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

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 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=(𝑥_𝑘,𝑙) has Pringsheim limit 𝐿 denoted by 𝑃−lim𝑥=𝐿, provided given 𝜖>0 there exists 𝑁∈ℕ such that |𝑥_𝑘,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={𝑘_𝑟} if {𝑘_𝑟}⊆{𝑘_𝑟} and studied many scholar [4–6]. Savaş and Patterson [7] gave some properties and theorem and also defined 𝑆_{𝜃𝑟,𝑠}−𝑃-convergence.

By a lacunary 𝜃=(𝑘_𝑟); 𝑟=0,1,2,… where 𝑘₀=0, we shall mean an increasing sequence of non-negative integers with 𝑘_𝑟−𝑘_𝑟−1→∞ as 𝑟→∞. The intervals determined by 𝜃 denoted by 𝐼_𝑟=(𝑘_𝑟−1,𝑘_𝑟] and ℎ_𝑟=𝑘_𝑟−𝑘_𝑟−1. The ratio 𝑘_𝑟/𝑘_𝑟−1 is denoted by 𝑞_𝑟 [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 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 (𝑥_𝑚𝑛)_{(𝑚,𝑛∈ℕ)} of elements of 𝑋 is called to be statistically convergent to 𝑥∈𝑋 if the set 𝐴(𝜀)={𝑚,𝑛∈ℕ∶‖𝑥_𝑚𝑛−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;(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 (𝑥_𝑘)_𝑘∈ℕ in a 2-normed space (𝑋,‖⋅,⋅‖) is said to be convergent to 𝐿 in 𝑋 if lim_𝑘→∞‖𝑥_𝑘−𝐿,𝑧‖=0 for every 𝑧∈𝑋. In this case, we write lim_𝑘→∞‖𝑥_𝑘,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=(𝑥_𝑘,𝑙) of real numbers is called almost 𝑃-convergent to a limit 𝐿 if 𝑃−lim_{𝑝,𝑞→∞}sup_{𝑚,𝑛≥0}1𝑝𝑞^{𝑚+𝑝−1}_{𝑘=𝑚𝑛+𝑞−1𝑙=𝑛}||𝑥_𝑘,𝑙−𝐿||=0.(2.1)
That is, the average (𝑥_𝑘,𝑙) take over any rectangle {(𝑘,𝑙)∶𝑚≤𝑘≤𝑚+𝑝−1,𝑛≤𝑙≤𝑛+𝑞−1},(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 [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=𝑥^𝑘,𝑙∶𝑃−lim_{𝑝,𝑞⟶∞}sup_{𝑚,𝑛≥0}1𝑝𝑞^{𝑚+𝑝−1}_{𝑘=𝑚𝑛+}^q−1_𝑙=𝑛forevery𝑧∈𝑋‖‖𝑥_𝑘,𝑙−𝐿,𝑧‖‖=0⎫⎪⎪⎬⎪⎪⎭.(2.3)

Definition 2.2. Let 𝑀 be an Orlicz function, 𝑝=(𝑝_𝑘,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=𝑥^𝑘,𝑙∶𝑃−lim_𝑝,𝑞1𝑝𝑞^𝑝,𝑞_{𝑘,𝑙=1,1}𝑀‖‖‖𝑥^{𝑘+𝑚,𝑙+𝑛}−𝐿𝜌,𝑧‖‖‖^{𝑝𝑘,𝑙}uniformlyin𝑚and𝑛,forsome𝜌>0and𝐿,andevery𝑧∈=𝑋0⎫⎪⎬⎪⎭.(2.4)
If we choose 𝑀(𝑥)=𝑥 and (𝑝_𝑘,𝑙)=1 for all 𝑘 and 𝑙, then [𝑐̂²,𝑀,𝑝,‖⋅,⋅‖]=[𝑐̂²,‖⋅,⋅‖] which was defined above.

Definition 2.3. The double sequnce 𝜃_𝑟,𝑠={𝑘_𝑟,𝑙_𝑠} is called double lacunary if there exist two increasing sequences of integers such that 𝑘𝑙₀=0,ℎ_𝑟=𝑘_𝑟−𝑘_𝑟−1⟶∞as𝑟⟶∞,₀=0,ℎ_𝑠=𝑙_𝑠−𝑙_𝑠−1⟶∞as𝑠⟶∞.(2.5)
Let 𝑘_𝑟,𝑠=𝑘_𝑟𝑙_𝑠, ℎ_𝑟,𝑠=ℎ_𝑟ℎ_𝑠 and 𝜃_𝑟,𝑠 is defined by 𝐼_𝑟,𝑠=(𝑘,𝑙)∶𝑘_𝑟−1<𝑘≤𝑘_𝑟and𝑙_𝑠−1<𝑙≤𝑙_𝑠,𝑞_𝑟=𝑘^𝑟𝑘_𝑟−1,𝑞_s=𝑙^𝑠𝑙_𝑠−1,𝑞_𝑟,𝑠=𝑞_𝑟𝑞_𝑠.(2.6)

Definition 2.4. Let 𝑀 be an Orlicz function, 𝑆^′(2−𝑋) denote all double sequence in (𝑋,‖⋅,⋅‖) 2-normed space, and 𝑝=(𝑝_𝑘,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: 𝐴𝐶^𝜃_𝑟,𝑠,‖⋅,⋅‖=⎧⎪⎨⎪⎩𝑥=𝑥^𝑘,𝑙∶𝑃−lim_𝑟,𝑠1ℎ_𝑟,𝑠_{𝑘,𝑙∈𝐼𝑟,𝑠}𝐴𝐶_{𝜃𝑟,𝑠}uniformlyin𝑚and𝑛,forsom‖‖e𝑥𝐿and_{𝑘+𝑚,𝑙+𝑛}e−ve𝐿ry,𝑧𝑧‖‖∈=𝑋0⎫⎪⎬⎪⎭,‖⋅,⋅‖₀=⎧⎪⎨⎪⎩𝑥=𝑥^𝑘,𝑙∶𝑃−lim_𝑟,𝑠1ℎ_𝑟,𝑠_{𝑘,𝑙∈𝐼𝑟,𝑠}𝐴𝐶_{𝜃𝑟,𝑠}uniformlyin𝑚and𝑛and‖‖e𝑥very𝑧_{𝑘+𝑚,𝑙+𝑛}∈,𝑧𝑋‖‖=0⎫⎪⎬⎪⎭,𝑀,𝑝,‖⋅,⋅‖=⎧⎪⎨⎪⎩𝑥=𝑥^𝑘,𝑙∶𝑃−lim_𝑟,𝑠1ℎ_𝑟,𝑠_{𝑘,𝑙∈𝐼𝑟,𝑠}𝑀‖‖‖𝑥^{𝑘+𝑚,𝑙+𝑛}−𝐿𝜌,𝑧‖‖‖^{𝑝𝑘,𝑙}𝐴𝐶_{𝜃𝑟,𝑠}uniformlyin𝑚and𝑛,forsome𝜌>0and𝐿,andevery𝑧∈=𝑋0⎫⎪⎬⎪⎭,𝑀,𝑝,‖⋅,⋅‖₀=⎧⎪⎨⎪⎩𝑥=𝑥^𝑘,𝑙∶𝑃−lim_𝑟,𝑠1ℎ_𝑟,𝑠_{𝑘,𝑙∈𝐼𝑟,𝑠}𝑀‖‖‖𝑥𝜌^{𝑘+𝑚,𝑙+𝑛},𝑧‖‖‖^{𝑝𝑘,𝑙}uniformlyin𝑚and𝑛,forsome𝜌>0andevery𝑧∈𝑋=0⎫⎪⎬⎪⎭.(2.7)
When (𝑝_𝑘,𝑙)=1 for all 𝑘 and 𝑙, we shall denote [𝐴𝐶_{𝜃𝑟,𝑠},𝑀,𝑝,‖⋅,⋅‖] and [𝐴𝐶_{𝜃𝑟,𝑠},𝑀,𝑝,‖⋅,⋅‖]₀ as [𝐴𝐶_{𝜃𝑟,𝑠},𝑀,‖⋅,⋅‖] and [𝐴𝐶_{𝜃𝑟,𝑠},𝑀,‖⋅,⋅‖]₀. That is, 𝐴𝐶^𝜃_𝑟,𝑠𝑀,‖⋅,⋅‖=⎧⎪⎨⎪⎩𝑥=𝑥^𝑘,𝑙∶𝑃−lim_𝑟,𝑠1ℎ_𝑟,𝑠_{𝑘,𝑙∈𝐼𝑟,𝑠}𝑀‖‖‖𝑥^{𝑘+𝑚,𝑙+𝑛}−𝐿𝐴𝐶_{𝜃𝑟,𝑠}uniformlyin𝑚and𝑛,forsome𝜌>0and𝐿𝜌,ande,v𝑧e‖‖‖ry𝑧∈=𝑋0⎫⎪⎬⎪⎭,𝑀,‖⋅,⋅‖₀=⎧⎪⎨⎪⎩𝑥=𝑥^𝑘,𝑙∶𝑃−lim_𝑟,𝑠1uniformlyin𝑚and𝑛,foℎrs_𝑟,𝑠ome_{𝑘,𝑙∈𝐼𝑟,𝑠}𝜌>𝑀0a‖‖‖n𝑥d𝐿𝜌,an^{𝑘+𝑚,𝑙+𝑛}d,𝑧e‖‖‖very𝑧=∈0𝑋⎫⎪⎬⎪⎭.(2.8)
If 𝑥∈[𝐴𝐶_{𝜃𝑟,𝑠}𝑀,‖⋅,⋅‖] 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 (𝑝_𝑘,𝑙)=1 for all 𝑘 and 𝑙, then [𝐴𝐶_{𝜃𝑟,𝑠},𝑀,𝑝,‖⋅,⋅‖]=[𝐴𝐶_{𝜃𝑟,𝑠},‖⋅,⋅‖] and [𝐴𝐶_{𝜃𝑟,𝑠},𝑀,𝑝,‖⋅,⋅‖]₀=[𝐴𝐶_{𝜃𝑟,𝑠},‖⋅,⋅‖]₀ which are defined above. Also note that if (𝑝_𝑘,𝑙)=1 for all 𝑘 and 𝑙, then [𝐴𝐶_{𝜃𝑟,𝑠},𝑀,𝑝,‖⋅,⋅‖]=[𝐴𝐶_{𝜃𝑟,𝑠},𝑀,‖⋅,⋅‖] and [𝐴𝐶_{𝜃𝑟,𝑠},𝑀,𝑝,‖⋅,⋅‖]₀=[𝐴𝐶_{𝜃𝑟,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 𝑝_𝑘,𝑙, [𝐴𝐶_{𝜃𝑟,𝑠},𝑀,𝑝,‖⋅,⋅‖] and [𝐴𝐶_{𝜃𝑟,𝑠},𝑀,𝑝,‖⋅,⋅‖]₀ are linear spaces.

Proof. Suppose that 𝑥, 𝑦∈[𝐴𝐶_{𝜃𝑟,𝑠},𝑀,𝑝,‖⋅,⋅‖]₀ and 𝛼, 𝛽∈ℝ. So we have 𝐴^1𝑟,𝑠=⎧⎪⎨⎪⎩𝑥=𝑥^𝑘,𝑙∶𝑃−lim_𝑟,𝑠1ℎ_𝑟,𝑠_{𝑘,𝑙∈𝐼𝑟,𝑠}𝑀‖‖‖𝑥𝜌^{𝑘+𝑚,𝑙+𝑛}₁uniformlyin𝑚and𝑛,forsome𝜌₁,𝑧‖‖‖^{𝑝𝑘,𝑙}=0𝐴>0andevery𝑧∈𝑋_2𝑟,𝑠⎫⎪⎬⎪⎭,=⎧⎪⎨⎪⎩𝑦=𝑦^𝑘,𝑙∶𝑃−lim_𝑟,𝑠1ℎ_𝑟,𝑠_{𝑘,𝑙∈𝐼𝑟,𝑠}𝑀‖‖‖𝑦𝜌^{𝑘+𝑚,𝑙+𝑛}₂uniformlyin𝑚and𝑛,forsome𝜌₂,𝑧‖‖‖^{𝑝𝑘,𝑙}=0>0andevery𝑧∈𝑋⎫⎪⎬⎪⎭.(3.1) Since 𝑀 is an Orlicz function we have the following inequality 1ℎ_𝑟,𝑠_{𝑘,𝑙∈𝐼𝑟,𝑠}𝑀‖‖‖𝛼𝑥^{𝑘+𝑚,𝑙+𝑛}+𝛽𝑦_{𝑘+𝑚,𝑙+𝑛}|𝛼|𝜌₁+||𝛽||𝜌₂,𝑧‖‖‖^{𝑝𝑘,𝑙}≤𝐷1ℎ_𝑟,𝑠_{𝑘,𝑙∈𝐼𝑟,𝑠}|𝛼|𝜌₁|+𝛼|||𝛽||𝜌₂𝑀‖‖‖𝑥^{𝑘+𝑚,𝑙+𝑛}𝜌₁,𝑧‖‖‖^{𝑝𝑘,𝑙}+𝐷1ℎ_𝑟,𝑠_{𝑘,𝑙∈𝐼𝑟,𝑠}|𝛼|𝜌₁||+𝛽||||𝛽||𝜌₂𝑀‖‖‖𝑦^{𝑘+𝑚,𝑙+𝑛}𝜌₂,𝑧‖‖‖^{𝑝𝑘,𝑙}≤𝐷𝐹1ℎ_𝑟,𝑠_{𝑘,𝑙∈𝐼𝑟,𝑠}𝑀‖‖‖𝑥𝜌^{𝑘+𝑚,𝑙+𝑛}₁,𝑧‖‖‖^{𝑝𝑘,𝑙}+𝐷𝐹1ℎ_𝑟,𝑠_{𝑘,𝑙∈𝐼𝑟,𝑠}𝑀‖‖‖𝑦𝜌^{𝑘+𝑚,𝑙+𝑛}₂,z‖‖‖^{𝑝𝑘,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,𝑠→∞⎧⎪⎨⎪⎩𝑥=𝑥_𝑘,𝑙,𝑦=𝑦_𝑘,𝑙∶𝑃−lim_𝑟,𝑠1ℎ_𝑟,𝑠_{𝑘,𝑙∈𝐼𝑟,𝑠}𝑀‖‖‖𝛼𝑥^{𝑘+𝑚,𝑙+𝑛}+𝛽𝑦_{𝑘+𝑚,𝑙+𝑛}|𝛼|𝜌₁+||𝛽||𝜌₂uniformlyin𝑚and𝑛,forsome𝜌_1,,𝑧‖‖‖^{𝑝𝑘,𝑙}=𝑜𝜌₂>0andeach𝑧∈𝑋⎫⎪⎬⎪⎭.(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)(𝑢≥0).(3.4)

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).(3.5)

Theorem 3.4. For any Orlicz function 𝑀 which satisfies Δ₂-condition, we have [𝐴𝐶_{𝜃𝑟,𝑠},‖⋅,⋅‖]⊆[𝐴𝐶_{𝜃𝑟,𝑠}𝑀,‖⋅,⋅‖]

Proof. Let 𝑥∈[𝐴𝐶_{𝜃𝑟,𝑠},‖⋅,⋅‖]. For each 𝑚 and 𝑛𝐴_𝑟,𝑠=⎧⎪⎨⎪⎩𝑥=𝑥^𝑘,𝑙∶𝑃−lim_𝑟,𝑠1ℎ_r,𝑠_{𝑘,𝑙∈𝐼𝑟,𝑠}forsome𝐿,andev‖‖er𝑥y𝑧∈_{𝑘+𝑚,𝑙+𝑛}𝑋−𝐿,𝑧‖‖=0⎫⎪⎬⎪⎭.(3.6) 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 1ℎ_𝑟,𝑠_{𝑘,𝑙∈𝐼𝑟,𝑠}𝑀‖‖𝑥_{𝑘+𝑚,𝑙+𝑛}−𝐿,𝑧‖‖=1ℎ_𝑟,𝑠_{𝑘,𝑙∈𝐼𝑟,𝑠,‖‖𝑥𝑘+𝑚,𝑙+𝑛‖‖−𝐿,𝑧≤𝛿}+1𝑀‖‖𝑥_{𝑘+𝑚,𝑙+𝑛}−𝐿,𝑧‖‖ℎ_𝑟,𝑠_{𝑘,𝑙∈𝐼𝑟,𝑠‖‖𝑥𝑘+𝑚,𝑙+𝑛‖‖−𝐿,𝑧>𝛿}≤1𝑀‖‖𝑥_{𝑘+𝑚,𝑙+𝑛}−𝐿,𝑧‖‖ℎ_𝑟,𝑠ℎ_𝑟,𝑠𝜀+1ℎ_𝑟,𝑠_{𝑘,𝑙∈𝐼𝑟,𝑠,‖‖𝑥𝑘+𝑚,𝑙+𝑛‖‖−𝐿,𝑧>𝛿}<1𝑀‖‖𝑥_{𝑘+𝑚,𝑙+𝑛}−𝐿,𝑧‖‖ℎ_𝑟,𝑠ℎ_𝑟,𝑠𝜀+1ℎ_𝑟,𝑠𝐾𝛿⁻¹𝑀(2)ℎ_𝑟,𝑠𝐴_𝑟,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∈[𝐴𝐶_{𝜃𝑟,𝑠}𝑀,‖⋅,⋅‖].

Theorem 3.5. Let 𝜃_𝑟,𝑠={𝑘_𝑟,𝑙_𝑠} be a double lacunary sequence with liminf_𝑟𝑞_𝑟>1 and liminf_𝑠𝑞_𝑠>1 then for any Orlicz function 𝑀, [𝑐̂²,𝑀,𝑝,‖⋅,⋅‖]⊂[𝐴𝐶_{𝜃𝑟,𝑠},𝑀,𝑝,‖⋅,⋅‖].

Proof. Since liminf_𝑟𝑞_𝑟>1 and liminf_𝑠𝑞_𝑠>1, then there exists 𝛿>0 such that 𝑞_𝑟>1+𝛿 and 𝑞_𝑠>1+𝛿. This mean ℎ_𝑟/𝑘_𝑟≥𝛿/(1+𝛿), ℎ_𝑠/𝑙_𝑠≥𝛿/(1+𝛿). Then for 𝑥∈[𝑐̂²,𝑀,𝑝,‖⋅,⋅‖], we can write for each 𝑚 and 𝑛, some 𝐿 and every 𝑧∈𝑋𝐵_𝑟,𝑠=1ℎ_𝑟,𝑠_{𝑘,𝑙∈𝐼𝑟,𝑠}𝑀‖‖‖𝑥^{𝑘+𝑚,𝑙+𝑛}−𝐿𝜌,𝑧‖‖‖^{𝑝𝑘,𝑙}=1ℎ_{𝑘𝑟,𝑠}^𝑟_𝑙𝑘=1^𝑠_𝑙=1𝑀‖‖‖𝑥^{𝑘+𝑚,𝑙+𝑛}−𝐿𝜌,𝑧‖‖‖^{𝑝𝑘,𝑙}−1ℎ_{𝑘𝑟,𝑠}^𝑟−1_𝑙𝑘=1^𝑠−1_𝑙=1𝑀‖‖‖𝑥^{𝑘+𝑚,𝑙+𝑛}−𝐿𝜌,𝑧‖‖‖^{𝑝𝑘,𝑙}−1ℎ_{𝑘𝑟,𝑠}^𝑟_{𝑘=𝑘𝑟−1𝑙+1}^𝑠−1_𝑙=1𝑀‖‖‖𝑥^{𝑘+𝑚,𝑙+𝑛}−𝐿𝜌,𝑧‖‖‖^{𝑝𝑘,𝑙}−1ℎ_{𝑘𝑟,𝑠}^𝑟−1_𝑙𝑘=1^𝑠_{𝑙=𝑙𝑠−1+1}𝑀‖‖‖𝑥^{𝑘+𝑚,𝑙+𝑛}−𝐿𝜌,𝑧‖‖‖^{𝑝𝑘,𝑙}−𝑘^𝑟𝑙_𝑠ℎ_𝑟,𝑠⎛⎜⎜⎝1𝑘_𝑟𝑙_𝑠𝑘^𝑟_𝑙𝑘=1^𝑠_𝑙=1𝑀‖‖‖𝑥^{𝑘+𝑚,𝑙+𝑛}−𝐿𝜌,𝑧‖‖‖^{𝑝𝑘,𝑙}−𝑘_𝑟⎞⎟⎟⎠𝑙_𝑠ℎ_𝑟,𝑠⎛⎜⎜⎝1𝑘_𝑟𝑙_𝑠𝑘^𝑟_𝑙𝑘=1^𝑠_𝑙=1𝑀‖‖‖𝑥^{𝑘+𝑚,𝑙+𝑛}−𝐿𝜌,𝑧‖‖‖^{𝑝𝑘,𝑙}=𝑘_r⎞⎟⎟⎠𝑙_𝑠ℎ_𝑟,𝑠⎛⎜⎜⎝1𝑘_𝑟𝑙_𝑠𝑘^𝑟_𝑙𝑘=1^𝑠_𝑙=1𝑀‖‖‖𝑥^{𝑘+𝑚,𝑙+𝑛}−𝐿𝜌,𝑧‖‖‖^{𝑝𝑘,𝑙}−𝑘_𝑟−1⎞⎟⎟⎠𝑙_𝑠−1ℎ_𝑟,𝑠⎛⎜⎜⎝1𝑘_𝑟−1𝑙_𝑘𝑠−1^𝑟−1_𝑙𝑘=1^𝑠−1_𝑙=1𝑀‖‖‖𝑥^{𝑘+𝑚,𝑙+𝑛}−𝐿𝜌,𝑧‖‖‖^{𝑝𝑘,𝑙}−1⎞⎟⎟⎠ℎ_𝑟𝑘^𝑟_{𝑘=𝑘𝑟−1+1}𝑙_𝑠−1ℎ_𝑠1𝑙_𝑙𝑠−1^𝑠−1_𝑙=1𝑀‖‖‖𝑥^{𝑘+𝑚,𝑙+𝑛}−𝐿𝜌,𝑧‖‖‖^{𝑝𝑘,𝑙}−1ℎ_𝑠𝑙^𝑠_{𝑙=𝑙𝑠−1+1}𝑘_𝑟−1ℎ_𝑟1𝑘_𝑘𝑟−1^𝑟−1_𝑘=1𝑀‖‖‖𝑥^{𝑘+𝑚,𝑙+𝑛}−𝐿𝜌,𝑧‖‖‖^{𝑝𝑘,𝑙}.(3.8) 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 𝑛𝐵_𝑟,𝑠=𝑘^𝑟𝑙_𝑠ℎ_𝑟,𝑠⎛⎜⎜⎝1𝑘_𝑟𝑙_𝑠𝑘^𝑟_𝑙𝑘=1^𝑠_𝑙=1𝑀‖‖‖𝑥^{𝑘+𝑚,𝑙+𝑛}−𝐿𝜌,𝑧‖‖‖^{𝑝𝑘,𝑙}−𝑘_𝑟−1⎞⎟⎟⎠𝑙_𝑠−1ℎ_𝑟,𝑠⎛⎜⎜⎝1𝑘_𝑟−1𝑙_𝑘𝑠−1^𝑟−1_𝑙𝑘=1^𝑠−1_𝑙=1𝑀‖‖‖𝑥^{𝑘+𝑚,𝑙+𝑛}−𝐿𝜌,𝑧‖‖‖^{𝑝𝑘,𝑙}⎞⎟⎟⎠+𝑜(1).(3.9) Since ℎ_𝑟,𝑠=𝑘_𝑟𝑙_𝑠−𝑘_𝑟−1𝑙_𝑠−1 we get for each 𝑚 and 𝑛 the following inequalities as follow: 𝑘_𝑟𝑙_𝑠ℎ_𝑟,𝑠≤1+𝛿𝛿,𝑘^𝑟−1𝑙_𝑠−1ℎ_𝑟,𝑠≤1𝛿.(3.10) Thus the terms 1𝑘_𝑟𝑙_𝑠𝑘^𝑟_𝑙𝑘=1^𝑠_𝑙=1𝑀‖‖‖𝑥^{𝑘+𝑚,𝑙+𝑛}−𝐿𝜌,𝑧‖‖‖^{𝑝𝑘,𝑙}1,𝑘_𝑟−1𝑙_𝑘𝑠−1^𝑟−1_𝑙𝑘=1^𝑠−1_𝑙=1𝑀‖‖‖𝑥^{𝑘+𝑚,𝑙+𝑛}−𝐿𝜌,𝑧‖‖‖^{𝑝𝑘,𝑙},(3.11) are both convergent to 𝐿 in Pringsheim sense for all 𝑚 and 𝑛, every 𝑧∈𝑋 and some 𝜌>0. Therefore 𝐵_𝑟,𝑠 is a convergent sequence in Pringsheim sense for each 𝑚 and 𝑛. So 𝑥∈[𝐴𝐶_{𝜃𝑟,𝑠},𝑀,𝑝,‖⋅,⋅‖] and this is the proof.

Theorem 3.6. Let 𝜃_𝑟,𝑠={𝑘_𝑟,𝑙_𝑠} be a double lacunary sequence with limsup_𝑟𝑞_𝑟<∞ and limsup_𝑠𝑞_𝑠<∞ then for any Orlicz function 𝑀, [𝐴𝐶_{𝜃𝑟,𝑠},𝑀,𝑝,‖⋅,⋅‖]⊂[𝑐̂²,𝑀,𝑝,‖⋅,⋅‖].

Proof. Since limsup_𝑟𝑞_𝑟<∞ and limsup_𝑠𝑞_𝑠<∞ there exists 𝐻>0 such that 𝑞_𝑟<𝐻 and 𝑞_𝑠<𝐻 for all 𝑟 and 𝑠. Let 𝜀>0 and 𝑥∈[𝐴𝐶_{𝜃𝑟,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∈𝑋𝐴^′_𝑖,𝑗=1ℎ_𝑖,𝑗_{𝑘,𝑙∈𝐼𝑖,𝑗}𝑀‖‖‖𝑥𝜌^{𝑘+𝑚,𝑙+𝑛},𝑧‖‖‖^{𝑝𝑘,𝑙}<𝜀.(3.12) Let 𝑀^=max𝐴^𝑖,𝑗∶1≤𝑖≤𝑟₀,1≤𝑗≤𝑠₀(3.13) and let 𝑘_𝑟−1<𝑝≤𝑘_𝑟 and 𝑙_𝑠−1<𝑞≤𝑙_𝑠. Hence we get 1𝑝𝑞^𝑝,𝑞_{𝑘,𝑙∈1,1}𝑀‖‖‖𝑥𝜌^{𝑘+𝑚,𝑙+𝑛},𝑧‖‖‖^{𝑝𝑘,𝑙}≤1𝑘_𝑟−1𝑙_𝑘𝑠−1^𝑟_𝑙𝑘=1^𝑠_𝑙=1𝑀‖‖‖𝑥𝜌^{𝑘+𝑚,𝑙+𝑛},𝑧‖‖‖^{𝑝𝑘,𝑙}≤1𝑘_𝑟−1𝑙_𝑟𝑠−1^0,𝑠0_{𝑡,𝑢=1,1}⎛⎜⎜⎝_{𝑘,𝑙∈𝐼𝑡,𝑢}𝑀‖‖‖𝑥𝜌^{𝑘+𝑚,𝑙+𝑛},𝑧‖‖‖^{𝑝𝑘,𝑙}=1⎞⎟⎟⎠𝑘_𝑟−1𝑙_{𝑠−1𝑟,𝑠𝑡,𝑢=1,1}ℎ_𝑡,𝑢𝐴^𝑡,𝑢+1𝑘_𝑟−1𝑙_𝑠−1_{𝑟0∪𝑠<𝑡≤𝑟0<𝑢≤𝑠}ℎ_𝑡,𝑢𝐴^𝑡,𝑢≤𝑀^𝑘_𝑟−1𝑙_𝑟𝑠−1^0,𝑠0_{𝑡,𝑢=1,1}ℎ_𝑡,𝑢+1𝑘_𝑟−1𝑙_𝑠−1_{𝑟0∪𝑠<𝑡≤𝑟0<𝑢≤𝑠}ℎ_𝑡,𝑢𝐴^𝑡,𝑢≤𝑀^𝑘_𝑟0𝑙_𝑠0𝑟₀𝑠₀𝑘_𝑟−1𝑙_𝑠−1+1𝑘_𝑟−1𝑙_𝑠−1_{𝑟0∪𝑠<𝑡≤𝑟0<𝑢≤𝑠}ℎ_𝑡,𝑢𝐴^𝑡,𝑢≤𝑀^𝑘_𝑟0𝑙_𝑠0𝑟₀𝑠₀𝑘_𝑟−1𝑙_𝑠−1+sup_{𝑡≥𝑟0∪𝑢≥𝑠0}𝐴^𝑡,𝑢𝑘_𝑟−11𝑙_𝑠−1_{𝑟0∪𝑠<𝑡≤𝑟0<𝑢≤𝑠}ℎ_𝑡,𝑢≤𝑀^𝑘_𝑟0𝑙_𝑠0𝑟₀𝑠₀𝑘_𝑟−1𝑙_𝑠−1+1𝑘_𝑟−1𝑙_𝑠−1𝜀_{𝑟0∪𝑠<𝑡≤𝑟0<𝑢≤𝑠}ℎ_𝑡,𝑢≤𝑀^𝑘_𝑟0𝑙_𝑠0𝑟₀𝑠₀𝑘_𝑟−1𝑙_𝑠−1+𝜀𝐻².(3.14) Since 𝑘_𝑟 and 𝑙_𝑠 both tends to infinity as both 𝑝 and 𝑞 tends to infinity, uniformly in 𝑚 and 𝑛, and for every 𝑧∈𝑋, 1𝑝𝑞^𝑝,𝑞_{𝑘,𝑙∈1,1}𝑀‖‖‖𝑥𝜌^{𝑘+𝑚,𝑙+𝑛},𝑧‖‖‖^{𝑝𝑘,𝑙}⟶0.(3.15) Therefore 𝑥∈[𝑐̂²,𝑀,𝑝,‖⋅,⋅‖].

The following theorem is a result of Theorems 3.4 and 3.5.

Theorem 3.7. Let 𝜃_𝑟,𝑠={𝑘_𝑟,𝑙_𝑠} be a double lacunary sequence with 1<liminf_𝑟,𝑠𝑞_𝑟,𝑠≤limsup_𝑟,𝑠𝑞_𝑟,𝑠<∞, then for any Orlicz function 𝑀[𝐴𝐶_{𝜃𝑟,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 𝜃_𝑟,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−lim_𝑟,𝑠1ℎ_𝑟,𝑠max||(𝑘,𝑙)∈𝐼_𝑟,𝑠∶‖‖𝑥_{𝑘+𝑚,𝑙+𝑛}−𝐿,𝑧‖‖≥𝜀||=0.(3.16) So we can write 𝑆_{𝜃𝑟,𝑠}−lim𝑥=𝐿.

Theorem 3.9. Let 𝜃_𝑟,𝑠 be a double lacunary sequence then(1)𝑥_{𝑃𝑘,𝑙}−→𝐿[𝐴𝐶_{𝜃𝑟,𝑠},‖⋅,⋅‖] implies 𝑥_{𝑃𝑘,𝑙}−→𝐿[𝑆_{𝜃𝑟,𝑠},‖⋅,⋅‖](2)[𝐴𝐶_{𝜃𝑟,𝑠},‖⋅,⋅‖] is a proper subset of [𝑆_{𝜃𝑟,𝑠},‖⋅,⋅‖](3)If 𝑥∈(𝑙_∞)₂ and 𝑥_{𝑃𝑘,𝑙}−→𝐿[𝑆_{𝜃𝑟,𝑠},‖⋅,⋅‖] then 𝑥_{𝑃𝑘,𝑙}−→𝐿[𝐴𝐶_{𝜃𝑟,𝑠},‖⋅,⋅‖](4)[𝑆_{𝜃𝑟,𝑠},‖⋅,⋅‖]∩(𝑙_∞)₂=[𝐴𝐶_{𝜃𝑟,𝑠},‖⋅,⋅‖]∩(𝑙_∞)₂where (𝑙_∞)₂ is the space of all bounded double sequence.

Proof. (1) Since for all 𝑚 and 𝑛, and every 𝑧∈𝑋||(𝑘,𝑙)∈𝐼_𝑟,𝑠∶‖‖𝑥_{𝑘+𝑚,𝑙+𝑛}−𝐿,𝑧‖‖≥𝜀||≤≤_{𝑘,𝑙∈𝐼𝑟,𝑠,‖‖𝑥𝑘+𝑚,𝑙+𝑛‖‖−𝐿,𝑧≥𝜀}‖‖𝑥_{𝑘+𝑚,𝑙+𝑛}−𝐿,𝑧_{𝑘,𝑙∈𝐼𝑟,𝑠}‖‖‖‖𝑥_{𝑘+𝑚,𝑙+𝑛}−𝐿,𝑧‖‖,(3.17) and for all 𝑚 and 𝑛𝑃−lim_𝑟,𝑠1ℎ_𝑟,𝑠_{𝑘,𝑙∈𝐼𝑟,𝑠}‖‖𝑥_{𝑘+𝑚,𝑙+𝑛}−𝐿,𝑧‖‖=0.(3.18) This show that for all 𝑚 and 𝑛𝑃−lim_𝑟,𝑠1ℎ_𝑟,𝑠||(𝑘,𝑙)∈𝐼_𝑟,𝑠∶‖‖𝑥_{𝑘+𝑚,𝑙+𝑛}−𝐿,𝑧‖‖≥𝜀||=0,(3.19) this completes the proof of (1)(2) Let 𝑥 be defined as follows:𝑥_𝑘,𝑙=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝123⋯³223⋯√₃ℎ_𝑟,𝑠0⋅⋅⋅2⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅√ℎ⋅⋅⋅_𝑟,𝑠0⋅⋅⋅⋅⋅⋅⋅⋅₃⋅0⋅⋅⋅√0ℎ⋅⋅⋅_𝑟,𝑠√³0ℎ⋅⋅⋅_𝑟,𝑠⋯0⋅⋅⋅√³0ℎ⋅⋅⋅_𝑟,𝑠00⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.(3.20)
It is obvious that 𝑥 is an unbounded double sequence and for 𝜀>0, for all 𝑚 and 𝑛, and for every 𝑧∈𝑋𝑃−lim_𝑟,𝑠1ℎ_𝑟,𝑠||(𝑘,𝑙)∈𝐼_𝑟,𝑠∶‖‖𝑥_{𝑘+𝑚,𝑙+𝑛}−𝐿,𝑧‖‖≥𝜀||=𝑃−lim_𝑟,𝑠√³ℎ_𝑟,𝑠ℎ_𝑟,𝑠=0.(3.21) Thus 𝑥_{𝑃𝑘,𝑙}−→0[𝑆_{𝜃𝑟,𝑠},‖⋅,⋅‖]. But 𝑃−lim_𝑟,𝑠1ℎ_𝑟,𝑠_{𝑘,𝑙∈𝐼𝑟,𝑠}‖‖𝑥_𝑘,𝑙,𝑧‖‖=𝑃−lim_𝑟,𝑠√³ℎ_𝑟,𝑠√³ℎ_𝑟,𝑠√³ℎ_𝑟,𝑠+12ℎ_𝑟,𝑠=12.(3.22) Therefore 𝑥_{𝑃𝑘,𝑙}↛𝐿[𝐴𝐶_{𝜃𝑟,𝑠},‖⋅,⋅‖] which is the proof of (2).(3)Let 𝑥∈(𝑙_∞)₂ and 𝑥_{𝑃𝑘,𝑙}−→𝐿[𝑆_{𝜃𝑟,𝑠},‖⋅,⋅‖]. Assuming that for all 𝑚 and 𝑛, and every 𝑧∈𝑋‖𝑥_{𝑘+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:1ℎ_𝑟,𝑠_{𝑘,𝑙∈𝐼𝑟,𝑠}=1‖‖𝑥_{𝑘+𝑚,𝑙+𝑛}−𝐿,𝑧‖‖ℎ_𝑟,𝑠_{𝑘,𝑙∈𝐼𝑟,𝑠,‖‖𝑥𝑘+𝑚,𝑙+𝑛‖‖−𝐿,𝑧≥𝜀}‖‖𝑥_{𝑘+𝑚,𝑙+𝑛}−𝐿,𝑧‖‖+1ℎ_𝑟,𝑠_{𝑘,𝑙∈𝐼𝑟,𝑠,‖‖𝑥𝑘+𝑚,𝑙+𝑛‖‖−𝐿,𝑧<𝜀}≤𝐾‖‖𝑥_{𝑘+𝑚,𝑙+𝑛}−𝐿,𝑧‖‖ℎ_𝑟,𝑠||(𝑘,𝑙)∈𝐼_𝑟,𝑠∶‖‖𝑥_{𝑘+𝑚,𝑙+𝑛}−𝐿,𝑧‖‖≥𝜀||+𝜀.(3.23) Therefore 𝑥∈(𝑙_∞)₂ and 𝑥_{𝑃𝑘,𝑙}−→𝐿[𝑆_{𝜃𝑟,𝑠},‖⋅,⋅‖], this shows that 𝑥_{𝑃𝑘,𝑙}−→𝐿[𝐴𝐶_{𝜃𝑟,𝑠},‖⋅,⋅‖].(4)from (1), (2) and (3), we get [𝑆_{𝜃𝑟,𝑠},‖⋅,⋅‖]∩(𝑙_∞)₂=[𝐴𝐶_{𝜃𝑟,𝑠},‖⋅,⋅‖]∩(𝑙_∞)₂.

Theorem 3.10. For any Orlicz function 𝑀, [𝐴𝐶_{𝜃𝑟,𝑠},𝑀,‖⋅,⋅‖]⊂[𝑆_{𝜃𝑟,𝑠},‖⋅,⋅‖]

Proof. Let 𝑥∈[𝐴𝐶_{𝜃𝑟,𝑠},𝑀,‖⋅,⋅‖] and 𝜀>0. Then for all 𝑚 and 𝑛, and every 𝑧∈𝑋1ℎ_𝑟,𝑠_{𝑘,𝑙∈𝐼𝑟,𝑠}𝑀‖‖‖𝑥^{𝑘+𝑚,𝑙+𝑛}−𝐿𝜌,𝑧‖‖‖≥ℎ1_𝑟,𝑠_{𝑘,𝑙∈𝐼𝑟,𝑠,‖‖𝑥𝑘+𝑚,𝑙+𝑛‖‖−𝐿≥𝜀}𝑀‖‖‖𝑥^{𝑘+𝑚,𝑙+𝑛}−𝐿>1𝜌,𝑧‖‖‖ℎ_𝑟,𝑠𝑀𝜀𝜌||(𝑘,𝑙)∈𝐼^𝑟,𝑠∶‖‖𝑥_{𝑘+𝑚,𝑙+𝑛}−𝐿,𝑧‖‖≥𝜀||.(3.24) This shows that 𝑥∈[𝑆_{𝜃𝑟,𝑠},‖⋅,⋅‖].