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ISRN Mathematical Analysis
Volume 2012 (2012), Article ID 378575, 11 pages
Research Article

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

Ishik University, Erbil, Iraq

Received 16 November 2011; Accepted 25 December 2011

Academic Editor: O. Miyagaki

Copyright © 2012 Orhan Tuğ. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 𝑃lim𝑥=𝐿, provided given 𝜖>0 there exists 𝑁 such that |𝑥𝑘,𝑙𝐿|<𝜖 whenever 𝑘,𝑙>𝑁. We write briefly as 𝑃-convergent [13].

Freedman et al. [4] presented a definition for lacunary refinement as follow: 𝑝={𝑘𝑟} is called a lacunary refinement of the lacunary sequence 𝜃={𝑘𝑟} if {𝑘𝑟}{𝑘𝑟} and studied many scholar [46]. Savaş and Patterson [7] gave some properties and theorem and also defined 𝑆𝜃𝑟,𝑠𝑃-convergence.

By a lacunary 𝜃=(𝑘𝑟); 𝑟=0,1,2, where 𝑘0=0, we shall mean an increasing sequence of non-negative integers with 𝑘𝑟𝑘𝑟1 as 𝑟. The intervals determined by 𝜃 denoted by 𝐼𝑟=(𝑘𝑟1,𝑘𝑟] and 𝑟=𝑘𝑟𝑘𝑟1. The ratio 𝑘𝑟/𝑘𝑟1 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)=𝑥0𝑝(𝑡)𝑑𝑡 where 𝑝 is the known kernel of 𝑀, right differential for 𝑡0, 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) of elements of 𝑋 is called to be statistically convergent to 𝑥𝑋 if the set 𝐴(𝜀)={𝑚,𝑛𝑥𝑚𝑛𝑥𝜀} has natural density zero for each 𝜀>0 [11].

Let 𝑋 be a real vector space of dimension 𝑑, where 2𝑑<. A 2-norm on 𝑋 is a function ,𝑋𝑥𝑋 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,𝛼𝑅(iv)𝑥,𝑦+𝑧𝑥,𝑦+𝑥,𝑧

the pair (𝑋,,) is then called a 2-normed spaces [12, 13].

The sequence (𝑥𝑘)𝑘 in a 2-normed space (𝑋,,) is said to be convergent to 𝐿 in 𝑋 if lim𝑘𝑥𝑘𝐿,𝑧=0 for every 𝑧𝑋. In this case, we write lim𝑘𝑥𝑘,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 𝑃lim𝑝,𝑞sup𝑚,𝑛01𝑝𝑞𝑚+𝑝1𝑘=𝑚𝑛+𝑞1𝑙=𝑛||𝑥𝑘,𝑙||𝐿=0.(2.1)
That is, the average (𝑥𝑘,𝑙) take over any rectangle {(𝑘,𝑙)𝑚𝑘𝑚+𝑝1,𝑛𝑙𝑛+𝑞1},(2.2) 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 [̂𝑐2] by Savaş and Paterson [16].
We can define the set of almost 𝑃-convergent double sequence in (𝑋,,) similar to above definition as follow: ̂𝑐2=𝑥,,𝑥=𝑘,𝑙𝑃lim𝑝,𝑞sup𝑚,𝑛01𝑝𝑞𝑚+𝑝1𝑘=𝑚𝑛+q1𝑙=𝑛𝑥𝑘,𝑙𝐿,𝑧=0forevery𝑧𝑋.(2.3)

Definition 2.2. Let 𝑀 be an Orlicz function, 𝑝=(𝑝𝑘,𝑙) be any factorable double sequence of strictly positive reel numbers and 𝑆(2𝑋) denote all double sequence in (𝑋,,)  2-normed space we can define the following double sequence space ̂𝑐2=𝑥,𝑀,𝑝,,𝑥=𝑘,𝑙𝑃lim𝑝,𝑞1𝑝𝑞𝑝,𝑞𝑘,𝑙=1,1𝑀𝑥𝑘+𝑚,𝑙+𝑛𝐿𝜌,𝑧𝑝𝑘,𝑙=0uniformlyin𝑚and𝑛,forsome𝜌>0and𝐿,andevery𝑧𝑋.(2.4)
If we choose 𝑀(𝑥)=𝑥 and (𝑝𝑘,𝑙)=1 for all 𝑘 and 𝑙, then [̂𝑐2,𝑀,𝑝,,]=[̂𝑐2,,] 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,𝑟=𝑘𝑟𝑘𝑟1𝑙as𝑟,0=0,𝑠=𝑙𝑠𝑙𝑠1as𝑠.(2.5)
Let 𝑘𝑟,𝑠=𝑘𝑟𝑙𝑠, 𝑟,𝑠=𝑟𝑠 and 𝜃𝑟,𝑠 is defined by 𝐼𝑟,𝑠=(𝑘,𝑙)𝑘𝑟1<𝑘𝑘𝑟and𝑙𝑠1<𝑙𝑙𝑠,𝑞𝑟=𝑘𝑟𝑘𝑟1,𝑞s=𝑙𝑠𝑙𝑠1,𝑞𝑟,𝑠=𝑞𝑟𝑞𝑠.(2.6)

Definition 2.4. Let 𝑀 be an Orlicz function, 𝑆(2𝑋) 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,𝑥=𝑘,𝑙𝑃lim𝑟,𝑠1𝑟,𝑠𝑘,𝑙𝐼𝑟,𝑠𝑥𝑘+𝑚,𝑙+𝑛𝐿,𝑧=0uniformlyin𝑚and𝑛,forsome𝐿andevery𝑧𝑋𝐴𝐶𝜃𝑟,𝑠,,0=𝑥𝑥=𝑘,𝑙𝑃lim𝑟,𝑠1𝑟,𝑠𝑘,𝑙𝐼𝑟,𝑠𝑥𝑘+𝑚,𝑙+𝑛,𝑧=0uniformlyin𝑚and𝑛andevery𝑧𝑋𝐴𝐶𝜃𝑟,𝑠=𝑥,𝑀,𝑝,,𝑥=𝑘,𝑙𝑃lim𝑟,𝑠1𝑟,𝑠𝑘,𝑙𝐼𝑟,𝑠𝑀𝑥𝑘+𝑚,𝑙+𝑛𝐿𝜌,𝑧𝑝𝑘,𝑙=0uniformlyin𝑚and𝑛,forsome𝜌>0and𝐿,andevery𝑧𝑋𝐴𝐶𝜃𝑟,𝑠,𝑀,𝑝,,0=𝑥𝑥=𝑘,𝑙𝑃lim𝑟,𝑠1𝑟,𝑠𝑘,𝑙𝐼𝑟,𝑠𝑀𝑥𝑘+𝑚,𝑙+𝑛𝜌,𝑧𝑝𝑘,𝑙.=0uniformlyin𝑚and𝑛,forsome𝜌>0andevery𝑧𝑋(2.7)
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,𝑠=𝑥𝑀,,𝑥=𝑘,𝑙𝑃lim𝑟,𝑠1𝑟,𝑠𝑘,𝑙𝐼𝑟,𝑠𝑀𝑥𝑘+𝑚,𝑙+𝑛𝐿𝜌,,𝑧=0uniformlyin𝑚and𝑛,forsome𝜌>0and𝐿,andevery𝑧𝑋𝐴𝐶𝜃𝑟,𝑠𝑀,,0=𝑥𝑥=𝑘,𝑙𝑃lim𝑟,𝑠1𝑟,𝑠𝑘,𝑙𝐼𝑟,𝑠𝑀𝑥𝑘+𝑚,𝑙+𝑛𝜌.,𝑧=0uniformlyin𝑚and𝑛,forsome𝜌>0and𝐿,andevery𝑧𝑋(2.8)
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 Results

Theorem 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. So we have 𝐴1𝑟,𝑠=𝑥𝑥=𝑘,𝑙𝑃lim𝑟,𝑠1𝑟,𝑠𝑘,𝑙𝐼𝑟,𝑠𝑀𝑥𝑘+𝑚,𝑙+𝑛𝜌1,𝑧𝑝𝑘,𝑙=0uniformlyin𝑚and𝑛,forsome𝜌1,𝐴>0andevery𝑧𝑋2𝑟,𝑠=𝑦𝑦=𝑘,𝑙𝑃lim𝑟,𝑠1𝑟,𝑠𝑘,𝑙𝐼𝑟,𝑠𝑀𝑦𝑘+𝑚,𝑙+𝑛𝜌2,𝑧𝑝𝑘,𝑙=0uniformlyin𝑚and𝑛,forsome𝜌2.>0andevery𝑧𝑋(3.1) Since 𝑀 is an Orlicz function we have the following inequality 1𝑟,𝑠𝑘,𝑙𝐼𝑟,𝑠𝑀𝛼𝑥𝑘+𝑚,𝑙+𝑛+𝛽𝑦𝑘+𝑚,𝑙+𝑛|𝛼|𝜌1+||𝛽||𝜌2,𝑧𝑝𝑘,𝑙1𝐷𝑟,𝑠𝑘,𝑙𝐼𝑟,𝑠|𝛼||𝛼|𝜌1+||𝛽||𝜌2𝑀𝑥𝑘+𝑚,𝑙+𝑛𝜌1,𝑧𝑝𝑘,𝑙1+𝐷𝑟,𝑠𝑘,𝑙𝐼𝑟,𝑠||𝛽|||𝛼|𝜌1+||𝛽||𝜌2𝑀𝑦𝑘+𝑚,𝑙+𝑛𝜌2,𝑧𝑝𝑘,𝑙1𝐷𝐹𝑟,𝑠𝑘,𝑙𝐼𝑟,𝑠𝑀𝑥𝑘+𝑚,𝑙+𝑛𝜌1,𝑧𝑝𝑘,𝑙1+𝐷𝐹𝑟,𝑠𝑘,𝑙𝐼𝑟,𝑠𝑀𝑦𝑘+𝑚,𝑙+𝑛𝜌2,z𝑝𝑘,𝑙,(3.2) 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,𝑠𝑥𝑥=𝑘,𝑙𝑦,𝑦=𝑘,𝑙𝑃lim𝑟,𝑠1𝑟,𝑠𝑘,𝑙𝐼𝑟,𝑠𝑀𝛼𝑥𝑘+𝑚,𝑙+𝑛+𝛽𝑦𝑘+𝑚,𝑙+𝑛|𝛼|𝜌1+||𝛽||𝜌2,𝑧𝑝𝑘,𝑙=𝑜uniformlyin𝑚and𝑛,forsome𝜌1,𝜌2>0andeach𝑧𝑋.(3.3) 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)(𝑢0).(3.4)

Lemma 3.3. Let 𝑀 be an Orlicz function which satisfies Δ2-condition and 0<𝛿<1. Then for each 𝑥𝛿 and some constant 𝐾>0 we have 𝑀(𝑥)𝐾𝛿1𝑀(2).(3.5)

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=𝑥𝑥=𝑘,𝑙𝑃lim𝑟,𝑠1r,𝑠𝑘,𝑙𝐼𝑟,𝑠𝑥𝑘+𝑚,𝑙+𝑛𝐿,𝑧=0forsome𝐿,andevery𝑧𝑋.(3.6) Let 𝜀>0 and choose 𝛿 with 0<𝛿<1 such that 𝑀(𝑡)<𝜀 for every 𝑡 with 0𝑡𝛿. For every 𝑧𝑋, we get 1𝑟,𝑠𝑘,𝑙𝐼𝑟,𝑠𝑀𝑥𝑘+𝑚,𝑙+𝑛=1𝐿,𝑧𝑟,𝑠𝑘,𝑙𝐼𝑟,𝑠,𝑥𝑘+𝑚,𝑙+𝑛𝐿,𝑧𝛿𝑀𝑥𝑘+𝑚,𝑙+𝑛+1𝐿,𝑧𝑟,𝑠𝑘,𝑙𝐼𝑟,𝑠𝑥𝑘+𝑚,𝑙+𝑛𝐿,𝑧>𝛿𝑀𝑥𝑘+𝑚,𝑙+𝑛1𝐿,𝑧𝑟,𝑠𝑟,𝑠𝜀+1𝑟,𝑠𝑘,𝑙𝐼𝑟,𝑠,𝑥𝑘+𝑚,𝑙+𝑛𝐿,𝑧>𝛿𝑀𝑥𝑘+𝑚,𝑙+𝑛<1𝐿,𝑧𝑟,𝑠𝑟,𝑠𝜀+1𝑟,𝑠𝐾𝛿1𝑀(2)𝑟,𝑠𝐴𝑟,𝑠.(3.7) 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, [̂𝑐2,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 𝑟/𝑘𝑟𝛿/(1+𝛿), 𝑠/𝑙𝑠𝛿/(1+𝛿). Then for 𝑥[̂𝑐2,𝑀,𝑝,,], we can write for each 𝑚 and 𝑛, some 𝐿 and every 𝑧𝑋𝐵𝑟,𝑠=1𝑟,𝑠𝑘,𝑙𝐼𝑟,𝑠𝑀𝑥𝑘+𝑚,𝑙+𝑛𝐿𝜌,𝑧𝑝𝑘,𝑙=1𝑘𝑟,𝑠𝑟𝑙𝑘=1𝑠𝑙=1𝑀𝑥𝑘+𝑚,𝑙+𝑛𝐿𝜌,𝑧𝑝𝑘,𝑙1𝑘𝑟,𝑠𝑟1𝑙𝑘=1𝑠1𝑙=1𝑀𝑥𝑘+𝑚,𝑙+𝑛𝐿𝜌,𝑧𝑝𝑘,𝑙1𝑘𝑟,𝑠𝑟𝑘=𝑘𝑟1𝑙+1𝑠1𝑙=1𝑀𝑥𝑘+𝑚,𝑙+𝑛𝐿𝜌,𝑧𝑝𝑘,𝑙1𝑘𝑟,𝑠𝑟1𝑙𝑘=1𝑠𝑙=𝑙𝑠1+1𝑀𝑥𝑘+𝑚,𝑙+𝑛𝐿𝜌,𝑧𝑝𝑘,𝑙𝑘𝑟𝑙𝑠𝑟,𝑠1𝑘𝑟𝑙𝑠𝑘𝑟𝑙𝑘=1𝑠𝑙=1𝑀𝑥𝑘+𝑚,𝑙+𝑛𝐿𝜌,𝑧𝑝𝑘,𝑙𝑘𝑟𝑙𝑠𝑟,𝑠1𝑘𝑟𝑙𝑠𝑘𝑟𝑙𝑘=1𝑠𝑙=1𝑀𝑥𝑘+𝑚,𝑙+𝑛𝐿𝜌,𝑧𝑝𝑘,𝑙=𝑘r𝑙𝑠𝑟,𝑠1𝑘𝑟𝑙𝑠𝑘𝑟𝑙𝑘=1𝑠𝑙=1𝑀𝑥𝑘+𝑚,𝑙+𝑛𝐿𝜌,𝑧𝑝𝑘,𝑙𝑘𝑟1𝑙𝑠1𝑟,𝑠1𝑘𝑟1𝑙𝑘𝑠1𝑟1𝑙𝑘=1𝑠1𝑙=1𝑀𝑥𝑘+𝑚,𝑙+𝑛𝐿𝜌,𝑧𝑝𝑘,𝑙1𝑟𝑘𝑟𝑘=𝑘𝑟1+1𝑙𝑠1𝑠1𝑙𝑙𝑠1𝑠1𝑙=1𝑀𝑥𝑘+𝑚,𝑙+𝑛𝐿𝜌,𝑧𝑝𝑘,𝑙1𝑠𝑙𝑠𝑙=𝑙𝑠1+1𝑘𝑟1𝑟1𝑘𝑘𝑟1𝑟1𝑘=1𝑀𝑥𝑘+𝑚,𝑙+𝑛𝐿𝜌,𝑧𝑝𝑘,𝑙.(3.8) Since 𝑥[̂𝑐2,𝑀,𝑝,,] the last two terms tends to 0 uniformly in 𝑚 and 𝑛 in Pringsheim sense. Thus for each 𝑚 and 𝑛𝐵𝑟,𝑠=𝑘𝑟𝑙𝑠𝑟,𝑠1𝑘𝑟𝑙𝑠𝑘𝑟𝑙𝑘=1𝑠𝑙=1𝑀𝑥𝑘+𝑚,𝑙+𝑛𝐿𝜌,𝑧𝑝𝑘,𝑙𝑘𝑟1𝑙𝑠1𝑟,𝑠1𝑘𝑟1𝑙𝑘𝑠1𝑟1𝑙𝑘=1𝑠1𝑙=1𝑀𝑥𝑘+𝑚,𝑙+𝑛𝐿𝜌,𝑧𝑝𝑘,𝑙+𝑜(1).(3.9) Since 𝑟,𝑠=𝑘𝑟𝑙𝑠𝑘𝑟1𝑙𝑠1 we get for each 𝑚 and 𝑛 the following inequalities as follow: 𝑘𝑟𝑙𝑠𝑟,𝑠1+𝛿𝛿,𝑘𝑟1𝑙𝑠1𝑟,𝑠1𝛿.(3.10) Thus the terms 1𝑘𝑟𝑙𝑠𝑘𝑟𝑙𝑘=1𝑠𝑙=1𝑀𝑥𝑘+𝑚,𝑙+𝑛𝐿𝜌,𝑧𝑝𝑘,𝑙,1𝑘𝑟1𝑙𝑘𝑠1𝑟1𝑙𝑘=1𝑠1𝑙=1𝑀𝑥𝑘+𝑚,𝑙+𝑛𝐿𝜌,𝑧𝑝𝑘,𝑙,(3.11) are both convergent to 𝐿 in Pringsheim sense for all 𝑚 and 𝑛, every 𝑧𝑋 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,,][̂𝑐2,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 𝑖𝑟0 and 𝑗𝑠0, and all 𝑚 and 𝑛, for every 𝑧𝑋𝐴𝑖,𝑗=1𝑖,𝑗𝑘,𝑙𝐼𝑖,𝑗𝑀𝑥𝑘+𝑚,𝑙+𝑛𝜌,𝑧𝑝𝑘,𝑙<𝜀.(3.12) Let 𝑀𝐴=max𝑖,𝑗1𝑖𝑟0,1𝑗𝑠0(3.13) and let 𝑘𝑟1<𝑝𝑘𝑟 and 𝑙𝑠1<𝑞𝑙𝑠. Hence we get 1𝑝𝑞𝑝,𝑞𝑘,𝑙1,1𝑀𝑥𝑘+𝑚,𝑙+𝑛𝜌,𝑧𝑝𝑘,𝑙1𝑘𝑟1𝑙𝑘𝑠1𝑟𝑙𝑘=1𝑠𝑙=1𝑀𝑥𝑘+𝑚,𝑙+𝑛𝜌,𝑧𝑝𝑘,𝑙1𝑘𝑟1𝑙𝑟𝑠10,𝑠0𝑡,𝑢=1,1𝑘,𝑙𝐼𝑡,𝑢𝑀𝑥𝑘+𝑚,𝑙+𝑛𝜌,𝑧𝑝𝑘,𝑙=1𝑘𝑟1𝑙𝑠1𝑟,𝑠𝑡,𝑢=1,1𝑡,𝑢𝐴𝑡,𝑢+1𝑘𝑟1𝑙𝑠1𝑟0𝑠<𝑡𝑟0<𝑢𝑠𝑡,𝑢𝐴𝑡,𝑢𝑀𝑘𝑟1𝑙𝑟𝑠10,𝑠0𝑡,𝑢=1,1𝑡,𝑢+1𝑘𝑟1𝑙𝑠1𝑟0𝑠<𝑡𝑟0<𝑢𝑠𝑡,𝑢𝐴𝑡,𝑢𝑀𝑘𝑟0𝑙𝑠0𝑟0𝑠0𝑘𝑟1𝑙𝑠1+1𝑘𝑟1𝑙𝑠1𝑟0𝑠<𝑡𝑟0<𝑢𝑠𝑡,𝑢𝐴𝑡,𝑢𝑀𝑘𝑟0𝑙𝑠0𝑟0𝑠0𝑘𝑟1𝑙𝑠1+sup𝑡𝑟0𝑢𝑠0𝐴𝑡,𝑢1𝑘𝑟1𝑙𝑠1𝑟0𝑠<𝑡𝑟0<𝑢𝑠𝑡,𝑢𝑀𝑘𝑟0𝑙𝑠0𝑟0𝑠0𝑘𝑟1𝑙𝑠1+1𝑘𝑟1𝑙𝑠1𝜀𝑟0𝑠<𝑡𝑟0<𝑢𝑠𝑡,𝑢𝑀𝑘𝑟0𝑙𝑠0𝑟0𝑠0𝑘𝑟1𝑙𝑠1+𝜀𝐻2.(3.14) Since 𝑘𝑟 and 𝑙𝑠 both tends to infinity as both 𝑝 and 𝑞 tends to infinity, uniformly in 𝑚 and 𝑛, and for every 𝑧𝑋, 1𝑝𝑞𝑝,𝑞𝑘,𝑙1,1𝑀𝑥𝑘+𝑚,𝑙+𝑛𝜌,𝑧𝑝𝑘,𝑙0.(3.15) Therefore 𝑥[̂𝑐2,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,𝑠limsup𝑟,𝑠𝑞𝑟,𝑠<, then for any Orlicz function 𝑀[𝐴𝐶𝜃𝑟,𝑠,𝑀,𝑝,,]=[̂𝑐2,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 𝑆𝜃𝑟,𝑠𝑃-convergent to 𝐿 provided that for every 𝜀>0 and 𝑧𝑋𝑃lim𝑟,𝑠1𝑟,𝑠||max(𝑘,𝑙)𝐼𝑟,𝑠𝑥𝑘+𝑚,𝑙+𝑛||𝐿,𝑧𝜀=0.(3.16) So we can write 𝑆𝜃𝑟,𝑠lim𝑥=𝐿.

Theorem 3.9. Let 𝜃𝑟,𝑠 be a double lacunary sequence then(1)𝑥𝑃𝑘,𝑙𝐿[𝐴𝐶𝜃𝑟,𝑠,,] implies 𝑥𝑃𝑘,𝑙𝐿[𝑆𝜃𝑟,𝑠,,](2)[𝐴𝐶𝜃𝑟,𝑠,,] is a proper subset of [𝑆𝜃𝑟,𝑠,,](3)If 𝑥(𝑙)2 and 𝑥𝑃𝑘,𝑙𝐿[𝑆𝜃𝑟,𝑠,,] then 𝑥𝑃𝑘,𝑙𝐿[𝐴𝐶𝜃𝑟,𝑠,,](4)[𝑆𝜃𝑟,𝑠,,](𝑙)2=[𝐴𝐶𝜃𝑟,𝑠,,](𝑙)2where (𝑙)2 is the space of all bounded double sequence.

Proof. (1) Since for all 𝑚 and 𝑛, and every 𝑧𝑋||(𝑘,𝑙)𝐼𝑟,𝑠𝑥𝑘+𝑚,𝑙+𝑛||𝐿,𝑧𝜀𝑘,𝑙𝐼𝑟,𝑠,𝑥𝑘+𝑚,𝑙+𝑛𝐿,𝑧𝜀𝑥𝑘+𝑚,𝑙+𝑛𝐿,𝑧𝑘,𝑙𝐼𝑟,𝑠𝑥𝑘+𝑚,𝑙+𝑛,𝐿,𝑧(3.17) and for all 𝑚 and 𝑛𝑃lim𝑟,𝑠1𝑟,𝑠𝑘,𝑙𝐼𝑟,𝑠𝑥𝑘+𝑚,𝑙+𝑛𝐿,𝑧=0.(3.18) This show that for all 𝑚 and 𝑛𝑃lim𝑟,𝑠1𝑟,𝑠||(𝑘,𝑙)𝐼𝑟,𝑠𝑥𝑘+𝑚,𝑙+𝑛||𝐿,𝑧𝜀=0,(3.19) this completes the proof of (1)(2) Let 𝑥 be defined as follows:𝑥𝑘,𝑙=1233𝑟,𝑠02233𝑟,𝑠203𝑟,𝑠3𝑟,𝑠3𝑟,𝑠0000000.(3.20)
It is obvious that 𝑥 is an unbounded double sequence and for 𝜀>0, for all 𝑚 and 𝑛, and for every 𝑧𝑋𝑃lim𝑟,𝑠1𝑟,𝑠||(𝑘,𝑙)𝐼𝑟,𝑠𝑥𝑘+𝑚,𝑙+𝑛||𝐿,𝑧𝜀=𝑃lim𝑟,𝑠3𝑟,𝑠𝑟,𝑠=0.(3.21) Thus 𝑥𝑃𝑘,𝑙0[𝑆𝜃𝑟,𝑠,,]. But 𝑃lim𝑟,𝑠1𝑟,𝑠𝑘,𝑙𝐼𝑟,𝑠𝑥𝑘,𝑙,𝑧=𝑃lim𝑟,𝑠3𝑟,𝑠3𝑟,𝑠3𝑟,𝑠+12𝑟,𝑠=12.(3.22) Therefore 𝑥𝑃𝑘,𝑙𝐿[𝐴𝐶𝜃𝑟,𝑠,,] which is the proof of (2).(3)Let 𝑥(𝑙)2 and 𝑥𝑃𝑘,𝑙𝐿[𝑆𝜃𝑟,𝑠,,]. Assuming that for all 𝑚 and 𝑛, and every 𝑧𝑋𝑥𝑘+𝑚,𝑙+𝑛𝐿,𝑧𝐾 for all 𝐾. And also for given 𝜀>0 and r and 𝑠 large for all 𝑚 and 𝑛, and every 𝑧𝑋 we get the following inequality as follow:1𝑟,𝑠𝑘,𝑙𝐼𝑟,𝑠𝑥𝑘+𝑚,𝑙+𝑛=1𝐿,𝑧𝑟,𝑠𝑘,𝑙𝐼𝑟,𝑠,𝑥𝑘+𝑚,𝑙+𝑛𝐿,𝑧𝜀𝑥𝑘+𝑚,𝑙+𝑛+1𝐿,𝑧𝑟,𝑠𝑘,𝑙𝐼𝑟,𝑠,𝑥𝑘+𝑚,𝑙+𝑛𝐿,𝑧<𝜀𝑥𝑘+𝑚,𝑙+𝑛𝐾𝐿,𝑧𝑟,𝑠||(𝑘,𝑙)𝐼𝑟,𝑠𝑥𝑘+𝑚,𝑙+𝑛||𝐿,𝑧𝜀+𝜀.(3.23) Therefore 𝑥(𝑙)2 and 𝑥𝑃𝑘,𝑙𝐿[𝑆𝜃𝑟,𝑠,,], this shows that 𝑥𝑃𝑘,𝑙𝐿[𝐴𝐶𝜃𝑟,𝑠,,].(4)from (1), (2) and (3), we get [𝑆𝜃𝑟,𝑠,,](𝑙)2=[𝐴𝐶𝜃𝑟,𝑠,,](𝑙)2.

Theorem 3.10. For any Orlicz function 𝑀, [𝐴𝐶𝜃𝑟,𝑠,𝑀,,][𝑆𝜃𝑟,𝑠,,]

Proof. Let 𝑥[𝐴𝐶𝜃𝑟,𝑠,𝑀,,] and 𝜀>0. Then for all 𝑚 and 𝑛, and every 𝑧𝑋1𝑟,𝑠𝑘,𝑙𝐼𝑟,𝑠𝑀𝑥𝑘+𝑚,𝑙+𝑛𝐿𝜌1,𝑧𝑟,𝑠𝑘,𝑙𝐼𝑟,𝑠,𝑥𝑘+𝑚,𝑙+𝑛𝐿𝜀𝑀𝑥𝑘+𝑚,𝑙+𝑛𝐿𝜌>1,𝑧𝑟,𝑠𝑀𝜀𝜌||(𝑘,𝑙)𝐼𝑟,𝑠𝑥𝑘+𝑚,𝑙+𝑛||.𝐿,𝑧𝜀(3.24) This shows that 𝑥[𝑆𝜃𝑟,𝑠,,].


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