s2={y=(yk,l):P-lim⁡r,s1hr,s∑k,l∈Ir,s[M(‖yk+m,l+nρ2,z‖)]pk,l=0uniformlyinmandn,forsomeρ2>0andeveryz∈X}.
Since M is an Orlicz function we have the following inequality
(3.2)1hr,s∑k,l∈Ir,s[M(‖αxk+m,l+n+βyk+m,l+n|α|ρ1+|β|ρ2,z‖)]pk,l≤D1hr,s∑k,l∈Ir,s[|α||α|ρ1+|β|ρ2M(‖xk+m,l+nρ1,z‖)]pk,l+D1hr,s∑k,l∈Ir,s[|β||α|ρ1+|β|ρ2M(‖yk+m,l+nρ2,z‖)]pk,l≤DF1hr,s∑k,l∈Ir,s[M(‖xk+m,l+nρ1,z‖)]pk,l+DF1hr,s∑k,l∈Ir,s[M(‖yk+m,l+nρ2,z‖)]pk,l,
where F=max[1,(|α|/(|α|ρ1+|β|ρ2))H,(|β|/(|α|ρ1+|β|ρ2))H]. When we take the limit of each side as r,s→∞(3.3){x=(xk,l),y=(yk,l):P-lim⁡r,s1hr,s∑k,l∈Ir,s[M(‖αxk+m,l+n+βyk+m,l+n|α|ρ1+|β|ρ2,z‖)]pk,l=ouniformlyinmandn,forsomeρ1,ρ2>0andeachz∈X}.
So this is the result.Definition 3.2.
An Orlicz functionM is said to be satisfy Δ2-condition for all values of u, if there exists a constant K>0 such that
(3.4)M(2u)≤KM(u)(u≥0).Lemma 3.3.
LetM be an Orlicz function which satisfies Δ2-condition and 0<δ<1. Then for each x≥δ and some constant K>0 we have
(3.5)M(x)≤Kδ-1M(2).Theorem 3.4.
For any Orlicz functionM which satisfies Δ2-condition, we have [ACθr,s,∥·,·∥]⊆[ACθr,sM,∥·,·∥]Proof.
Letx∈[ACθr,s,∥·,·∥]. For each m and n(3.6)Ar,s={x=(xk,l):P-lim⁡r,s1hr,s∑k,l∈Ir,s‖xk+m,l+n-L,z‖=0forsomeL,andeveryz∈X}.
Let ε>0 and choose δ with 0<δ<1 such that M(t)<ε for every t with 0≤t≤δ. For every z∈X, we get
(3.7)1hr,s∑k,l∈Ir,sM(‖xk+m,l+n-L,z‖)=1hr,s∑k,l∈Ir,s,‖xk+m,l+n-L,z‖≤δM(‖xk+m,l+n-L,z‖)+1hr,s∑k,l∈Ir,s‖xk+m,l+n-L,z‖>δM(‖xk+m,l+n-L,z‖)≤1hr,s(hr,sε)+1hr,s∑k,l∈Ir,s,‖xk+m,l+n-L,z‖>δM(‖xk+m,l+n-L,z‖)<1hr,s(hr,sε)+1hr,sKδ-1M(2)hr,sAr,s.
From Lemma 3.3 as r and s goes to infinity in Pringsheim sense, for each m and n we are granted x∈[ACθr,sM,∥·,·∥].Theorem 3.5.
Letθr,s={kr,ls} be a double lacunary sequence with liminfrqr>1 and liminfsqs¯>1 then for any Orlicz function M, [ĉ2,M,p,∥·,·∥]⊂[ACθr,s,M,p,∥·,·∥].Proof.
Sinceliminfrqr>1 and liminfsqs¯>1, then there exists δ>0 such that qr>1+δ and qs¯>1+δ. This mean hr/kr≥δ/(1+δ), h¯s/ls≥δ/(1+δ). Then for x∈[ĉ2,M,p,∥·,·∥], we can write for each m and n, some L and every z∈X(3.8)Br,s=1hr,s∑k,l∈Ir,s[M(‖xk+m,l+n-Lρ,z‖)]pk,l=1hr,s∑k=1kr∑l=1ls[M(‖xk+m,l+n-Lρ,z‖)]pk,l-1hr,s∑k=1kr-1∑l=1ls-1[M(‖xk+m,l+n-Lρ,z‖)]pk,l-1hr,s∑k=kr-1+1kr∑l=1ls-1[M(‖xk+m,l+n-Lρ,z‖)]pk,l-1hr,s∑k=1kr-1∑l=ls-1+1ls[M(‖xk+m,l+n-Lρ,z‖)]pk,l-krlshr,s(1krls∑k=1kr∑l=1ls[M(‖xk+m,l+n-Lρ,z‖)]pk,l)-krlshr,s(1krls∑k=1kr∑l=1ls[M(‖xk+m,l+n-Lρ,z‖)]pk,l)=krlshr,s(1krls∑k=1kr∑l=1ls[M(‖xk+m,l+n-Lρ,z‖)]pk,l)-kr-1ls-1hr,s(1kr-1ls-1∑k=1kr-1∑l=1ls-1[M(‖xk+m,l+n-Lρ,z‖)]pk,l)-1hr∑k=kr-1+1krls-1hs1ls-1∑l=1ls-1[M(‖xk+m,l+n-Lρ,z‖)]pk,l-1hs∑l=ls-1+1lskr-1hr1kr-1∑k=1kr-1[M(‖xk+m,l+n-Lρ,z‖)]pk,l.
Since x∈[ĉ2,M,p,∥·,·∥] the last two terms tends to 0 uniformly in m and n in Pringsheim sense. Thus for each m and n(3.9)Br,s=krlshr,s(1krls∑k=1kr∑l=1ls[M(‖xk+m,l+n-Lρ,z‖)]pk,l)-kr-1ls-1hr,s(1kr-1ls-1∑k=1kr-1∑l=1ls-1[M(‖xk+m,l+n-Lρ,z‖)]pk,l)+o(1).
Since hr,s=krls-kr-1ls-1 we get for each m and n the following inequalities as follow:
(3.10)krlshr,s≤1+δδ,kr-1ls-1hr,s≤1δ.
Thus the terms
(3.11)1krls∑k=1kr∑l=1ls[M(‖xk+m,l+n-Lρ,z‖)]pk,l,1kr-1ls-1∑k=1kr-1∑l=1ls-1[M(‖xk+m,l+n-Lρ,z‖)]pk,l,
are both convergent to L in Pringsheim sense for all m and n, every z∈X and some ρ>0. Therefore Br,s is a convergent sequence in Pringsheim sense for each m and n. So x∈[ACθr,s,M,p,∥·,·∥] and this is the proof.Theorem 3.6.
Letθr,s={kr,ls} be a double lacunary sequence with limsuprqr<∞ and limsupsqs¯<∞ then for any Orlicz function M, [ACθr,s,M,p,∥·,·∥]⊂[ĉ2,M,p,∥·,·∥].Proof.
Sincelimsuprqr<∞ and limsupsqs¯<∞ there exists H>0 such that qr<H and qs¯<H for all r and s. Let ε>0 and x∈[ACθr,s,M,p,∥·,·∥]. There exists r0>0 and s0>0 such that for every i≥r0 and j≥s0, and all m and n, for every z∈X(3.12)Ai,j′=1hi,j∑k,l∈Ii,j[M(‖xk+m,l+nρ,z‖)]pk,l<ε.
Let
(3.13)M′=max⁡{Ai,j′:1≤i≤r0,1≤j≤s0}
and let kr-1<p≤kr and ls-1<q≤ls. Hence we get
(3.14)1pq∑k,l∈1,1p,q[M(‖xk+m,l+nρ,z‖)]pk,l≤1kr-1ls-1∑k=1kr∑l=1ls[M(‖xk+m,l+nρ,z‖)]pk,l≤1kr-1ls-1∑t,u=1,1r0,s0(∑k,l∈It,u[M(‖xk+m,l+nρ,z‖)]pk,l)=1kr-1ls-1∑t,u=1,1r,sht,uAt,u′+1kr-1ls-1∑(r0<t≤r)∪(s0<u≤s)ht,uAt,u′≤M′kr-1ls-1∑t,u=1,1r0,s0ht,u+1kr-1ls-1∑(r0<t≤r)∪(s0<u≤s)ht,uAt,u′≤M′kr0ls0r0s0kr-1ls-1+1kr-1ls-1∑(r0<t≤r)∪(s0<u≤s)ht,uAt,u′≤M′kr0ls0r0s0kr-1ls-1+(sup⁡t≥r0∪u≥s0At,u′)1kr-1ls-1∑(r0<t≤r)∪(s0<u≤s)ht,u≤M′kr0ls0r0s0kr-1ls-1+1kr-1ls-1ε∑(r0<t≤r)∪(s0<u≤s)ht,u≤M′kr0ls0r0s0kr-1ls-1+εH2.
Since kr and ls both tends to infinity as both p and q tends to infinity, uniformly in m and n, and for every z∈X,
(3.15)1pq∑k,l∈1,1p,q[M(‖xk+m,l+nρ,z‖)]pk,l⟶0.
Therefore x∈[ĉ2,M,p,∥·,·∥].The following theorem is a result of Theorems3.4 and 3.5.Theorem 3.7.
Letθr,s={kr,ls} be a double lacunary sequence with 1<liminfr,sqr,s≤limsupr,sqr,s<∞, then for any Orlicz function M[ACθr,s,M,p,∥·,·∥]=[ĉ2,M,p,∥·,·∥].Gähler [12] defined almost lacunary statistical convergence for single sequence, then Savaş and Patterson [16] defined almost lacunary statistical convergence for double sequence by combining lacunary sequence and almost convergence. Now we can define this definition in 2-normed space as follow:Definition 3.8.
Letθr,s be a double lacunary sequence; the double number sequence x is S¯θr,s-P-convergent to L provided that for every ε>0 and z∈X(3.16)P-lim⁡r,s1hr,smax⁡|{(k,l)∈Ir,s:‖xk+m,l+n-L,z‖≥ε}|=0.
So we can write S¯θr,s-limx=L.Theorem 3.9.
Letθr,s be a double lacunary sequence then(1)
xk,l→PL[ACθr,s,∥·,·∥] implies  xk,l→PL[S¯θr,s,∥·,·∥](2)
[ACθr,s,∥·,·∥] is a proper subset of  [S¯θr,s,∥·,·∥](3)
Ifx∈(l∞)2 and  xk,l→PL[S¯θr,s,∥·,·∥] then  xk,l→PL[ACθr,s,∥·,·∥](4)
[S¯θr,s,∥·,·∥]∩(l∞)2=[ACθr,s,∥·,·∥]∩(l∞)2where  (l∞)2 is the space of all bounded double sequence.Proof.
(1) Since for allm and n, and every z∈X(3.17)|{(k,l)∈Ir,s:‖xk+m,l+n-L,z‖≥ε}|≤∑k,l∈Ir,s,‖xk+m,l+n-L,z‖≥ε‖xk+m,l+n-L,z‖≤∑k,l∈Ir,s‖xk+m,l+n-L,z‖,
and for all m and n(3.18)P-lim⁡r,s1hr,s∑k,l∈Ir,s‖xk+m,l+n-L,z‖=0.
This show that for all m and n(3.19)P-lim⁡r,s1hr,s|{(k,l)∈Ir,s:‖xk+m,l+n-L,z‖≥ε}|=0,
this completes the proof of (1)(2)
Letx be defined as follows:(3.20)xk,l=(123⋯[hr,s3]0⋅⋅⋅223⋯[hr,s3]0⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅2[hr,s3][hr,s3]⋯[hr,s3]0⋅⋅⋅000000⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅).
It is obvious thatx is an unbounded double sequence and for ε>0, for all m and n, and for every z∈X(3.21)P-lim⁡r,s1hr,s|{(k,l)∈Ir,s:‖xk+m,l+n-L,z‖≥ε}|=P-lim⁡r,s[hr,s3]hr,s=0.
Thus xk,l→P0[S¯θr,s,∥·,·∥]. But
(3.22)P-lim⁡r,s1hr,s∑k,l∈Ir,s‖xk,l,z‖=P-lim⁡r,s[hr,s3]([hr,s3]([hr,s3]+1))2hr,s=12.
Therefore xk,l↛PL[ACθr,s,∥·,·∥] which is the proof of (2).(3)
Letx∈(l∞)2 and xk,l→PL[S¯θr,s,∥·,·∥]. Assuming that for all m and n, and every z∈X∥xk+m,l+n-L,z∥≤K for all K. And also for given ε>0 and r and s large for all m and n, and every z∈X we get the following inequality as follow:(3.23)1hr,s∑k,l∈Ir,s‖xk+m,l+n-L,z‖=1hr,s∑k,l∈Ir,s,‖xk+m,l+n-L,z‖≥ε‖xk+m,l+n-L,z‖+1hr,s∑k,l∈Ir,s,‖xk+m,l+n-L,z‖<ε‖xk+m,l+n-L,z‖≤Khr,s|{(k,l)∈Ir,s:‖xk+m,l+n-L,z‖≥ε}|+ε.
Therefore x∈(l∞)2 and xk,l→PL[S¯θr,s,∥·,·∥], this shows that xk,l→PL[ACθr,s,∥·,·∥].(4)
from (1), (2) and (3), we get[S¯θr,s,∥·,·∥]∩(l∞)2=[ACθr,s,∥·,·∥]∩(l∞)2.Theorem 3.10.
For any Orlicz functionM, [ACθr,s,M,∥·,·∥]⊂[S¯θr,s,∥·,·∥]Proof.
Letx∈[ACθr,s,M,∥·,·∥] and ε>0. Then for all m and n, and every z∈X(3.24)1hr,s∑k,l∈Ir,sM(‖xk+m,l+n-Lρ,z‖)≥1hr,s∑k,l∈Ir,s,‖xk+m,l+n-L‖≥εM(‖xk+m,l+n-Lρ,z‖)>1hr,sM(ερ)|{(k,l)∈Ir,s:‖xk+m,l+n-L,z‖≥ε}|.
This shows that x∈[S¯θr,s,∥·,·∥].



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