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 almost P-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.
1. Introduction
By the convergence of double sequence which is known the convergence on the Pringsheim sense, that is, a double sequence x=(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 lacunary θ=(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 function M 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].
Let X 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;
∥x,y∥=0 if and only if x and y are linear dependent.
∥x,y∥=∥y,x∥
∥αx,y∥=|α|∥x,y∥,α∈R
∥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
Almost P-convergent sequences have been defined by Móricz and Rhoades [15] as follow:
Definition 2.1.
A double sequence x=(xk,l) of real numbers is called almost P-convergent to a limit L if
P-limp,q→∞supm,n≥01pq∑k=mm+p-1∑l=nn+q-1|xk,l-L|=0.
That is, the average (xk,l) take over any rectangle
{(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 almost P-convergent double sequence in (X,∥·,·∥) similar to above definition as follow:
[ĉ2,‖⋅,⋅‖]={x=(xk,l):P-limp,q⟶∞supm,n≥01pq∑k=mm+p-1∑l=nn+q-1‖xk,l-L,z‖=0foreveryz∈X}.
Definition 2.2.
Let M 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
[ĉ2,M,p,‖⋅,⋅‖]={x=(xk,l):P-limp,q1pq∑k,l=1,1p,q[M(‖xk+m,l+n-Lρ,z‖)]pk,l=0uniformlyinmandn,for someρ>0andL,andeveryz∈X}.
If we choose M(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 sequnce θr,s={kr,ls} is called double lacunary if there exist two increasing sequences of integers such that
k0=0,hr=kr-kr-1⟶∞asr⟶∞,l0=0,hs¯=ls-ls-1⟶∞ass⟶∞.
Let kr,s=krls, hr,s=hrhs¯ and θr,s is defined by
Ir,s={(k,l):kr-1<k≤krandls-1<l≤ls},qr=krkr-1,qs¯=lsls-1,qr,s=qrqs¯.
Definition 2.4.
Let M 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:
[ACθr,s,‖⋅,⋅‖]={x=(xk,l):P-limr,s1hr,s∑k,l∈Ir,s‖xk+m,l+n-L,z‖=0uniformlyinmandn,forsomeLandeveryz∈X}[ACθr,s,‖⋅,⋅‖]0={x=(xk,l):P-limr,s1hr,s∑k,l∈Ir,s‖xk+m,l+n,z‖=0uniformlyinmandnandeveryz∈X}[ACθr,s,M,p,‖⋅,⋅‖]={x=(xk,l):P-limr,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-limr,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,
[ACθr,sM,‖⋅,⋅‖]={x=(xk,l):P-limr,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-limr,s1hr,s∑k,l∈Ir,s[M(‖xk+m,l+nρ,z‖)]=0uniformlyinmandn,for someρ>0andL,andeveryz∈X}.
If x∈[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 almost P-convergent double sequence to Orlicz function in 2-normed spaces.
3. Main ResultsTheorem 3.1.
For any Orlicz function M and a bounded factorable positive double sequence pk,l, [ACθr,s,M,p,∥·,·∥] and [ACθr,s,M,p,∥·,·∥]0 are linear spaces.
Proof.
Suppose that x, y∈[ACθr,s,M,p,∥·,·∥]0 and α, β∈ℝ. So we have
Ar,s1={x=(xk,l):P-limr,s1hr,s∑k,l∈Ir,s[M(‖xk+m,l+nρ1,z‖)]pk,l=0uniformlyinmandn,forsomeρ1>0andeveryz∈X},Ar,s2={y=(yk,l):P-limr,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
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→∞{x=(xk,l),y=(yk,l):P-limr,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 function M is said to be satisfy Δ2-condition for all values of u, if there exists a constant K>0 such that
M(2u)≤KM(u)(u≥0).
Lemma 3.3.
Let M be an Orlicz function which satisfies Δ2-condition and 0<δ<1. Then for each x≥δ and some constant K>0 we have
M(x)≤Kδ-1M(2).
Theorem 3.4.
For any Orlicz function M which satisfies Δ2-condition, we have [ACθr,s,∥·,·∥]⊆[ACθr,sM,∥·,·∥]
Proof.
Let x∈[ACθr,s,∥·,·∥]. For each m and nAr,s={x=(xk,l):P-limr,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
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.
Let θr,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.
Since liminfrqr>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∈XBr,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 nBr,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:
krlshr,s≤1+δδ,kr-1ls-1hr,s≤1δ.
Thus the terms
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.
Let θr,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.
Since limsuprqr<∞ 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∈XAi,j′=1hi,j∑k,l∈Ii,j[M(‖xk+m,l+nρ,z‖)]pk,l<ε.
Let
M′=max{Ai,j′:1≤i≤r0,1≤j≤s0}
and let kr-1<p≤kr and ls-1<q≤ls. Hence we get
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+(supt≥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,
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 Theorems 3.4 and 3.5.
Theorem 3.7.
Let θr,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.
Let θr,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∈XP-limr,s1hr,smax|{(k,l)∈Ir,s:‖xk+m,l+n-L,z‖≥ε}|=0.
So we can write S¯θr,s-limx=L.
[ACθr,s,∥·,·∥] is a proper subset of [S¯θr,s,∥·,·∥]
If x∈(l∞)2 and xk,l→PL[S¯θr,s,∥·,·∥] then xk,l→PL[ACθr,s,∥·,·∥]
[S¯θr,s,∥·,·∥]∩(l∞)2=[ACθr,s,∥·,·∥]∩(l∞)2
where (l∞)2 is the space of all bounded double sequence.
Proof.
(1) Since for all m and n, and every z∈X|{(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 nP-limr,s1hr,s∑k,l∈Ir,s‖xk+m,l+n-L,z‖=0.
This show that for all m and nP-limr,s1hr,s|{(k,l)∈Ir,s:‖xk+m,l+n-L,z‖≥ε}|=0,
this completes the proof of (1)
It is obvious that x is an unbounded double sequence and for ε>0, for all m and n, and for every z∈XP-limr,s1hr,s|{(k,l)∈Ir,s:‖xk+m,l+n-L,z‖≥ε}|=P-limr,s[hr,s3]hr,s=0.
Thus xk,l→P0[S¯θr,s,∥·,·∥]. But
P-limr,s1hr,s∑k,l∈Ir,s‖xk,l,z‖=P-limr,s[hr,s3]([hr,s3]([hr,s3]+1))2hr,s=12.
Therefore xk,l↛PL[ACθr,s,∥·,·∥] which is the proof of (2).
Let x∈(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:
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,∥·,·∥].
from (1), (2) and (3), we get [S¯θr,s,∥·,·∥]∩(l∞)2=[ACθr,s,∥·,·∥]∩(l∞)2.
Theorem 3.10.
For any Orlicz function M, [ACθr,s,M,∥·,·∥]⊂[S¯θr,s,∥·,·∥]
Proof.
Let x∈[ACθr,s,M,∥·,·∥] and ε>0. Then for all m and n, and every z∈X1hr,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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