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ISRN Mathematical Analysis
Volume 2012 (2012), Article ID 378575, 11 pages
doi:10.5402/2012/378575
Research Article

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

Orhan Tuğ

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.

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 − l i m 𝑥 = 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 𝜃 𝑟 , 𝑠 − 𝑃 -convergence.

By a lacunary 𝜃 = ( 𝑘 𝑟 ) ; 𝑟 = 0 , 1 , 2 , … where 𝑘 0 = 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 ) = 𝑥 0 𝑝 ( 𝑡 ) 𝑑 𝑡 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 𝑚 𝑛 ) ( 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 𝑚 𝑛 − 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 ‖ ⋅ , ⋅ ‖ ∶ 𝑋 𝑥 𝑋 → 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) ‖ 𝛼 𝑥 , 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 l i m 𝑘 → ∞ ‖ 𝑥 𝑘 − 𝐿 , 𝑧 ‖ = 0 for every 𝑧 ∈ 𝑋 . In this case, we write l i m 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 − l i m 𝑝 , 𝑞 → ∞ s u p 𝑚 , 𝑛 ≥ 0 1 𝑝 𝑞 𝑚 + 𝑝 − 1 𝑘 = 𝑚 𝑛 + 𝑞 − 1 𝑙 = 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 [ ̂ 𝑐 2 ] by Savaş and Paterson [16].
We can define the set of almost 𝑃 -convergent double sequence in ( 𝑋 , ‖ ⋅ , ⋅ ‖ ) similar to above definition as follow: ̂ 𝑐 2 = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 𝑥 , ‖ ⋅ , ⋅ ‖ 𝑥 = 𝑘 , 𝑙 ∶ 𝑃 − l i m 𝑝 , 𝑞 ⟶ ∞ s u p 𝑚 , 𝑛 ≥ 0 1 𝑝 𝑞 𝑚 + 𝑝 − 1 𝑘 = 𝑚 𝑛 + q − 1 𝑙 = 𝑛 ‖ ‖ 𝑥 𝑘 , 𝑙 ‖ ‖ ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ − 𝐿 , 𝑧 = 0 f o r e v e r y 𝑧 ∈ 𝑋 . ( 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 ̂ 𝑐 2 = ⎧ ⎪ ⎨ ⎪ ⎩ 𝑥 , 𝑀 , 𝑝 , ‖ ⋅ , ⋅ ‖ 𝑥 = 𝑘 , 𝑙 ∶ 𝑃 − l i m 𝑝 , 𝑞 1 𝑝 𝑞 𝑝 , 𝑞 𝑘 , 𝑙 = 1 , 1 𝑀 ‖ ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 − 𝐿 𝜌 ‖ ‖ ‖ , 𝑧 𝑝 𝑘 , 𝑙 ⎫ ⎪ ⎬ ⎪ ⎭ = 0 u n i f o r m l y i n 𝑚 a n d 𝑛 , f o r s o m e 𝜌 > 0 a n d 𝐿 , a n d e v e r y 𝑧 ∈ 𝑋 . ( 2 . 4 )
If we choose 𝑀 ( 𝑥 ) = 𝑥 and ( 𝑝 𝑘 , 𝑙 ) = 1 for all 𝑘 and 𝑙 , then [ ̂ 𝑐 2 , 𝑀 , 𝑝 , ‖ ⋅ , ⋅ ‖ ] = [ ̂ 𝑐 2 , ‖ ⋅ , ⋅ ‖ ] 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 𝑘 0 = 0 , ℎ 𝑟 = 𝑘 𝑟 − 𝑘 𝑟 − 1 𝑙 ⟶ ∞ a s 𝑟 ⟶ ∞ , 0 = 0 , ℎ 𝑠 = 𝑙 𝑠 − 𝑙 𝑠 − 1 ⟶ ∞ a s 𝑠 ⟶ ∞ . ( 2 . 5 )
Let 𝑘 𝑟 , 𝑠 = 𝑘 𝑟 𝑙 𝑠 , ℎ 𝑟 , 𝑠 = ℎ 𝑟 ℎ 𝑠 and 𝜃 𝑟 , 𝑠 is defined by 𝐼 𝑟 , 𝑠 = ( 𝑘 , 𝑙 ) ∶ 𝑘 𝑟 − 1 < 𝑘 ≤ 𝑘 𝑟 a n d 𝑙 𝑠 − 1 < 𝑙 ≤ 𝑙 𝑠 , 𝑞 𝑟 = 𝑘 𝑟 𝑘 𝑟 − 1 , 𝑞 s = 𝑙 𝑠 𝑙 𝑠 − 1 , 𝑞 𝑟 , 𝑠 = 𝑞 𝑟 𝑞 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: 𝐴 𝐶 𝜃 𝑟 , 𝑠 , = ⎧ ⎪ ⎨ ⎪ ⎩ 𝑥 ‖ ⋅ , ⋅ ‖ 𝑥 = 𝑘 , 𝑙 ∶ 𝑃 − l i m 𝑟 , 𝑠 1 ℎ 𝑟 , 𝑠 𝑘 , 𝑙 ∈ 𝐼 𝑟 , 𝑠 ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 ‖ ‖ ⎫ ⎪ ⎬ ⎪ ⎭ − 𝐿 , 𝑧 = 0 u n i f o r m l y i n 𝑚 a n d 𝑛 , f o r s o m e 𝐿 a n d e v e r y 𝑧 ∈ 𝑋 𝐴 𝐶 𝜃 𝑟 , 𝑠 , ‖ ⋅ , ⋅ ‖ 0 = ⎧ ⎪ ⎨ ⎪ ⎩ 𝑥 𝑥 = 𝑘 , 𝑙 ∶ 𝑃 − l i m 𝑟 , 𝑠 1 ℎ 𝑟 , 𝑠 𝑘 , 𝑙 ∈ 𝐼 𝑟 , 𝑠 ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 ‖ ‖ ⎫ ⎪ ⎬ ⎪ ⎭ , 𝑧 = 0 u n i f o r m l y i n 𝑚 a n d 𝑛 a n d e v e r y 𝑧 ∈ 𝑋 𝐴 𝐶 𝜃 𝑟 , 𝑠 = ⎧ ⎪ ⎨ ⎪ ⎩ 𝑥 , 𝑀 , 𝑝 , ‖ ⋅ , ⋅ ‖ 𝑥 = 𝑘 , 𝑙 ∶ 𝑃 − l i m 𝑟 , 𝑠 1 ℎ 𝑟 , 𝑠 𝑘 , 𝑙 ∈ 𝐼 𝑟 , 𝑠 𝑀 ‖ ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 − 𝐿 𝜌 ‖ ‖ ‖ , 𝑧 𝑝 𝑘 , 𝑙 ⎫ ⎪ ⎬ ⎪ ⎭ = 0 u n i f o r m l y i n 𝑚 a n d 𝑛 , f o r s o m e 𝜌 > 0 a n d 𝐿 , a n d e v e r y 𝑧 ∈ 𝑋 𝐴 𝐶 𝜃 𝑟 , 𝑠 , 𝑀 , 𝑝 , ‖ ⋅ , ⋅ ‖ 0 = ⎧ ⎪ ⎨ ⎪ ⎩ 𝑥 𝑥 = 𝑘 , 𝑙 ∶ 𝑃 − l i m 𝑟 , 𝑠 1 ℎ 𝑟 , 𝑠 𝑘 , 𝑙 ∈ 𝐼 𝑟 , 𝑠 𝑀 ‖ ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 𝜌 ‖ ‖ ‖ , 𝑧 𝑝 𝑘 , 𝑙 ⎫ ⎪ ⎬ ⎪ ⎭ . = 0 u n i f o r m l y i n 𝑚 a n d 𝑛 , f o r s o m e 𝜌 > 0 a n d e v e r y 𝑧 ∈ 𝑋 ( 2 . 7 )
When ( 𝑝 𝑘 , 𝑙 ) = 1 for all 𝑘 and 𝑙 , we shall denote [ 𝐴 𝐶 𝜃 𝑟 , 𝑠 , 𝑀 , 𝑝 , ‖ ⋅ , ⋅ ‖ ] and [ 𝐴 𝐶 𝜃 𝑟 , 𝑠 , 𝑀 , 𝑝 , ‖ ⋅ , ⋅ ‖ ] 0 as [ 𝐴 𝐶 𝜃 𝑟 , 𝑠 , 𝑀 , ‖ ⋅ , ⋅ ‖ ] and [ 𝐴 𝐶 𝜃 𝑟 , 𝑠 , 𝑀 , ‖ ⋅ , ⋅ ‖ ] 0 . That is, 𝐴 𝐶 𝜃 𝑟 , 𝑠 = ⎧ ⎪ ⎨ ⎪ ⎩ 𝑥 𝑀 , ‖ ⋅ , ⋅ ‖ 𝑥 = 𝑘 , 𝑙 ∶ 𝑃 − l i m 𝑟 , 𝑠 1 ℎ 𝑟 , 𝑠 𝑘 , 𝑙 ∈ 𝐼 𝑟 , 𝑠 𝑀 ‖ ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 − 𝐿 𝜌 ‖ ‖ ‖ ⎫ ⎪ ⎬ ⎪ ⎭ , , 𝑧 = 0 u n i f o r m l y i n 𝑚 a n d 𝑛 , f o r s o m e 𝜌 > 0 a n d 𝐿 , a n d e v e r y 𝑧 ∈ 𝑋 𝐴 𝐶 𝜃 𝑟 , 𝑠 𝑀 , ‖ ⋅ , ⋅ ‖ 0 = ⎧ ⎪ ⎨ ⎪ ⎩ 𝑥 𝑥 = 𝑘 , 𝑙 ∶ 𝑃 − l i m 𝑟 , 𝑠 1 ℎ 𝑟 , 𝑠 𝑘 , 𝑙 ∈ 𝐼 𝑟 , 𝑠 𝑀 ‖ ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 𝜌 ‖ ‖ ‖ ⎫ ⎪ ⎬ ⎪ ⎭ . , 𝑧 = 0 u n i f o r m l y i n 𝑚 a n d 𝑛 , f o r s o m e 𝜌 > 0 a n d 𝐿 , a n d e v e r y 𝑧 ∈ 𝑋 ( 2 . 8 )
If 𝑥 ∈ [ 𝐴 𝐶 𝜃 𝑟 , 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 [ 𝐴 𝐶 𝜃 𝑟 , 𝑠 , 𝑀 , 𝑝 , ‖ ⋅ , ⋅ ‖ ] = [ 𝐴 𝐶 𝜃 𝑟 , 𝑠 , ‖ ⋅ , ⋅ ‖ ] and [ 𝐴 𝐶 𝜃 𝑟 , 𝑠 , 𝑀 , 𝑝 , ‖ ⋅ , ⋅ ‖ ] 0 = [ 𝐴 𝐶 𝜃 𝑟 , 𝑠 , ‖ ⋅ , ⋅ ‖ ] 0 which are defined above. Also note that if ( 𝑝 𝑘 , 𝑙 ) = 1 for all 𝑘 and 𝑙 , then [ 𝐴 𝐶 𝜃 𝑟 , 𝑠 , 𝑀 , 𝑝 , ‖ ⋅ , ⋅ ‖ ] = [ 𝐴 𝐶 𝜃 𝑟 , 𝑠 , 𝑀 , ‖ ⋅ , ⋅ ‖ ] and [ 𝐴 𝐶 𝜃 𝑟 , 𝑠 , 𝑀 , 𝑝 , ‖ ⋅ , ⋅ ‖ ] 0 = [ 𝐴 𝐶 𝜃 𝑟 , 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 , [ 𝐴 𝐶 𝜃 𝑟 , 𝑠 , 𝑀 , 𝑝 , ‖ ⋅ , ⋅ ‖ ] and [ 𝐴 𝐶 𝜃 𝑟 , 𝑠 , 𝑀 , 𝑝 , ‖ ⋅ , ⋅ ‖ ] 0 are linear spaces.

Proof. Suppose that 𝑥 , 𝑦 ∈ [ 𝐴 𝐶 𝜃 𝑟 , 𝑠 , 𝑀 , 𝑝 , ‖ ⋅ , ⋅ ‖ ] 0 and 𝛼 , 𝛽 ∈ ℝ . So we have 𝐴 1 𝑟 , 𝑠 = ⎧ ⎪ ⎨ ⎪ ⎩ 𝑥 𝑥 = 𝑘 , 𝑙 ∶ 𝑃 − l i m 𝑟 , 𝑠 1 ℎ 𝑟 , 𝑠 𝑘 , 𝑙 ∈ 𝐼 𝑟 , 𝑠 𝑀 ‖ ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 𝜌 1 ‖ ‖ ‖ , 𝑧 𝑝 𝑘 , 𝑙 = 0 u n i f o r m l y i n 𝑚 a n d 𝑛 , f o r s o m e 𝜌 1 ⎫ ⎪ ⎬ ⎪ ⎭ , 𝐴 > 0 a n d e v e r y 𝑧 ∈ 𝑋 2 𝑟 , 𝑠 = ⎧ ⎪ ⎨ ⎪ ⎩ 𝑦 𝑦 = 𝑘 , 𝑙 ∶ 𝑃 − l i m 𝑟 , 𝑠 1 ℎ 𝑟 , 𝑠 𝑘 , 𝑙 ∈ 𝐼 𝑟 , 𝑠 𝑀 ‖ ‖ ‖ 𝑦 𝑘 + 𝑚 , 𝑙 + 𝑛 𝜌 2 ‖ ‖ ‖ , 𝑧 𝑝 𝑘 , 𝑙 = 0 u n i f o r m l y i n 𝑚 a n d 𝑛 , f o r s o m e 𝜌 2 ⎫ ⎪ ⎬ ⎪ ⎭ . > 0 a n d e v e r y 𝑧 ∈ 𝑋 ( 3 . 1 ) Since 𝑀 is an Orlicz function we have the following inequality 1 ℎ 𝑟 , 𝑠 𝑘 , 𝑙 ∈ 𝐼 𝑟 , 𝑠 𝑀 ‖ ‖ ‖ 𝛼 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 + 𝛽 𝑦 𝑘 + 𝑚 , 𝑙 + 𝑛 | 𝛼 | 𝜌 1 + | | 𝛽 | | 𝜌 2 ‖ ‖ ‖ , 𝑧 𝑝 𝑘 , 𝑙 1 ≤ 𝐷 ℎ 𝑟 , 𝑠 𝑘 , 𝑙 ∈ 𝐼 𝑟 , 𝑠 | 𝛼 | | 𝛼 | 𝜌 1 + | | 𝛽 | | 𝜌 2 𝑀 ‖ ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 𝜌 1 ‖ ‖ ‖ , 𝑧 𝑝 𝑘 , 𝑙 1 + 𝐷 ℎ 𝑟 , 𝑠 𝑘 , 𝑙 ∈ 𝐼 𝑟 , 𝑠 | | 𝛽 | | | 𝛼 | 𝜌 1 + | | 𝛽 | | 𝜌 2 𝑀 ‖ ‖ ‖ 𝑦 𝑘 + 𝑚 , 𝑙 + 𝑛 𝜌 2 ‖ ‖ ‖ , 𝑧 𝑝 𝑘 , 𝑙 1 ≤ 𝐷 𝐹 ℎ 𝑟 , 𝑠 𝑘 , 𝑙 ∈ 𝐼 𝑟 , 𝑠 𝑀 ‖ ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 𝜌 1 ‖ ‖ ‖ , 𝑧 𝑝 𝑘 , 𝑙 1 + 𝐷 𝐹 ℎ 𝑟 , 𝑠 𝑘 , 𝑙 ∈ 𝐼 𝑟 , 𝑠 𝑀 ‖ ‖ ‖ 𝑦 𝑘 + 𝑚 , 𝑙 + 𝑛 𝜌 2 ‖ ‖ ‖ , z 𝑝 𝑘 , 𝑙 , ( 3 . 2 ) where 𝐹 = m a x [ 1 , ( | 𝛼 | / ( | 𝛼 | 𝜌 1 + | 𝛽 | 𝜌 2 ) ) 𝐻 , ( | 𝛽 | / ( | 𝛼 | 𝜌 1 + | 𝛽 | 𝜌 2 ) ) 𝐻 ] . When we take the limit of each side as 𝑟 , 𝑠 → ∞ ⎧ ⎪ ⎨ ⎪ ⎩ 𝑥 𝑥 = 𝑘 , 𝑙 𝑦 , 𝑦 = 𝑘 , 𝑙 ∶ 𝑃 − l i m 𝑟 , 𝑠 1 ℎ 𝑟 , 𝑠 𝑘 , 𝑙 ∈ 𝐼 𝑟 , 𝑠 𝑀 ‖ ‖ ‖ 𝛼 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 + 𝛽 𝑦 𝑘 + 𝑚 , 𝑙 + 𝑛 | 𝛼 | 𝜌 1 + | | 𝛽 | | 𝜌 2 ‖ ‖ ‖ , 𝑧 𝑝 𝑘 , 𝑙 = 𝑜 u n i f o r m l y i n 𝑚 a n d 𝑛 , f o r s o m e 𝜌 1 , 𝜌 2 ⎫ ⎪ ⎬ ⎪ ⎭ > 0 a n d e a c h 𝑧 ∈ 𝑋 . ( 3 . 3 ) So this is the result.

Definition 3.2. An Orlicz function 𝑀 is said to be satisfy Δ 2 -condition for all values of 𝑢 , if there exists a constant 𝐾 > 0 such that 𝑀 ( 2 𝑢 ) ≤ 𝐾 𝑀 ( 𝑢 ) ( 𝑢 ≥ 0 ) . ( 3 . 4 )

Lemma 3.3. Let 𝑀 be an Orlicz function which satisfies Δ 2 -condition and 0 < 𝛿 < 1 . Then for each 𝑥 ≥ 𝛿 and some constant 𝐾 > 0 we have 𝑀 ( 𝑥 ) ≤ 𝐾 𝛿 − 1 𝑀 ( 2 ) . ( 3 . 5 )

Theorem 3.4. For any Orlicz function 𝑀 which satisfies Δ 2 -condition, we have [ 𝐴 𝐶 𝜃 𝑟 , 𝑠 , ‖ ⋅ , ⋅ ‖ ] ⊆ [ 𝐴 𝐶 𝜃 𝑟 , 𝑠 𝑀 , ‖ ⋅ , ⋅ ‖ ]

Proof. Let 𝑥 ∈ [ 𝐴 𝐶 𝜃 𝑟 , 𝑠 , ‖ ⋅ , ⋅ ‖ ] . For each 𝑚 and 𝑛 𝐴 𝑟 , 𝑠 = ⎧ ⎪ ⎨ ⎪ ⎩ 𝑥 𝑥 = 𝑘 , 𝑙 ∶ 𝑃 − l i m 𝑟 , 𝑠 1 ℎ r , 𝑠 𝑘 , 𝑙 ∈ 𝐼 𝑟 , 𝑠 ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 ‖ ‖ ⎫ ⎪ ⎬ ⎪ ⎭ − 𝐿 , 𝑧 = 0 f o r s o m e 𝐿 , a n d e v e r y 𝑧 ∈ 𝑋 . ( 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 1 ℎ 𝑟 , 𝑠 𝑘 , 𝑙 ∈ 𝐼 𝑟 , 𝑠 𝑀 ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 ‖ ‖ = 1 − 𝐿 , 𝑧 ℎ 𝑟 , 𝑠 𝑘 , 𝑙 ∈ 𝐼 𝑟 , 𝑠 , ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 ‖ ‖ − 𝐿 , 𝑧 ≤ 𝛿 𝑀 ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 ‖ ‖ + 1 − 𝐿 , 𝑧 ℎ 𝑟 , 𝑠 𝑘 , 𝑙 ∈ 𝐼 𝑟 , 𝑠 ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 ‖ ‖ − 𝐿 , 𝑧 > 𝛿 𝑀 ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 ‖ ‖ ≤ 1 − 𝐿 , 𝑧 ℎ 𝑟 , 𝑠 ℎ 𝑟 , 𝑠 𝜀 + 1 ℎ 𝑟 , 𝑠 𝑘 , 𝑙 ∈ 𝐼 𝑟 , 𝑠 , ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 ‖ ‖ − 𝐿 , 𝑧 > 𝛿 𝑀 ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 ‖ ‖ < 1 − 𝐿 , 𝑧 ℎ 𝑟 , 𝑠 ℎ 𝑟 , 𝑠 𝜀 + 1 ℎ 𝑟 , 𝑠 𝐾 𝛿 − 1 𝑀 ( 2 ) ℎ 𝑟 , 𝑠 𝐴 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 ∈ [ 𝐴 𝐶 𝜃 𝑟 , 𝑠 𝑀 , ‖ ⋅ , ⋅ ‖ ] .

Theorem 3.5. Let 𝜃 𝑟 , 𝑠 = { 𝑘 𝑟 , 𝑙 𝑠 } be a double lacunary sequence with l i m i n f 𝑟 𝑞 𝑟 > 1 and l i m i n f 𝑠 𝑞 𝑠 > 1 then for any Orlicz function 𝑀 , [ ̂ 𝑐 2 , 𝑀 , 𝑝 , ‖ ⋅ , ⋅ ‖ ] ⊂ [ 𝐴 𝐶 𝜃 𝑟 , 𝑠 , 𝑀 , 𝑝 , ‖ ⋅ , ⋅ ‖ ] .

Proof. Since l i m i n f 𝑟 𝑞 𝑟 > 1 and l i m i n f 𝑠 𝑞 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 ∈ [ ̂ 𝑐 2 , 𝑀 , 𝑝 , ‖ ⋅ , ⋅ ‖ ] , we can write for each 𝑚 and 𝑛 , some 𝐿 and every 𝑧 ∈ 𝑋 𝐵 𝑟 , 𝑠 = 1 ℎ 𝑟 , 𝑠 𝑘 , 𝑙 ∈ 𝐼 𝑟 , 𝑠 𝑀 ‖ ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 − 𝐿 𝜌 ‖ ‖ ‖ , 𝑧 𝑝 𝑘 , 𝑙 = 1 ℎ 𝑘 𝑟 , 𝑠 𝑟 𝑙 𝑘 = 1 𝑠 𝑙 = 1 𝑀 ‖ ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 − 𝐿 𝜌 ‖ ‖ ‖ , 𝑧 𝑝 𝑘 , 𝑙 − 1 ℎ 𝑘 𝑟 , 𝑠 𝑟 − 1 𝑙 𝑘 = 1 𝑠 − 1 𝑙 = 1 𝑀 ‖ ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 − 𝐿 𝜌 ‖ ‖ ‖ , 𝑧 𝑝 𝑘 , 𝑙 − 1 ℎ 𝑘 𝑟 , 𝑠 𝑟 𝑘 = 𝑘 𝑟 − 1 𝑙 + 1 𝑠 − 1 𝑙 = 1 𝑀 ‖ ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 − 𝐿 𝜌 ‖ ‖ ‖ , 𝑧 𝑝 𝑘 , 𝑙 − 1 ℎ 𝑘 𝑟 , 𝑠 𝑟 − 1 𝑙 𝑘 = 1 𝑠 𝑙 = 𝑙 𝑠 − 1 + 1 𝑀 ‖ ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 − 𝐿 𝜌 ‖ ‖ ‖ , 𝑧 𝑝 𝑘 , 𝑙 − 𝑘 𝑟 𝑙 𝑠 ℎ 𝑟 , 𝑠 ⎛ ⎜ ⎜ ⎝ 1 𝑘 𝑟 𝑙 𝑠 𝑘 𝑟 𝑙 𝑘 = 1 𝑠 𝑙 = 1 𝑀 ‖ ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 − 𝐿 𝜌 ‖ ‖ ‖ , 𝑧 𝑝 𝑘 , 𝑙 ⎞ ⎟ ⎟ ⎠ − 𝑘 𝑟 𝑙 𝑠 ℎ 𝑟 , 𝑠 ⎛ ⎜ ⎜ ⎝ 1 𝑘 𝑟 𝑙 𝑠 𝑘 𝑟 𝑙 𝑘 = 1 𝑠 𝑙 = 1 𝑀 ‖ ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 − 𝐿 𝜌 ‖ ‖ ‖ , 𝑧 𝑝 𝑘 , 𝑙 ⎞ ⎟ ⎟ ⎠ = 𝑘 r 𝑙 𝑠 ℎ 𝑟 , 𝑠 ⎛ ⎜ ⎜ ⎝ 1 𝑘 𝑟 𝑙 𝑠 𝑘 𝑟 𝑙 𝑘 = 1 𝑠 𝑙 = 1 𝑀 ‖ ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 − 𝐿 𝜌 ‖ ‖ ‖ , 𝑧 𝑝 𝑘 , 𝑙 ⎞ ⎟ ⎟ ⎠ − 𝑘 𝑟 − 1 𝑙 𝑠 − 1 ℎ 𝑟 , 𝑠 ⎛ ⎜ ⎜ ⎝ 1 𝑘 𝑟 − 1 𝑙 𝑘 𝑠 − 1 𝑟 − 1 𝑙 𝑘 = 1 𝑠 − 1 𝑙 = 1 𝑀 ‖ ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 − 𝐿 𝜌 ‖ ‖ ‖ , 𝑧 𝑝 𝑘 , 𝑙 ⎞ ⎟ ⎟ ⎠ − 1 ℎ 𝑟 𝑘 𝑟 𝑘 = 𝑘 𝑟 − 1 + 1 𝑙 𝑠 − 1 ℎ 𝑠 1 𝑙 𝑙 𝑠 − 1 𝑠 − 1 𝑙 = 1 𝑀 ‖ ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 − 𝐿 𝜌 ‖ ‖ ‖ , 𝑧 𝑝 𝑘 , 𝑙 − 1 ℎ 𝑠 𝑙 𝑠 𝑙 = 𝑙 𝑠 − 1 + 1 𝑘 𝑟 − 1 ℎ 𝑟 1 𝑘 𝑘 𝑟 − 1 𝑟 − 1 𝑘 = 1 𝑀 ‖ ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 − 𝐿 𝜌 ‖ ‖ ‖ , 𝑧 𝑝 𝑘 , 𝑙 . ( 3 . 8 ) Since 𝑥 ∈ [ ̂ 𝑐 2 , 𝑀 , 𝑝 , ‖ ⋅ , ⋅ ‖ ] the last two terms tends to 0 uniformly in 𝑚 and 𝑛 in Pringsheim sense. Thus for each 𝑚 and 𝑛 𝐵 𝑟 , 𝑠 = 𝑘 𝑟 𝑙 𝑠 ℎ 𝑟 , 𝑠 ⎛ ⎜ ⎜ ⎝ 1 𝑘 𝑟 𝑙 𝑠 𝑘 𝑟 𝑙 𝑘 = 1 𝑠 𝑙 = 1 𝑀 ‖ ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 − 𝐿 𝜌 ‖ ‖ ‖ , 𝑧 𝑝 𝑘 , 𝑙 ⎞ ⎟ ⎟ ⎠ − 𝑘 𝑟 − 1 𝑙 𝑠 − 1 ℎ 𝑟 , 𝑠 ⎛ ⎜ ⎜ ⎝ 1 𝑘 𝑟 − 1 𝑙 𝑘 𝑠 − 1 𝑟 − 1 𝑙 𝑘 = 1 𝑠 − 1 𝑙 = 1 𝑀 ‖ ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 − 𝐿 𝜌 ‖ ‖ ‖ , 𝑧 𝑝 𝑘 , 𝑙 ⎞ ⎟ ⎟ ⎠ + 𝑜 ( 1 ) . ( 3 . 9 ) Since ℎ 𝑟 , 𝑠 = 𝑘 𝑟 𝑙 𝑠 − 𝑘 𝑟 − 1 𝑙 𝑠 − 1 we get for each 𝑚 and 𝑛 the following inequalities as follow: 𝑘 𝑟 𝑙 𝑠 ℎ 𝑟 , 𝑠 ≤ 1 + 𝛿 𝛿 , 𝑘 𝑟 − 1 𝑙 𝑠 − 1 ℎ 𝑟 , 𝑠 ≤ 1 𝛿 . ( 3 . 1 0 ) Thus the terms 1 𝑘 𝑟 𝑙 𝑠 𝑘 𝑟 𝑙 𝑘 = 1 𝑠 𝑙 = 1 𝑀 ‖ ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 − 𝐿 𝜌 ‖ ‖ ‖ , 𝑧 𝑝 𝑘 , 𝑙 , 1 𝑘 𝑟 − 1 𝑙 𝑘 𝑠 − 1 𝑟 − 1 𝑙 𝑘 = 1 𝑠 − 1 𝑙 = 1 𝑀 ‖ ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 − 𝐿 𝜌 ‖ ‖ ‖ , 𝑧 𝑝 𝑘 , 𝑙 , ( 3 . 1 1 ) 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 ∈ [ 𝐴 𝐶 𝜃 𝑟 , 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 l i m s u p 𝑟 𝑞 𝑟 < ∞ and l i m s u p 𝑠 𝑞 𝑠 < ∞ then for any Orlicz function 𝑀 , [ 𝐴 𝐶 𝜃 𝑟 , 𝑠 , 𝑀 , 𝑝 , ‖ ⋅ , ⋅ ‖ ] ⊂ [ ̂ 𝑐 2 , 𝑀 , 𝑝 , ‖ ⋅ , ⋅ ‖ ] .

Proof. Since l i m s u p 𝑟 𝑞 𝑟 < ∞ and l i m s u p 𝑠 𝑞 𝑠 < ∞ there exists 𝐻 > 0 such that 𝑞 𝑟 < 𝐻 and 𝑞 𝑠 < 𝐻 for all 𝑟 and 𝑠 . Let 𝜀 > 0 and 𝑥 ∈ [ 𝐴 𝐶 𝜃 𝑟 , 𝑠 , 𝑀 , 𝑝 , ‖ ⋅ , ⋅ ‖ ] . There exists 𝑟 0 > 0 and 𝑠 0 > 0 such that for every 𝑖 ≥ 𝑟 0 and 𝑗 ≥ 𝑠 0 , and all 𝑚 and 𝑛 , for every 𝑧 ∈ 𝑋 𝐴 ′ 𝑖 , 𝑗 = 1 ℎ 𝑖 , 𝑗 𝑘 , 𝑙 ∈ 𝐼 𝑖 , 𝑗 𝑀 ‖ ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 𝜌 ‖ ‖ ‖ , 𝑧 𝑝 𝑘 , 𝑙 < 𝜀 . ( 3 . 1 2 ) Let 𝑀 𝐴 = m a x 𝑖 , 𝑗 ∶ 1 ≤ 𝑖 ≤ 𝑟 0 , 1 ≤ 𝑗 ≤ 𝑠 0 ( 3 . 1 3 ) and let 𝑘 𝑟 − 1 < 𝑝 ≤ 𝑘 𝑟 and 𝑙 𝑠 − 1 < 𝑞 ≤ 𝑙 𝑠 . Hence we get 1 𝑝 𝑞 𝑝 , 𝑞 𝑘 , 𝑙 ∈ 1 , 1 𝑀 ‖ ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 𝜌 ‖ ‖ ‖ , 𝑧 𝑝 𝑘 , 𝑙 ≤ 1 𝑘 𝑟 − 1 𝑙 𝑘 𝑠 − 1 𝑟 𝑙 𝑘 = 1 𝑠 𝑙 = 1 𝑀 ‖ ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 𝜌 ‖ ‖ ‖ , 𝑧 𝑝 𝑘 , 𝑙 ≤ 1 𝑘 𝑟 − 1 𝑙 𝑟 𝑠 − 1 0 , 𝑠 0 𝑡 , 𝑢 = 1 , 1 ⎛ ⎜ ⎜ ⎝ 𝑘 , 𝑙 ∈ 𝐼 𝑡 , 𝑢 𝑀 ‖ ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 𝜌 ‖ ‖ ‖ , 𝑧 𝑝 𝑘 , 𝑙 ⎞ ⎟ ⎟ ⎠ = 1 𝑘 𝑟 − 1 𝑙 𝑠 − 1 𝑟 , 𝑠 𝑡 , 𝑢 = 1 , 1 ℎ 𝑡 , 𝑢 𝐴 𝑡 , 𝑢 + 1 𝑘 𝑟 − 1 𝑙 𝑠 − 1 𝑟 0 ∪ 𝑠 < 𝑡 ≤ 𝑟 0 < 𝑢 ≤ 𝑠 ℎ 𝑡 , 𝑢 𝐴 𝑡 , 𝑢 ≤ 𝑀 𝑘 𝑟 − 1 𝑙 𝑟 𝑠 − 1 0 , 𝑠 0 𝑡 , 𝑢 = 1 , 1 ℎ 𝑡 , 𝑢 + 1 𝑘 𝑟 − 1 𝑙 𝑠 − 1 𝑟 0 ∪ 𝑠 < 𝑡 ≤ 𝑟 0 < 𝑢 ≤ 𝑠 ℎ 𝑡 , 𝑢 𝐴 𝑡 , 𝑢 ≤ 𝑀 𝑘 𝑟 0 𝑙 𝑠 0 𝑟 0 𝑠 0 𝑘 𝑟 − 1 𝑙 𝑠 − 1 + 1 𝑘 𝑟 − 1 𝑙 𝑠 − 1 𝑟 0 ∪ 𝑠 < 𝑡 ≤ 𝑟 0 < 𝑢 ≤ 𝑠 ℎ 𝑡 , 𝑢 𝐴 𝑡 , 𝑢 ≤ 𝑀 𝑘 𝑟 0 𝑙 𝑠 0 𝑟 0 𝑠 0 𝑘 𝑟 − 1 𝑙 𝑠 − 1 + s u p 𝑡 ≥ 𝑟 0 ∪ 𝑢 ≥ 𝑠 0 𝐴 𝑡 , 𝑢 1 𝑘 𝑟 − 1 𝑙 𝑠 − 1 𝑟 0 ∪ 𝑠 < 𝑡 ≤ 𝑟 0 < 𝑢 ≤ 𝑠 ℎ 𝑡 , 𝑢 ≤ 𝑀 𝑘 𝑟 0 𝑙 𝑠 0 𝑟 0 𝑠 0 𝑘 𝑟 − 1 𝑙 𝑠 − 1 + 1 𝑘 𝑟 − 1 𝑙 𝑠 − 1 𝜀 𝑟 0 ∪ 𝑠 < 𝑡 ≤ 𝑟 0 < 𝑢 ≤ 𝑠 ℎ 𝑡 , 𝑢 ≤ 𝑀 𝑘 𝑟 0 𝑙 𝑠 0 𝑟 0 𝑠 0 𝑘 𝑟 − 1 𝑙 𝑠 − 1 + 𝜀 𝐻 2 . ( 3 . 1 4 ) 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 , 1 𝑝 𝑞 𝑝 , 𝑞 𝑘 , 𝑙 ∈ 1 , 1 𝑀 ‖ ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 𝜌 ‖ ‖ ‖ , 𝑧 𝑝 𝑘 , 𝑙 ⟶ 0 . ( 3 . 1 5 ) Therefore 𝑥 ∈ [ ̂ 𝑐 2 , 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 < l i m i n f 𝑟 , 𝑠 𝑞 𝑟 , 𝑠 ≤ l i m s u p 𝑟 , 𝑠 𝑞 𝑟 , 𝑠 < ∞ , then for any Orlicz function 𝑀 [ 𝐴 𝐶 𝜃 𝑟 , 𝑠 , 𝑀 , 𝑝 , ‖ ⋅ , ⋅ ‖ ] = [ ̂ 𝑐 2 , 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 𝜃 𝑟 , 𝑠 − 𝑃 -convergent to 𝐿 provided that for every 𝜀 > 0 and 𝑧 ∈ 𝑋 𝑃 − l i m 𝑟 , 𝑠 1 ℎ 𝑟 , 𝑠 | | m a x ( 𝑘 , 𝑙 ) ∈ 𝐼 𝑟 , 𝑠 ∶ ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 ‖ ‖ | | − 𝐿 , 𝑧 ≥ 𝜀 = 0 . ( 3 . 1 6 ) So we can write 𝑆 𝜃 𝑟 , 𝑠 − l i m 𝑥 = 𝐿 .

Theorem 3.9. Let 𝜃 𝑟 , 𝑠 be a double lacunary sequence then(1) 𝑥 𝑃 𝑘 , 𝑙 − → 𝐿 [ 𝐴 𝐶 𝜃 𝑟 , 𝑠 , ‖ ⋅ , ⋅ ‖ ] implies 𝑥 𝑃 𝑘 , 𝑙 − → 𝐿 [ 𝑆 𝜃 𝑟 , 𝑠 , ‖ ⋅ , ⋅ ‖ ] (2) [ 𝐴 𝐶 𝜃 𝑟 , 𝑠 , ‖ ⋅ , ⋅ ‖ ] is a proper subset of [ 𝑆 𝜃 𝑟 , 𝑠 , ‖ ⋅ , ⋅ ‖ ] (3)If 𝑥 ∈ ( 𝑙 ∞ ) 2 and 𝑥 𝑃 𝑘 , 𝑙 − → 𝐿 [ 𝑆 𝜃 𝑟 , 𝑠 , ‖ ⋅ , ⋅ ‖ ] then 𝑥 𝑃 𝑘 , 𝑙 − → 𝐿 [ 𝐴 𝐶 𝜃 𝑟 , 𝑠 , ‖ ⋅ , ⋅ ‖ ] (4) [ 𝑆 𝜃 𝑟 , 𝑠 , ‖ ⋅ , ⋅ ‖ ] ∩ ( 𝑙 ∞ ) 2 = [ 𝐴 𝐶 𝜃 𝑟 , 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 ) ∈ 𝐼 𝑟 , 𝑠 ∶ ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 ‖ ‖ | | ≤ − 𝐿 , 𝑧 ≥ 𝜀 𝑘 , 𝑙 ∈ 𝐼 𝑟 , 𝑠 , ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 ‖ ‖ − 𝐿 , 𝑧 ≥ 𝜀 ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 ‖ ‖ ≤ − 𝐿 , 𝑧 𝑘 , 𝑙 ∈ 𝐼 𝑟 , 𝑠 ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 ‖ ‖ , − 𝐿 , 𝑧 ( 3 . 1 7 ) and for all 𝑚 and 𝑛 𝑃 − l i m 𝑟 , 𝑠 1 ℎ 𝑟 , 𝑠 𝑘 , 𝑙 ∈ 𝐼 𝑟 , 𝑠 ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 ‖ ‖ − 𝐿 , 𝑧 = 0 . ( 3 . 1 8 ) This show that for all 𝑚 and 𝑛 𝑃 − l i m 𝑟 , 𝑠 1 ℎ 𝑟 , 𝑠 | | ( 𝑘 , 𝑙 ) ∈ 𝐼 𝑟 , 𝑠 ∶ ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 ‖ ‖ | | − 𝐿 , 𝑧 ≥ 𝜀 = 0 , ( 3 . 1 9 ) this completes the proof of (1)(2) Let 𝑥 be defined as follows: 𝑥 𝑘 , 𝑙 = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 2 3 ⋯ 3 √ ℎ 𝑟 , 𝑠 0 ⋅ ⋅ ⋅ 2 2 3 ⋯ 3 √ ℎ 𝑟 , 𝑠 2 0 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 3 √ ℎ 𝑟 , 𝑠 3 √ ℎ 𝑟 , 𝑠 ⋯ 3 √ ℎ 𝑟 , 𝑠 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ 0 ⋅ ⋅ ⋅ 0 0 0 0 0 0 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ . ( 3 . 2 0 )
It is obvious that 𝑥 is an unbounded double sequence and for 𝜀 > 0 , for all 𝑚 and 𝑛 , and for every 𝑧 ∈ 𝑋 𝑃 − l i m 𝑟 , 𝑠 1 ℎ 𝑟 , 𝑠 | | ( 𝑘 , 𝑙 ) ∈ 𝐼 𝑟 , 𝑠 ∶ ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 ‖ ‖ | | − 𝐿 , 𝑧 ≥ 𝜀 = 𝑃 − l i m 𝑟 , 𝑠 3 √ ℎ 𝑟 , 𝑠 ℎ 𝑟 , 𝑠 = 0 . ( 3 . 2 1 ) Thus 𝑥 𝑃 𝑘 , 𝑙 − → 0 [ 𝑆 𝜃 𝑟 , 𝑠 , ‖ ⋅ , ⋅ ‖ ] . But 𝑃 − l i m 𝑟 , 𝑠 1 ℎ 𝑟 , 𝑠 𝑘 , 𝑙 ∈ 𝐼 𝑟 , 𝑠 ‖ ‖ 𝑥 𝑘 , 𝑙 ‖ ‖ , 𝑧 = 𝑃 − l i m 𝑟 , 𝑠 3 √ ℎ 𝑟 , 𝑠 3 √ ℎ 𝑟 , 𝑠 3 √ ℎ 𝑟 , 𝑠 + 1 2 ℎ 𝑟 , 𝑠 = 1 2 . ( 3 . 2 2 ) Therefore 𝑥 𝑃 𝑘 , 𝑙 ↛ 𝐿 [ 𝐴 𝐶 𝜃 𝑟 , 𝑠 , ‖ ⋅ , ⋅ ‖ ] which is the proof of (2).(3)Let 𝑥 ∈ ( 𝑙 ∞ ) 2 and 𝑥 𝑃 𝑘 , 𝑙 − → 𝐿 [ 𝑆 𝜃 𝑟 , 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: 1 ℎ 𝑟 , 𝑠 𝑘 , 𝑙 ∈ 𝐼 𝑟 , 𝑠 ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 ‖ ‖ = 1 − 𝐿 , 𝑧 ℎ 𝑟 , 𝑠 𝑘 , 𝑙 ∈ 𝐼 𝑟 , 𝑠 , ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 ‖ ‖ − 𝐿 , 𝑧 ≥ 𝜀 ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 ‖ ‖ + 1 − 𝐿 , 𝑧 ℎ 𝑟 , 𝑠 𝑘 , 𝑙 ∈ 𝐼 𝑟 , 𝑠 , ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 ‖ ‖ − 𝐿 , 𝑧 < 𝜀 ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 ‖ ‖ ≤ 𝐾 − 𝐿 , 𝑧 ℎ 𝑟 , 𝑠 | | ( 𝑘 , 𝑙 ) ∈ 𝐼 𝑟 , 𝑠 ∶ ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 ‖ ‖ | | − 𝐿 , 𝑧 ≥ 𝜀 + 𝜀 . ( 3 . 2 3 ) Therefore 𝑥 ∈ ( 𝑙 ∞ ) 2 and 𝑥 𝑃 𝑘 , 𝑙 − → 𝐿 [ 𝑆 𝜃 𝑟 , 𝑠 , ‖ ⋅ , ⋅ ‖ ] , this shows that 𝑥 𝑃 𝑘 , 𝑙 − → 𝐿 [ 𝐴 𝐶 𝜃 𝑟 , 𝑠 , ‖ ⋅ , ⋅ ‖ ] .(4)from (1), (2) and (3), we get [ 𝑆 𝜃 𝑟 , 𝑠 , ‖ ⋅ , ⋅ ‖ ] ∩ ( 𝑙 ∞ ) 2 = [ 𝐴 𝐶 𝜃 𝑟 , 𝑠 , ‖ ⋅ , ⋅ ‖ ] ∩ ( 𝑙 ∞ ) 2 .

Theorem 3.10. For any Orlicz function 𝑀 , [ 𝐴 𝐶 𝜃 𝑟 , 𝑠 , 𝑀 , ‖ ⋅ , ⋅ ‖ ] ⊂ [ 𝑆 𝜃 𝑟 , 𝑠 , ‖ ⋅ , ⋅ ‖ ]

Proof. Let 𝑥 ∈ [ 𝐴 𝐶 𝜃 𝑟 , 𝑠 , 𝑀 , ‖ ⋅ , ⋅ ‖ ] and 𝜀 > 0 . Then for all 𝑚 and 𝑛 , and every 𝑧 ∈ 𝑋 1 ℎ 𝑟 , 𝑠 𝑘 , 𝑙 ∈ 𝐼 𝑟 , 𝑠 𝑀 ‖ ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 − 𝐿 𝜌 ‖ ‖ ‖ ≥ 1 , 𝑧 ℎ 𝑟 , 𝑠 𝑘 , 𝑙 ∈ 𝐼 𝑟 , 𝑠 , ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 ‖ ‖ − 𝐿 ≥ 𝜀 𝑀 ‖ ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 − 𝐿 𝜌 ‖ ‖ ‖ > 1 , 𝑧 ℎ 𝑟 , 𝑠 𝑀 𝜀 𝜌 | | ( 𝑘 , 𝑙 ) ∈ 𝐼 𝑟 , 𝑠 ∶ ‖ ‖ 𝑥 𝑘 + 𝑚 , 𝑙 + 𝑛 ‖ ‖ | | . − 𝐿 , 𝑧 ≥ 𝜀 ( 3 . 2 4 ) This shows that 𝑥 ∈ [ 𝑆 𝜃 𝑟 , 𝑠 , ‖ ⋅ , ⋅ ‖ ] .

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