Some Paranormed Double Difference Sequence Spaces for Orlicz

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 419064, 10 pages http://dx.doi.org/10.1155/2014/419064

Research Article Some Paranormed Double Difference Sequence Spaces for Orlicz Functions and Bounded-Regular Matrices S. A. Mohiuddine,1 Kuldip Raj,2 and Abdullah Alotaibi1 1

Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia 2 School of Mathematics, Shri Mata Vaishno Devi University, Katra, Jammu and Kashmir 182320, India Correspondence should be addressed to S. A. Mohiuddine; [email protected] Received 23 November 2013; Accepted 14 January 2014; Published 10 March 2014 Academic Editor: M. Mursaleen Copyright © 2014 S. A. Mohiuddine et al. 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. The aim of this paper is to introduce some new double difference sequence spaces with the help of the Musielak-Orlicz function F = (𝐹𝑗𝑘 ) and four-dimensional bounded-regular (shortly, RH-regular) matrices 𝐴 = (𝑎𝑛𝑚𝑗𝑘 ). We also make an effort to study some topological properties and inclusion relations between these double difference sequence spaces.

1. Introduction, Notations, and Preliminaries In [1], Hardy introduced the concept of regular convergence for double sequences. Some important work on double sequences is also found by Bromwich [2]. Later on, it was studied by various authors, for example, Móricz [3], Móricz and Rhoades [4], Bas¸arır and Sonalcan [5], Mursaleen and Mohiuddine [6–8], and many others. Mursaleen [9] has defined and characterized the notion of almost strong regularity of four-dimensional matrices and applied these matrices to establish a core theorem (also see [10, 11]). Altay and Bas¸ar [12] have recently introduced the double sequence spaces BS, BS(𝑡), CS𝑝 , CS𝑏𝑝 , CS𝑟 , and BV consisting of all double series whose sequence of partial sums are in the spaces M𝑢 , M𝑢 (𝑡), C𝑝 , C𝑏𝑝 , C𝑟 , and L𝑢 , respectively. Bas¸ar and Sever [13] extended the well-known space ℓ𝑞 from single sequence to double sequences, denoted by L𝑞 , and established its interesting properties. The authors of [14] defined some convex and paranormed sequences spaces and presented some interesting characterization. Most recently, Mohiuddine and Alotaibi [15] introduced some new double sequences spaces for 𝜎-convergence of double sequences and invariant mean and also determined some inclusion results for these spaces. For more details on these concepts, one can be referred to [16–18].

The notion of difference sequence spaces was introduced by Kızmaz [19], who studied the difference sequence spaces 𝑙∞ (Δ), 𝑐(Δ), and 𝑐0 (Δ). The notion was further generalized by Et and C ¸ olak [20] by introducing the spaces 𝑙∞ (Δ𝑟 ), 𝑐(Δ𝑟 ), and 𝑐0 (Δ𝑟 ). Let 𝑤 be the space of all complex or real sequences 𝑥 = (𝑥𝑘 ) and let 𝑟 and 𝑠 be two nonnegative integers. Then for 𝑍 = 𝑙∞ , 𝑐, 𝑐0 , we have the following sequence spaces: 𝑍 (Δ𝑟𝑠 ) = {𝑥 = (𝑥𝑘 ) ∈ 𝑤 : (Δ𝑟𝑠 𝑥𝑘 ) ∈ 𝑍} ,

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𝑟−1 0 where Δ𝑟𝑠 𝑥 = (Δ𝑟𝑠 𝑥𝑘 ) = (Δ𝑟−1 𝑠 𝑥𝑘 − Δ 𝑠 𝑥𝑘+1 ) and Δ 𝑥𝑘 = 𝑥𝑘 for all 𝑘 ∈ N, which is equivalent to the following binomial representation: 𝑟

𝑟 Δ𝑟𝑠 𝑥𝑘 = ∑ (−1)V ( ) 𝑥𝑘+𝑠V . V V=0

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We remark that for 𝑠 = 1 and 𝑟 = 𝑠 = 1, we obtain the sequence spaces which were introduced and studied by Et and C ¸ olak [20] and Kızmaz [19], respectively. For more details about sequence spaces see [21–27] and references therein.

2

Abstract and Applied Analysis

An Orlicz function 𝐹 : [0, ∞) → [0, ∞) is continuous, nondecreasing, and convex such that 𝐹(0) = 0, 𝐹(𝑥) > 0 for 𝑥 > 0 and 𝐹(𝑥) → ∞ as 𝑥 → ∞. If convexity of Orlicz function is replaced by 𝐹(𝑥 + 𝑦) ≤ 𝐹(𝑥) + 𝐹(𝑦), then this function is called modulus function. Lindenstrauss and Tzafriri [28] used the idea of Orlicz function to define the following sequence space: 󵄨󵄨 󵄨󵄨 󵄨𝑥 󵄨 ℓ𝐹 = {𝑥 = (𝑥𝑘 ) ∈ 𝑤 : ∑ 𝐹 ( 󵄨 𝑘 󵄨 ) < ∞, 𝜌 > 0} , 𝜌 𝑘=1 ∞

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which is known as an Orlicz sequence space. The space ℓ𝐹 is a Banach space with the norm 󵄨󵄨 󵄨󵄨 ∞ 󵄨𝑥𝑘 󵄨 ‖𝑥‖ = inf {𝜌 > 0 : ∑ 𝐹 ( 󵄨 󵄨 ) ≤ 1} . 𝜌 𝑘=1

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Also it was shown in [28] that every Orlicz sequence space ℓ𝐹 contains a subspace isomorphic to ℓ𝑝 (𝑝 ≥ 1). An Orlicz function 𝐹 can always be represented in the following integral form: 𝑥

𝐹 (𝑥) = ∫ 𝜂 (𝑡) 𝑑𝑡,

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0

where 𝜂 is known as the kernel of 𝐹, is a right differentiable for 𝑡 ≥ 0, 𝜂(0) = 0, 𝜂(𝑡) > 0, 𝜂 is nondecreasing, and 𝜂(𝑡) → ∞ as 𝑡 → ∞. A sequence F = (𝐹𝑘 ) of Orlicz functions is said to be a Musielak-Orlicz function (see [29, 30]). A sequence N = (𝑁𝑘 ) is defined by 𝑁𝑘 (V) = sup {|V| 𝑢 − 𝐹𝑘 (𝑢) : 𝑢 ≥ 0} ,

𝑘 = 1, 2, . . . ,

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which is called the complementary function of a MusielakOrlicz function F. For a given Musielak-Orlicz function F, the Musielak-Orlicz sequence space 𝑡F and its subspace ℎF are defined as follows: 𝑡F = {𝑥 ∈ 𝑤 : 𝐼F (𝑐𝑥) < ∞ for some 𝑐 > 0} , ℎF = {𝑥 ∈ 𝑤 : 𝐼F (𝑐𝑥) < ∞ ∀𝑐 > 0} ,

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where 𝐼F is a convex modular defined by ∞

𝐼F (𝑥) = ∑ 𝐹𝑘 (𝑥𝑘 ) , 𝑘=1

𝑥 = (𝑥𝑘 ) ∈ 𝑡F .

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We consider 𝑡F equipped with the Luxemburg norm 𝑥 ‖𝑥‖ = inf {𝑘 > 0 : 𝐼F ( ) ≤ 1} 𝑘

𝐹𝑘 (2𝑢) ≤ 𝐾𝐹𝑘 (𝑢) + 𝑐𝑘

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∞,∞

𝑦𝑛𝑚 = ∑ 𝑎𝑛𝑚𝑗𝑘 𝑥𝑗𝑘 𝑗,𝑘=0,0

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is called the 𝐴-𝑚𝑒𝑎𝑛𝑠 of the double sequence (𝑥𝑗𝑘 ). A double sequence (𝑥𝑗𝑘 ) is said to be 𝐴-𝑠𝑢𝑚𝑚𝑎𝑏𝑙𝑒 to the limit 𝐿 if the 𝐴-means exist for all 𝑚, 𝑛 in the sense of Pringsheim’s convergence: 𝑝,𝑞

𝑃- lim

𝑝,𝑞 → ∞

∑ 𝑎𝑛𝑚𝑗𝑘 𝑥𝑗𝑘 = 𝑦𝑛𝑚 ,

𝑗,𝑘=0,0

𝑃- lim 𝑦𝑛𝑚 = 𝐿. (13) 𝑛,𝑚 → ∞

A four-dimensional matrix 𝐴 is said to be boundedregular (or RH-regular) if every bounded 𝑃-convergent sequence is 𝐴-summable to the same limit and the 𝐴-means are also bounded. The following is a four-dimensional analogue of the wellknown Silverman-Toeplitz theorem [32]. Theorem 1 (Robison [33] and Hamilton [34]). The fourdimensional matrix 𝐴 is RH-regular if and only if

(RH2 ) 𝑃-lim𝑛,𝑚 ∑∞,∞ 𝑗,𝑘=0,0 |𝑎𝑛𝑚𝑗𝑘 | = 1, (RH3 ) 𝑃-lim𝑛,𝑚 ∑∞ 𝑗=0 |𝑎𝑛𝑚𝑗𝑘 | = 0 for each 𝑘, (RH4 ) 𝑃-lim𝑛,𝑚 ∑∞ 𝑘=0 |𝑎𝑛𝑚𝑗𝑘 | = 0 for each 𝑗,

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holds for all 𝑘 ∈ N and 𝑢 ∈ R+ , whenever 𝐹𝑘 (𝑢) ≤ 𝑎. A double sequence 𝑥 = (𝑥𝑗𝑘 ) is said to be bounded if 2 ‖𝑥‖(∞,2) = sup𝑗,𝑘 |𝑥𝑗𝑘 | < ∞. We denote by 𝑙∞ the space of all bounded double sequences. By the convergence of double sequence 𝑥 = (𝑥𝑗𝑘 ) we mean the convergence in the Pringsheim sense; that is, a double sequence 𝑥 = (𝑥𝑗𝑘 ) is said to converge to the limit 𝐿 in Pringsheim sense (denoted by, 𝑃-lim 𝑥 = 𝐿) provided that given 𝜖 > 0 there exists 𝑛 ∈ N such that |𝑥𝑗𝑘 − 𝐿| < 𝜖 whenever 𝑗, 𝑘 > 𝑛 (see [31]). We will write more briefly as 2 𝑃-convergent. If, in addition, 𝑥 ∈ 𝑙∞ , then 𝑥 is said to be boundedly P-convergent to 𝐿. We will denote the space of all bounded convergent double sequences (or boundedly 𝑃2 . convergent) by 𝑐∞ Let 𝑆 ⊆ N × N and let 𝜖 > 0 be given. By 𝜒𝑆(𝑥;𝜖) , we denote the characteristic function of the set 𝑆(𝑥; 𝜖) = {(𝑗, 𝑘) ∈ N×N : |𝑥𝑗𝑘 | ≥ 𝜖}. Let 𝐴 = (𝑎𝑛𝑚𝑗𝑘 ) be a four-dimensional infinite matrix of scalers. For all 𝑚, 𝑛 ∈ N0 , where N0 := N ∪ {0}, the sum

(RH1 ) 𝑃-lim𝑛,𝑚 𝑎𝑛𝑚𝑗𝑘 = 0 for each 𝑗 and 𝑘,

or equipped with the Orlicz norm 1 ‖𝑥‖0 = inf { (1 + 𝐼F (𝑘𝑥)) : 𝑘 > 0} . 𝑘

A Musielak-Orlicz function F = (𝐹𝑘 ) is said to satisfy Δ 2 -𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛 if there exist constants 𝑎, 𝐾 > 0 and a sequence 1 1 𝑐 = (𝑐𝑘 )∞ 𝑘=1 ∈ 𝑙+ (the positive cone of 𝑙 ) such that the inequality

(RH5 ) ∑∞,∞ 𝑗,𝑘=0,0 |𝑎𝑛𝑚𝑗𝑘 | < ∞ for all 𝑛, 𝑚 ∈ N0 .


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2. The Double Difference Sequence Spaces In this section, we define some new paranormed double difference sequence spaces with the help of Musielak-Orlicz functions and four-dimensional bounded-regular matrices. Before proceeding further, first we recall the notion of paranormed space as follows. A linear topological space 𝑋 over the real field R (the set of real numbers) is said to be a paranormed space if there is a subadditive function 𝑔 : 𝑋 → R such that 𝑔(𝜃) = 0, 𝑔(𝑥) = 𝑔(−𝑥), and scalar multiplication is continuous; that is, |𝛼𝑛 − 𝛼| → 0 and 𝑔(𝑥𝑛 − 𝑥) → 0 imply 𝑔(𝛼𝑛 𝑥𝑛 − 𝛼𝑥| → 0 for all 𝛼’s in R and all 𝑥’s in 𝑋, where 𝜃 is the zero vector in the linear space 𝑋. The linear spaces 𝑙∞ (𝑝), 𝑐(𝑝), and 𝑐0 (𝑝) were defined by Maddox [35] (also, see Simons [36]). Let F = (𝐹𝑗𝑘 ) be a Musielak-Orlicz function; that is, F is a sequence of Orlicz functions and let 𝐴 = (𝑎𝑛𝑚𝑗𝑘 ) be a nonnegative four-dimensional bounded-regular matrix. Then, we define the following:

𝑊02

Remark 2. If we take F(𝑥) = 𝑥 in 𝑊02 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝) and 𝑊2 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝), then we have the following spaces: 𝑊02 (𝐴, 𝑢, Δ𝑟𝑠 , 𝑝) { = {𝑥 = (𝑥𝑗𝑘 ) : { ∞,∞ } 󵄨 󵄨 𝑝𝑗𝑘 𝑃-lim ∑ 𝑎𝑛𝑚𝑗𝑘 [(𝑢𝑗𝑘 󵄨󵄨󵄨󵄨Δ𝑟𝑠 𝑥𝑗𝑘 󵄨󵄨󵄨󵄨) ] = 0} , 𝑛,𝑚 𝑗,𝑘=0,0 }

𝑊2 (𝐴, 𝑢, Δ𝑟𝑠 , 𝑝) { = {𝑥 = (𝑥𝑗𝑘 ) : { ∞,∞ 󵄨 󵄨 𝑝𝑗𝑘 𝑃-lim ∑ 𝑎𝑛𝑚𝑗𝑘 [(𝑢𝑗𝑘 󵄨󵄨󵄨󵄨Δ𝑟𝑠 𝑥𝑗𝑘 − 𝐿󵄨󵄨󵄨󵄨) ] 𝑛,𝑚 𝑗,𝑘=0,0

} = 0 for some 𝐿 ∈ C} . }

(𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝)

(15) { = {𝑥 = (𝑥𝑗𝑘 ) : {

Remark 3. Let 𝑝 = (𝑝𝑗𝑘 ) = 1 for all 𝑗, 𝑘. Then 𝑊02 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝) and 𝑊2 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝) are reduced to 𝑊02 (𝐴, F, 𝑢, Δ𝑟𝑠 )

∞,∞

} 󵄨 󵄨 𝑝𝑗𝑘 𝑃-lim ∑ 𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨Δ𝑟𝑠 𝑥𝑗𝑘 󵄨󵄨󵄨󵄨) ] = 0} , 𝑛,𝑚 𝑗,𝑘=0,0 } 𝑊2 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝)

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{ = {𝑥 = (𝑥𝑗𝑘 ) : { ∞,∞

󵄨 󵄨 𝑝𝑗𝑘 𝑃-lim ∑ 𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨Δ𝑟𝑠 𝑥𝑗𝑘 − 𝐿󵄨󵄨󵄨󵄨) ] 𝑛,𝑚 𝑗,𝑘=0,0

} = 0 for some 𝐿 ∈ C} , }

where 𝑝 = (𝑝𝑗𝑘 ) is a double sequence of real numbers such that 𝑝𝑗𝑘 > 0 for 𝑗, 𝑘, sup𝑗,𝑘 𝑝𝑗𝑘 = 𝐻 < ∞, and 𝑢 = (𝑢𝑗𝑘 ) is a double sequence of strictly positive real numbers.

{ = {𝑥 = (𝑥𝑗𝑘 ) : { ∞,∞ } 󵄨󵄨 𝑟 󵄨󵄨 󵄨󵄨Δ 𝑠 𝑥𝑗𝑘 󵄨󵄨)] = 0} , 𝑃-lim 𝑎 [𝐹 (𝑢 ∑ 𝑛𝑚𝑗𝑘 𝑗𝑘 𝑗𝑘 󵄨 󵄨 𝑛,𝑚 𝑗,𝑘=0,0 }

𝑊2 (𝐴, F, 𝑢, Δ𝑟𝑠 ) { = {𝑥 = (𝑥𝑗𝑘 ) : { ∞,∞ 󵄨 󵄨 𝑃-lim ∑ 𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨Δ𝑟𝑠 𝑥𝑗𝑘 − 𝐿󵄨󵄨󵄨󵄨)] 𝑛,𝑚 𝑗,𝑘=0,0

} = 0 for some 𝐿 ∈ C} , } respectively.

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4


Remark 4. Let 𝑢 = (𝑢𝑗𝑘 ) = 1 for all 𝑗, 𝑘. Then, the spaces 𝑊02 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝) and 𝑊2 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝) are reduced to 𝑊02 (𝐴, F, Δ𝑟𝑠 , 𝑝)

Remark 6. If we take 𝐴 = (𝐶, 1, 1) and F(𝑥) = 𝑥 in 𝑊02 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝) and 𝑊2 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝), then we have the following spaces: 𝑊02 (𝑢, Δ𝑟𝑠 , 𝑝)

{ = {𝑥 = (𝑥𝑗𝑘 ) : {

{ = {𝑥 = (𝑥𝑗𝑘 ) : {

∞,∞ } 󵄨 𝑝𝑗𝑘 󵄨 𝑃-lim ∑ 𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (󵄨󵄨󵄨󵄨Δ𝑟𝑠 𝑥𝑗𝑘 󵄨󵄨󵄨󵄨) ] = 0} , 𝑛,𝑚 𝑗,𝑘=0,0 }

𝑚−1,𝑛−1 } 󵄨 󵄨 𝑝𝑗𝑘 𝑃-lim ∑ [(𝑢𝑗𝑘 󵄨󵄨󵄨󵄨Δ𝑟𝑠 𝑥𝑗𝑘 󵄨󵄨󵄨󵄨) ] = 0} , 𝑛,𝑚 𝑗,𝑘=0,0 }

𝑊2 (𝑢, Δ𝑟𝑠 , 𝑝)

𝑊2 (𝐴, F, Δ𝑟𝑠 , 𝑝)

{ = {𝑥 = (𝑥𝑗𝑘 ) : {

{ = {𝑥 = (𝑥𝑗𝑘 ) : {

𝑚−1,𝑛−1 󵄨 󵄨 𝑝𝑗𝑘 𝑃-lim ∑ [(𝑢𝑗𝑘 󵄨󵄨󵄨󵄨Δ𝑟𝑠 𝑥𝑗𝑘 − 𝐿󵄨󵄨󵄨󵄨) ] 𝑛,𝑚

∞,∞

󵄨 󵄨 𝑝𝑗𝑘 𝑃-lim ∑ 𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (󵄨󵄨󵄨󵄨Δ𝑟𝑠 𝑥𝑗𝑘 − 𝐿󵄨󵄨󵄨󵄨) ] 𝑛,𝑚

𝑗,𝑘=0,0

𝑗,𝑘=0,0

} = 0 for some 𝐿 ∈ C} . }

} = 0 for some 𝐿 ∈ C} , }

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respectively. Remark 5. If we take 𝐴 = (𝐶, 1, 1) in 𝑊02 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝) and 𝑊2 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝), then we have the following spaces: 𝑊02 (F, 𝑢, Δ𝑟𝑠 , 𝑝)

} { 󵄨 󵄨 𝑊02 (𝐴, 𝐹) = {𝑥 = (𝑥𝑗𝑘 ) : 𝑃-lim ∑𝑎𝑛𝑚𝑗𝑘 𝐹 (󵄨󵄨󵄨󵄨𝑥𝑗𝑘 󵄨󵄨󵄨󵄨) = 0} , 𝑛,𝑚 𝑗,𝑘 } {

{ = {𝑥 = (𝑥𝑗𝑘 ) : {

{ 󵄨 󵄨 𝑊2 (𝐴, 𝐹) = {𝑥 = (𝑥𝑗𝑘 ) : 𝑃-lim ∑𝑎𝑛𝑚𝑗𝑘 𝐹 (󵄨󵄨󵄨󵄨𝑥𝑗𝑘 − 𝐿󵄨󵄨󵄨󵄨) 𝑛,𝑚 𝑗,𝑘 {

𝑚−1,𝑛−1 } 󵄨󵄨 𝑟 󵄨󵄨 𝑝𝑗𝑘 󵄨󵄨Δ 𝑠 𝑥𝑗𝑘 󵄨󵄨) ] = 0} , 𝑃-lim [𝐹 (𝑢 ∑ 𝑗𝑘 𝑗𝑘 󵄨 󵄨 𝑛,𝑚 𝑗,𝑘=0,0 }

𝑊2 (F, 𝑢, Δ𝑟𝑠 , 𝑝) { = {𝑥 = (𝑥𝑗𝑘 ) : { 𝑚−1,𝑛−1 󵄨 󵄨 𝑝𝑗𝑘 𝑃-lim ∑ [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨Δ𝑟𝑠 𝑥𝑗𝑘 − 𝐿󵄨󵄨󵄨󵄨) ] 𝑛,𝑚 𝑗,𝑘=0,0

} = 0 for some 𝐿 ∈ C} . }

Remark 7. Let 𝑝𝑗𝑘 = 𝑢𝑗𝑘 = 1 for all 𝑗, 𝑘. If, in addition, F(𝑥) = 𝐹(𝑥) and 𝑟 = 0, then the spaces 𝑊02 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝) and 𝑊2 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝) are reduced to 𝑊02 (𝐴, 𝐹) and 𝑊2 (𝐴, 𝐹) which were introduced and studied by Yurdakadim and Tas [37] as below:

(18)

} = 0 for some 𝐿 ∈ C} . }

(20)

Throughout the paper, we will use the following inequality: let (𝑎𝑗𝑘 ) and (𝑏𝑗𝑘 ) be two double sequences. Then 󵄨󵄨 󵄨𝑝 󵄨 󵄨𝑝 󵄨 󵄨𝑝 󵄨󵄨𝑎𝑗𝑘 + 𝑏𝑗𝑘 󵄨󵄨󵄨 𝑗𝑘 ≤ 𝐾 (󵄨󵄨󵄨𝑎𝑗𝑘 󵄨󵄨󵄨 𝑗𝑘 + 󵄨󵄨󵄨𝑏𝑗𝑘 󵄨󵄨󵄨 𝑗𝑘 ) , 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨

(21)

where 𝐾 = max(1, 2𝐻−1 ) and sup𝑗,𝑘 𝑝𝑗𝑘 = 𝐻 (see [15]). We will also assume throughout this paper that the symbol F will denote the sublinear Musielak-Orlicz function.


5

3. Main Results Theorem 8. Let F = (𝐹𝑗𝑘 ) be a sublinear Musielak-Orlicz function, 𝐴 = (𝑎𝑛𝑚𝑗𝑘 ) a nonnegative four-dimensional 𝑅𝐻regular matrix, 𝑝 = (𝑝𝑗𝑘 ) a bounded sequence of positive real numbers, and 𝑢 = (𝑢𝑗𝑘 ) a sequence of strictly positive real numbers. Then 𝑊02 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝) and 𝑊2 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝) are linear spaces over the complex field C. Proof. Let 𝑥 = (𝑥𝑗𝑘 ), 𝑦 = (𝑦𝑗𝑘 ) ∈ 𝑊02 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝) and 𝛼, 𝛽 ∈ C. Then there exist integers 𝑀𝛼 and 𝑁𝛽 such that |𝛼| ≤ 𝑀𝛼 and |𝛽| ≤ 𝑁𝛽 . Since F = (𝐹𝑗𝑘 ) is a nondecreasing function, so by inequality (21), we have ∞,∞

󵄨 󵄨 𝑝𝑗𝑘 ∑ 𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨Δ𝑟𝑠 (𝛼𝑥𝑗𝑘 + 𝛽𝑦𝑗𝑘 )󵄨󵄨󵄨󵄨) ]

∞,∞ 󵄨 󵄨 𝑝𝑗𝑘 𝑔 (𝜆𝑥) = sup ∑ 𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨𝜆Δ𝑟𝑠 𝑥𝑗𝑘 󵄨󵄨󵄨󵄨) ] 𝑛𝑚

𝑗,𝑘=0,0

where [|𝜆|] denotes the integer part of |𝜆|. It is also clear that if 𝑥 → 0 and 𝜆 → 0 implies 𝑔(𝜆𝑥) → 0. For fixed 𝜆, if 𝑥 → 0, then 𝑔(𝜆𝑥) → 0. We need to show that for fixed 𝑥, 𝜆 → 0 implies 𝑔(𝜆𝑥) → 0. Let 𝑥 ∈ 𝑊2 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝). Thus

𝑗,𝑘=0,0

∞,∞ 󵄨 󵄨 𝑝𝑗𝑘 ≤ ∑ 𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨𝛼Δ𝑟𝑠 𝑥𝑗𝑘 + 𝛽Δ𝑟𝑠 𝑦𝑗𝑘 󵄨󵄨󵄨󵄨) ]

(24)

≤ (1 + [|𝜆|]) 𝑔 (𝑥) ,

∞,∞ 󵄨 󵄨 𝑝𝑗𝑘 𝑃-lim ∑ 𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨Δ𝑟𝑠 𝑥𝑗𝑘 − 𝐿󵄨󵄨󵄨󵄨) ] = 0. 𝑛,𝑚

𝑗,𝑘=0,0

(25)

Then, for 𝜖 > 0 there exists 𝑁 ∈ N such that

𝑗,𝑘=0,0

∞,∞ 󵄨 󵄨 𝑝𝑗𝑘 ≤ 𝐾 ∑ 𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 𝑀𝛼 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨Δ𝑟𝑠 𝑥𝑗𝑘 󵄨󵄨󵄨󵄨) ]

∞,∞

𝑗,𝑘=0,0

∞,∞ 󵄨 󵄨 𝑝𝑗𝑘 + 𝐾 ∑ 𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 𝑁𝛽 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨Δ𝑟𝑠 𝑦𝑗𝑘 󵄨󵄨󵄨󵄨) ]

We now show that the scalar multiplication is continuous. First observe the following:

(22)

𝜖 󵄨 󵄨 𝑝𝑗𝑘 ∑ 𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨Δ𝑟𝑠 𝑥𝑗𝑘 − 𝐿󵄨󵄨󵄨󵄨) ] < 4 𝑗,𝑘=0,0

(26)

for 𝑚, 𝑛 > 𝑁. Also, for each 𝑚, 𝑛 with 1 ≤ 𝑚, 𝑛 ≤ 𝑁, since

𝑗,𝑘=0,0

∞,∞ 󵄨 󵄨 𝑝𝑗𝑘 ≤ 𝐾𝑀𝛼𝐻 ∑ 𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨Δ𝑟𝑠 𝑥𝑗𝑘 󵄨󵄨󵄨󵄨) ]

∞,∞

󵄨 󵄨 𝑝𝑗𝑘 ∑ 𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨Δ𝑟𝑠 𝑥𝑗𝑘 − 𝐿󵄨󵄨󵄨󵄨) ] < ∞,

𝑗,𝑘=0,0

𝑗,𝑘=0,0

∞,∞ 󵄨 󵄨 𝑝𝑗𝑘 + 𝐾𝑁𝛽𝐻 ∑ 𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨Δ𝑟𝑠 𝑦𝑗𝑘 󵄨󵄨󵄨󵄨) ] 󳨀→ 0.

(27)

there exists an integer 𝑀𝑚,𝑛 such that

𝑗,𝑘=0,0

Thus 𝛼𝑥 + 𝛽𝑦 ∈ 𝑊02 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝). This proves that 𝑊02 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝) is a linear space. Similarly we can prove that 𝑊2 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝) is also a linear space.

𝜖 󵄨 󵄨 𝑝𝑗𝑘 ∑ 𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨Δ𝑟𝑠 𝑥𝑗𝑘 − 𝐿󵄨󵄨󵄨󵄨) ] < . 4 𝑗,𝑘>𝑀

Theorem 9. Let F = (𝐹𝑗𝑘 ) be a sublinear Musielak-Orlicz function, 𝐴 = (𝑎𝑛𝑚𝑗𝑘 ) a nonnegative four-dimensional 𝑅𝐻regular matrix, 𝑝 = (𝑝𝑗𝑘 ) a bounded sequence of positive real numbers, and 𝑢 = (𝑢𝑗𝑘 ) a sequence of strictly positive real numbers. Then 𝑊02 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝) and 𝑊2 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝) are paranormed spaces with the paranorm

Let 𝑀 = max1≤(𝑚,𝑛)≤𝑁{𝑀𝑚,𝑛 }. We have for each 𝑚, 𝑛 with 1 ≤ 𝑚, 𝑛 ≤ 𝑁

∞,∞ 󵄨 󵄨 𝑝𝑗𝑘 1/𝑀 𝑔 (𝑥) = sup ∑ {𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨Δ𝑟𝑠 𝑥𝑗𝑘 󵄨󵄨󵄨󵄨) ]} , 𝑛,𝑚

𝑗,𝑘=0,0

(28)

𝑚,𝑛


(29)

Also from (26), for 𝑚, 𝑛 > 𝑁, we have (23)

where 0 < 𝑝𝑗𝑘 ≤ sup 𝑝𝑗𝑘 = 𝐻 < ∞ and 𝑀 = max(1, 𝐻). Proof. We will prove the result for 𝑊02 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝). Let 𝑥 = (𝑥𝑗𝑘 ) ∈ 𝑊02 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝). Then for each 𝑥 = (𝑥𝑗𝑘 ) ∈ 𝑊02 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝), 𝑔(𝑥) exists. Also it is clear that 𝑔(0) = 0, 𝑔(−𝑥) = 𝑔(𝑥), and 𝑔(𝑥 + 𝑦) ≤ 𝑔(𝑥) + 𝑔(𝑦).


(30)

Thus 𝑀 is an integer independent of 𝑚, 𝑛 such that 𝜖 󵄨 󵄨 𝑝𝑗𝑘 ∑ 𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨Δ𝑟𝑠 𝑥𝑗𝑘 − 𝐿󵄨󵄨󵄨󵄨) ] < . 4 𝑗,𝑘>𝑀

(31)

6


Since |𝜆|𝑝𝑗𝑘 ≤ max(1, |𝜆|𝐻), therefore

Thus 𝑔(𝜆𝑥) → 0 as 𝜆 → 0. Therefore 𝑊02 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝) is a paranormed space. Similarly, we can prove that 𝑊2 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝) is a paranormed space. This completes the proof.

∞,∞

󵄨 󵄨 𝑝𝑗𝑘 ∑ 𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨𝜆Δ𝑟𝑠 𝑥𝑗𝑘 󵄨󵄨󵄨󵄨) ]

𝑗,𝑘=0,0

∞,∞ 󵄨 󵄨 𝑝𝑗𝑘 = ∑ 𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨𝜆Δ𝑟𝑠 𝑥𝑗𝑘 − 𝜆𝐿 + 𝜆𝐿󵄨󵄨󵄨󵄨) ]

Theorem 10. Let F = (𝐹𝑗𝑘 ) be a sublinear Musielak-Orlicz function, 𝐴 = (𝑎𝑛𝑚𝑗𝑘 ) a nonnegative four-dimensional 𝑅𝐻regular matrix, 𝑝 = (𝑝𝑗𝑘 ) a bounded sequence of positive real numbers, and 𝑢 = (𝑢𝑗𝑘 ) a sequence of strictly positive real numbers. Then 𝑊02 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝) and 𝑊2 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝) are complete topological linear spaces.

𝑗,𝑘=0,0

∞,∞ 󵄨 󵄨 𝑝𝑗𝑘 ≤ ∑ 𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨𝜆Δ𝑟𝑠 𝑥𝑗𝑘 − 𝜆𝐿󵄨󵄨󵄨󵄨) ] 𝑗,𝑘=0,0

∞,∞

𝑝𝑗𝑘

𝑞

Proof. Let (𝑥𝑗𝑘 ) be a Cauchy sequence in 𝑊02 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝); that is, 𝑔(𝑥𝑞 − 𝑥𝑡 ) → 0 as 𝑞, 𝑡 → ∞. Then, we have

+ ∑ 𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 |𝜆𝐿|) ] 𝑗,𝑘=0,0

󵄨 󵄨 𝑝𝑗𝑘 ≤ ∑ 𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨𝜆Δ𝑟𝑠 𝑥𝑗𝑘 − 𝜆𝐿󵄨󵄨󵄨󵄨) ]

(32)

𝑗,𝑘>𝑀

󵄨 󵄨 𝑝𝑗𝑘 + ∑ 𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨𝜆Δ𝑟𝑠 𝑥𝑗𝑘 − 𝜆𝐿󵄨󵄨󵄨󵄨) ] 𝑗,𝑘≤𝑀

+

∑

∞,∞

𝑝 󵄨󵄨 𝑞 𝑡 󵄨󵄨󵄨 𝑗𝑘 ∑ 𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨Δ𝑟𝑠 𝑥𝑗𝑘 − Δ𝑟𝑠 𝑥𝑗𝑘 󵄨󵄨) ] 󳨀→ 0. 󵄨 󵄨 𝑗,𝑘=0,0

Thus for each fixed 𝑗 and 𝑘 as 𝑞, 𝑡 → ∞, since 𝐴 = (𝑎𝑛𝑚𝑗𝑘 ) is nonnegative, we are granted that

󵄨 󵄨 𝑝𝑗𝑘 𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨𝜆Δ𝑟𝑠 𝑥𝑗𝑘 − 𝜆𝐿󵄨󵄨󵄨󵄨) ]

𝑗≥𝑀,𝑘𝑁

󵄨 󵄨 𝑝𝑗𝑘 ∑ 𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨𝜆Δ𝑟𝑠 𝑥𝑗𝑘 − 𝜆𝐿󵄨󵄨󵄨󵄨) ]

∑

𝑗,𝑘≤𝑀

𝑝𝑗𝑘

Since for any fixed natural number 𝑀, from (38) we have

𝑗,𝑘=0,0

in Pringsheim’s sense. Now choose 𝛿 < 1 such that |𝜆| < 𝛿 implies 󵄨 󵄨 𝑝𝑗𝑘 ∑ 𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨𝜆Δ𝑟𝑠 𝑥𝑗𝑘 − 𝜆𝐿󵄨󵄨󵄨󵄨) ]

∞,∞

𝑝 󵄨󵄨 𝑞 𝑡 󵄨󵄨󵄨 𝑗𝑘 𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨Δ𝑟𝑠 𝑥𝑗𝑘 − Δ𝑟𝑠 𝑥𝑗𝑘 󵄨󵄨) ] < 𝜖 ∀𝑚, 𝑛. 󵄨 󵄨 𝑗,𝑘≤𝑀 𝑞,𝑡>𝑁

∑

(41)

𝑗,𝑘≤𝑀

𝜖 + ∑ 𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 |𝜆𝐿|) ] < . 4 𝑗,𝑘=0,0 𝑝𝑗𝑘

In the same manner, we have 𝜖 󵄨 󵄨 𝑝𝑗𝑘 ∑ 𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨𝜆Δ𝑟𝑠 𝑥𝑗𝑘 − 𝜆𝐿󵄨󵄨󵄨󵄨) ] < , 4 𝑗≥𝑀,𝑘𝛿 𝐹𝑗𝑘

󵄨 󵄨 𝑝𝑗𝑘 ≤𝐾 (2) ∑𝑎𝑛𝑚𝑗𝑘 [(𝑢𝑗𝑘 󵄨󵄨󵄨󵄨Δ𝑟𝑠 𝑥𝑗𝑘 − 𝐿󵄨󵄨󵄨󵄨) ] . 𝛿 𝑗,𝑘

∞,∞

󵄨󵄨 󵄨󵄨 𝑝𝑗𝑘 𝑞 ∑ 𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨Δ𝑟𝑠 𝑥𝑗𝑘 − Δ𝑟𝑠 𝐿𝑞 󵄨󵄨󵄨) ] 󳨀→ 0 󵄨 󵄨 𝑗,𝑘=0,0

(44)

as 𝑚, 𝑛 󳨀→ ∞; whence from the fact that sup𝑛𝑚 ∑∞,∞ 𝑗,𝑘=0,0 𝑎𝑛𝑚𝑗𝑘 < ∞ and from the definition of Musielak-Orlicz function, we have 𝐹𝑗𝑘 |Δ𝑟𝑠 𝐿𝑞 − Δ𝑟𝑠 𝐿| → 0 as 𝑞 → ∞ and so 𝐿𝑞 converges to 𝐿. Thus ∞,∞

󵄨 󵄨 𝑝𝑗𝑘 ∑ 𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨Δ𝑟𝑠 𝑥𝑗𝑘 − 𝐿󵄨󵄨󵄨󵄨) ] 󳨀→ 0

𝑗,𝑘=0,0

(45)

as 𝑚, 𝑛 󳨀→ ∞. Hence 𝑥 ∈ 𝑊2 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝) and this completes the proof. Theorem 11. Let F = (𝐹𝑗𝑘 ) be a sublinear MusielakOrlicz function which satisfies the Δ 2 -condition. Then 𝑊2 (𝐴, 𝑢, Δ𝑟𝑠 , 𝑝) ⊆ 𝑊2 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝).

Since 𝐴 is 𝑅𝐻-regular and 𝑥 ∈ 𝑊2 (𝐴, 𝑢, Δ𝑟𝑠 , 𝑝), we get 𝑥 ∈ 𝑊2 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝). Theorem 12. Let F = (𝐹𝑗𝑘 ) be a sublinear MusielakOrlicz function and let 𝐴 = (𝑎𝑛𝑚𝑗𝑘 ) be a nonnegative four-dimensional RH-regular matrix. Suppose that 𝛽 = lim𝑡 → ∞ (𝐹𝑗𝑘 (𝑡)/𝑡) < ∞. Then 𝑊2 (𝐴, 𝑢, Δ𝑟𝑠 , 𝑝) = 𝑊2 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝) .

∞,∞

󵄨 󵄨 𝑝𝑗𝑘 ∑ 𝑎𝑛𝑚𝑗𝑘 [(𝑢𝑗𝑘 󵄨󵄨󵄨󵄨Δ𝑟𝑠 𝑥𝑗𝑘 − 𝐿󵄨󵄨󵄨󵄨) ]

𝑗,𝑘=0,0

Let 𝜖 > 0 and choose 𝛿 with 0 < 𝛿 < 1 such that 𝐹𝑗𝑘 (𝑡) < 𝜖 for 0 ≤ 𝑡 ≤ 𝛿. Write 𝑦𝑗𝑘 = (𝑢𝑗𝑘 |Δ𝑟𝑠 𝑥𝑗𝑘 − 𝐿|) and consider 𝑝𝑗𝑘

∑𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑦𝑗𝑘 ) ] = 𝑗,𝑘

𝑗,𝑘:|𝑦𝑗𝑘 |≤𝛿

𝑊2 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝) ⊆ 𝑊2 (𝐴, F, 𝑢, Δ𝑟𝑠 ) .

∑

𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑦𝑗𝑘 ) ]

𝑊2 (𝐴, F, 𝑢, Δ𝑟𝑠 ) ⊆ 𝑊2 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝) .

∞,∞

𝛿

∑ 𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑦𝑗𝑘 ) ] . 𝑗,𝑘:|𝑦𝑗𝑘 |>𝛿 (47)

)
𝛿, we use the fact that 𝑦𝑗𝑘 < 𝑦𝑗𝑘 /𝛿 < 1 + 𝑦𝑗𝑘 /𝛿. Hence 𝑦𝑗𝑘

𝐹𝑗𝑘 (2) + 𝐾

𝑦𝑗𝑘 2𝛿

𝐹𝑗𝑘 (2) = 𝐾

(54)

Proof. (i) Let 𝑥 = (𝑥𝑗𝑘 ) ∈ 𝑊2 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝). Then since 0 < inf 𝑝𝑗𝑘 < 𝑝𝑗𝑘 ≤ 1, we obtain the following:

∑ 𝑎𝑛𝑚𝑗𝑘 𝑗,𝑘:|𝑦𝑗𝑘 |≤𝛿

𝐹𝑗𝑘 (𝑦𝑗𝑘 ) < 𝐹𝑗𝑘 (1 +

(53)

(ii) Let 1 ≤ 𝑝𝑗𝑘 ≤ sup 𝑝𝑗𝑘 < ∞. Then

𝑝𝑗𝑘

=𝜖

+

Theorem 13. (i) Let 0 < inf 𝑝𝑗𝑘 < 𝑝𝑗𝑘 ≤ 1. Then

𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑦𝑗𝑘 ) ]

𝑗,𝑘:|𝑦𝑗𝑘 |>𝛿

(52)

which implies that 𝑥 = (𝑥𝑗𝑘 ) ∈ 𝑊2 (𝐴, 𝑢, Δ𝑟𝑠 , 𝑝). This completes the proof.

𝑝𝑗𝑘

∑ +

1 ∞,∞ 󵄨 󵄨 𝑝𝑗𝑘 ≤ [𝐹 (𝑢 󵄨󵄨󵄨Δ𝑟 𝑥 − 𝐿󵄨󵄨󵄨󵄨) ] , ∑ 𝑎 𝛽 𝑗,𝑘=0,0 𝑛𝑚𝑗𝑘 𝑗𝑘 𝑗𝑘 󵄨 𝑠 𝑗𝑘

(46)

𝑗,𝑘

(51)

Proof. In order to prove that 𝑊2 (𝐴, 𝑢, Δ𝑟𝑠 , 𝑝) = 𝑊2 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝), it is sufficient to show that 𝑊2 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝) ⊂ 𝑊2 (𝐴, 𝑢, Δ𝑟𝑠 , 𝑝). Now, let 𝛽 > 0. By definition of 𝛽, we have 𝐹𝑗𝑘 (𝑡) ≥ 𝛽𝑡 for all 𝑡 ≥ 0. Since 𝛽 > 0, we have 𝑡 ≤ (1/𝛽)𝐹𝑗𝑘 (𝑡) for all 𝑡 ≥ 0. Let 𝑥 = (𝑥𝑗𝑘 ) ∈ 𝑊2 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝). Thus, we have

Proof. Let 𝑥 = (𝑥𝑘 ) ∈ 𝑊2 (𝐴, 𝑢, Δ𝑟𝑠 , 𝑝); that is, 󵄨 󵄨 𝑝𝑗𝑘 lim ∑𝑎𝑛𝑚𝑗𝑘 [(𝑢𝑗𝑘 󵄨󵄨󵄨󵄨Δ𝑟𝑠 𝑥𝑗𝑘 − 𝐿󵄨󵄨󵄨󵄨) ] = 0. 𝑛,𝑚

(50)

𝑦𝑗𝑘 𝛿

𝑗,𝑘=0,0

∞,∞

󵄨 󵄨 𝑝𝑗𝑘 ≤ ∑ 𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨Δ𝑟𝑠 𝑥𝑗𝑘 − 𝐿󵄨󵄨󵄨󵄨) ] .

(55)

𝑗,𝑘=0,0

Thus 𝑥 = (𝑥𝑗𝑘 ) ∈ 𝑊2 (𝐴, F, 𝑢, Δ𝑟𝑠 ). (ii) Let 𝑝𝑗𝑘 ≥ 1 for each 𝑗 and 𝑘 and sup 𝑝𝑗𝑘 < ∞. Let 𝑥 = (𝑥𝑗𝑘 ) ∈ 𝑊2 (𝐴, F, 𝑢, Δ𝑟𝑠 ). Then for each 0 < 𝜖 < 1 there exists a positive integer 𝑁 such that ∞,∞

𝐹𝑗𝑘 (2) , (49)

󵄨 󵄨 ∑ 𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨Δ𝑟𝑠 𝑥𝑗𝑘 − 𝐿󵄨󵄨󵄨󵄨)] ≤ 𝜖 < 1 ∀𝑚, 𝑛 ≥ 𝑁.

𝑗,𝑘=0,0

(56)

8

Abstract and Applied Analysis 2 2 ⊂ 𝑙∞ and it is Proof. We have 𝑊02 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝) ∩ 𝑙∞ 2 𝑟 2 clear that 𝑊0 (𝐴, F, 𝑢, Δ 𝑠 , 𝑝) ∩ 𝑙∞ ≠ 0. For 𝑥, 𝑦 ∈ 𝑊02 (𝐴, F, 𝑢, 2 , we get |𝑥𝑗𝑘 + 𝑦𝑗𝑘 | < |𝑥𝑗𝑘 | + |𝑦𝑗𝑘 |. Now, we have Δ𝑟𝑠 , 𝑝) ∩ 𝑙∞

This implies that ∞,∞

󵄨 󵄨 𝑝𝑗𝑘 ∑ 𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨Δ𝑟𝑠 𝑥𝑗𝑘 − 𝐿󵄨󵄨󵄨󵄨) ]

𝑗,𝑘=0,0

∞,∞

󵄨 󵄨 ≤ ∑ 𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨Δ𝑟𝑠 𝑥𝑗𝑘 − 𝐿󵄨󵄨󵄨󵄨)] .

(57)

󵄨 󵄨 𝑝𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨Δ𝑟𝑠 (𝑥𝑗𝑘 + 𝑦𝑗𝑘 )󵄨󵄨󵄨󵄨) ] 󵄨 󵄨 󵄨 󵄨 𝑝𝑗𝑘 ≤ [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨Δ𝑟𝑠 𝑥𝑗𝑘 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨Δ𝑟𝑠 𝑦𝑗𝑘 󵄨󵄨󵄨󵄨) ]

𝑗,𝑘=0,0

Therefore 𝑥 = (𝑥𝑗𝑘 ) ∈ 𝑊2 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝). This completes the proof. Lemma 14. Let F = (𝐹𝑗𝑘 ) be a sublinear Musielak-Orlicz function which satisfies the Δ 2 -condition and let 𝐴 = (𝑎𝑛𝑚𝑗𝑘 ) be a nonnegative four-dimensional 𝑅𝐻-regular matrix. Then 2 2 𝑊02 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝) ∩ 𝑙∞ is an ideal in 𝑙∞ . 2 2 Proof. Let 𝑥 ∈ 𝑊02 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝) ∩ 𝑙∞ and 𝑦 ∈ 𝑙∞ . We need 2 𝑟 2 2 , to show that 𝑥𝑦 ∈ 𝑊0 (𝐴, F, 𝑢, Δ 𝑠 , 𝑝) ∩ 𝑙∞ . Since 𝑦 ∈ 𝑙∞ there exists 𝑇1 > 1 such that ‖𝑦‖ < 𝑇1 . In this case |𝑥𝑗𝑘 𝑦𝑗𝑘 | < 𝑇1 |𝑥𝑗𝑘 | for all 𝑗, 𝑘. Since F is nondecreasing and satisfies Δ 2 condition, we have

0. Therefore lim𝑛,𝑚 ∑𝑗,𝑘 𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 2 (𝑢𝑗𝑘 |Δ𝑟𝑠 (𝑥𝑗𝑘 𝑦𝑗𝑘 )|)𝑝𝑗𝑘 ] = 0. Thus 𝑥𝑦 ∈ 𝑊02 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝) ∩ 𝑙∞ . This completes the proof. 2 2 Lemma 15. Let 𝐺 be an ideal in 𝑙∞ and let 𝑥 = (𝑥𝑗𝑘 ) ∈ 𝑙∞ . 2 Then 𝑥 is in the closure of 𝐺 in 𝑙∞ if and only if 𝜒𝑆(𝑥;𝜖) ∈ 𝐺 for all 𝜖 > 0.

Proof. Let 𝑥 be in the closure of 𝐺 and let 𝜖 > 0 be given. Suppose that 𝑧 = (𝑧𝑗𝑘 ) ∈ 𝐺 such that ‖𝑥 − 𝑧‖ < 𝜖/2 and observe that 𝑆(𝑥; 𝜖) ⊆ 𝑆(𝑧; 𝜖/2). Define a double sequence 𝑦 = 2 (𝑦𝑗𝑘 ) ∈ 𝑙∞ by 1 󵄨 󵄨 𝜖 { , if 󵄨󵄨󵄨󵄨𝑧𝑗𝑘 󵄨󵄨󵄨󵄨 ≧ , { { 𝑧𝑗𝑘 2 𝑦𝑗𝑘 = { { { otherwise. {0,

(59)

(61)

1 󵄨 󵄨 𝑝𝑗𝑘 + 𝐾∑𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨Δ𝑟𝑠 𝑦𝑗𝑘 󵄨󵄨󵄨󵄨) ] , 2 𝑗,𝑘 where 𝐾 = max{𝐾1 , 𝐾2 }, so 𝑥+𝑦, 𝑥−𝑦 ∈ 𝑊02 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝)∩ 2 . 𝑙∞ 2 2 and 𝑦 ∈ 𝑙∞ . Thus, there Let 𝑥 ∈ 𝑊02 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝) ∩ 𝑙∞ exists a positive integer 𝐾, so that, for every 𝑗, 𝑘, we have |𝑥𝑗𝑘 𝑦𝑗𝑘 | ≤ 𝐾|𝑥𝑗𝑘 |. Therefore 󵄨 󵄨 𝑝𝑗𝑘 󵄨 𝑝𝑗𝑘 󵄨 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨Δ𝑟𝑠 (𝑥𝑗𝑘 𝑦𝑗𝑘 )󵄨󵄨󵄨󵄨) ] ≤ [𝐹𝑗𝑘 (𝑢𝑗𝑘 𝐾 󵄨󵄨󵄨󵄨Δ𝑟𝑠 𝑥𝑗𝑘 󵄨󵄨󵄨󵄨) ] 󵄨 󵄨 𝑝𝑗𝑘 ≤ 𝑇 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨Δ𝑟𝑠 𝑥𝑗𝑘 󵄨󵄨󵄨󵄨) ] ,

(62)

and so 󵄨 󵄨 𝑝𝑗𝑘 ∑𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨Δ𝑟𝑠 (𝑥𝑗𝑘 𝑦𝑗𝑘 )󵄨󵄨󵄨󵄨) ] 𝑗,𝑘

󵄨 󵄨 𝑝𝑗𝑘 ≤ 𝑇∑𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨Δ𝑟𝑠 𝑥𝑗𝑘 󵄨󵄨󵄨󵄨) ] .

(63)

𝑗,𝑘

Clearly 𝑦𝑧 = 𝜒𝑆(𝑧;𝜖/2) ∈ 𝐺. Since 𝑆(𝑥; 𝜖) ⊆ 𝑆(𝑧; 𝜖/2) and 2 𝜒𝑆(𝑥;𝜖) ∈ 𝑙∞ , hence 𝜒𝑆(𝑥;𝜖) 𝜒𝑆(𝑧;𝜖/2) = 𝜒𝑆(𝑥;𝜖) ∈ 𝐺. 2 Conversely, if 𝑥 ∈ 𝑙∞ then ‖𝑥 − 𝑥𝜒𝑆(𝑥;𝜖) ‖ < 𝜖. It follows that 𝜒𝑆(𝑥;𝜖) ∈ 𝐺 for all 𝜖 > 0; then 𝑥 is in the closure of 𝐺. Lemma 16. If 𝐴 is a nonnegative four-dimensional 𝑅𝐻2 regular matrix, then 𝑊02 (𝐴, 𝑢, Δ𝑟𝑠 , 𝑝) ∩ 𝑙∞ is a closed ideal in 2 𝑙∞ .

2 Hence 𝑥𝑦 ∈ 𝑊02 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝) ∩ 𝑙∞ . So 𝑊02 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝) ∩ 2 2 𝑙∞ is an ideal in 𝑙∞ for a Musielak-Orlicz function which satisfies the Δ 2 -condition. 2 is Now, we have to show that 𝑊02 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝) ∩ 𝑙∞ 2 ; there exists 𝑥𝑐𝑑 = closed. Let 𝑥 ∈ 𝑊02 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝) ∩ 𝑙∞ 𝑐𝑑 2 2 𝑥𝑗𝑘 ∈ 𝑊02 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝) ∩ 𝑙∞ such that 𝑥𝑐𝑑 → 𝑥 ∈ 𝑙∞ .


9

For every 𝜖 > 0 there exists 𝑁1 (𝜖) ∈ N such that, for all 𝑐, 𝑑 > 𝑁1 (𝜖), |Δ𝑟𝑠 𝑥𝑐𝑑 − Δ𝑟𝑠 𝑥| < 𝜖. Now, for 𝜖 > 0, we have 󵄨 󵄨 𝑝𝑗𝑘 ∑𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨Δ𝑟𝑠 𝑥𝑗𝑘 󵄨󵄨󵄨󵄨) ] 𝑗,𝑘

The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia.

𝑗,𝑘

󵄨 𝑐𝑑 󵄨󵄨 󵄨󵄨 𝑟 𝑐𝑑 󵄨󵄨 𝑝𝑗𝑘 󵄨󵄨 + 󵄨󵄨Δ 𝑠 𝑥𝑗𝑘 󵄨󵄨) ] ≤ ∑𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨Δ𝑟𝑠 𝑥𝑗𝑘 − Δ𝑟𝑠 𝑥𝑗𝑘 󵄨 󵄨 󵄨

References

𝑗,𝑘

1 󵄨 𝑐𝑑 󵄨󵄨 𝑝𝑗𝑘 󵄨󵄨) ] ≤ ∑𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 2 󵄨󵄨󵄨󵄨Δ𝑟𝑠 𝑥𝑗𝑘 − Δ𝑟𝑠 𝑥𝑗𝑘 󵄨 2 𝑗,𝑘 1 󵄨 𝑐𝑑 󵄨󵄨 𝑝𝑗𝑘 󵄨󵄨) ] + ∑𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 2 󵄨󵄨󵄨󵄨Δ𝑟𝑠 𝑥𝑗𝑘 󵄨 2 𝑗,𝑘 1 1 󵄨 𝑐𝑑 󵄨󵄨 𝑝𝑗𝑘 󵄨󵄨) ] . ≤ 𝐾𝐹𝑗𝑘 (𝜖) ∑𝑎𝑛𝑚𝑗𝑘 + 𝐾∑𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨Δ𝑟𝑠 𝑥𝑗𝑘 󵄨 2 2 𝑗,𝑘 𝑗,𝑘 (64) 2 Since 𝑥𝑐𝑑 ∈ 𝑊02 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝) ∩ 𝑙∞ and 𝐴 is RH-regular, we get

󵄨 󵄨 𝑝𝑗𝑘 lim ∑𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨Δ𝑟𝑠 𝑥𝑗𝑘 󵄨󵄨󵄨󵄨) ] = 0; 𝑛,𝑚

(65)

𝑗,𝑘

2 . This completes the proof. so 𝑥 ∈ 𝑊02 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝)∩𝑙∞

Theorem 17. Let 𝑥 = (𝑥𝑗𝑘 ) be a bounded sequence, F = (𝐹𝑗𝑘 ) a sublinear Musielak-Orlicz function which satisfies the Δ 2 condition, and 𝐴 a nonnegative four-dimensional 𝑅𝐻-regular 2 2 matrix. Then 𝑊2 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝) ∩ 𝑙∞ = 𝑊2 (𝐴, 𝑢, Δ𝑟𝑠 , 𝑝) ∩ 𝑙∞ . Proof. Without loss of generality we may take 𝐿 = 0 and establish (66)

Since 𝑊02 (𝐴, 𝑢, Δ𝑟𝑠 , 𝑝) ⊆ 𝑊02 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝), therefore 𝑊02 (𝐴, 2 2 ⊆ 𝑊02 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝) ∩ 𝑙∞ . We need to show 𝑢, Δ𝑟𝑠 , 𝑝) ∩ 𝑙∞ 2 𝑟 2 2 2 . Notice that 𝑊0 (𝐴, F, 𝑢, Δ 𝑠 , 𝑝) ∩ 𝑙∞ ⊆ 𝑊0 (𝐴, 𝑢, Δ𝑟𝑠 , 𝑝) ∩ 𝑙∞ that if 𝑆 ⊂ N × N, then 𝑝

𝑝

∑𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝜒𝑆 (𝑗, 𝑘)) 𝑗𝑘 ] = 𝐹𝑗𝑘 (1) ∑𝑎𝑛𝑚𝑗𝑘 (𝜒𝑆 (𝑗, 𝑘)) 𝑗𝑘 , 𝑗,𝑘

𝑗,𝑘

(67) 2 for all 𝑛, 𝑚. Observe that 𝜒𝑆 (𝑗, 𝑘) ∈ 𝑊02 (𝐴, 𝑢, Δ𝑟𝑠 , 𝑝) ∩ 𝑙∞ 2 whenever 𝑥 ∈ 𝑊02 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝) ∩ 𝑙∞ by Lemmas 14 and 15, so 2 2 ⊆ 𝑊02 (𝐴, 𝑢, Δ𝑟𝑠 , 𝑝) ∩ 𝑙∞ . 𝑊02 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝) ∩ 𝑙∞

The proof is complete.

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

󵄨 𝑐𝑑 𝑐𝑑 󵄨󵄨 𝑝𝑗𝑘 󵄨󵄨) ] = ∑𝑎𝑛𝑚𝑗𝑘 [𝐹𝑗𝑘 (𝑢𝑗𝑘 󵄨󵄨󵄨󵄨Δ𝑟𝑠 𝑥𝑗𝑘 − Δ𝑟𝑠 𝑥𝑗𝑘 + Δ𝑟𝑠 𝑥𝑗𝑘 󵄨

2 2 = 𝑊02 (𝐴, 𝑢, Δ𝑟𝑠 , 𝑝) ∩ 𝑙∞ . 𝑊02 (𝐴, F, 𝑢, Δ𝑟𝑠 , 𝑝) ∩ 𝑙∞

Conflict of Interests

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