On Some Classes of Double Difference Sequences of Interval Numbers

Abstract

The aim of this paper is to introduce some interval valued double difference sequence spaces by means of Musielak-Orlicz function ℳ=(𝑀_𝑖𝑗). We also determine some topological properties and inclusion relations between these double difference sequence spaces.

1. Introduction

Interval arithmetic was first suggested by Dwyer [1] in 1951. Development of interval arithmetic as a formal system and evidence of its value as a computational device was provided by Moore [2] in 1959 and also by Moore and Yang [3] in 1962. Further works on interval numbers can be found in Dwyer [4] and Markov [5]. Furthermore, Moore and Yang [6] have developed applications of interval number sequences to differential equations. Chiao in [7] introduced sequences of interval numbers and defined usual convergence of sequences of interval number. Şengönül and Eryilmaz in [8] introduced and studied bounded and convergent sequence spaces of interval numbers and showed that these spaces are complete metric spaces. Recently, Esi in [9, 10] introduced and studied strongly almost 𝜆-convergence and statistically almost 𝜆-convergence of interval numbers and lacunary sequence spaces of interval numbers, respectively (also see [11–17]).

A set consisting of a closed interval of real numbers 𝑥 such that 𝑎≤𝑥≤𝑏 is called an interval number. A real interval can also be considered as a set. Thus we can investigate some properties of interval numbers, for instance, arithmetic properties or analysis properties. We denote the set of all real valued closed intervals by ℝ. Any elements of ℝ are called closed interval and denoted by 𝑥. That is, 𝑥={𝑥∈ℝ∶𝑎≤𝑥≤𝑏}. An interval number 𝑥 is a closed subset of real numbers [7]. Let 𝑥_𝑙 and 𝑥_𝑟 be first and last points of 𝑥 interval number, respectively. For 𝑥₁,𝑥₂∈ℝ, we have 𝑥₁=𝑥₂⇔𝑥_1𝑙=𝑥_2𝑙, 𝑥_1𝑟=𝑥_2𝑟. Consider 𝑥₁+𝑥₂={𝑥∈ℝ∶𝑥_1𝑙+𝑥_2𝑙≤𝑥≤𝑥_1𝑟+𝑥_2𝑟}, and if 𝛼≥0, then 𝛼𝑥={𝑥∈ℝ∶𝛼𝑥_1𝑙≤𝑥≤𝛼𝑥_1𝑟} and if 𝛼<0, then 𝛼𝑥={𝑥∈ℝ∶𝛼𝑥_1𝑟≤𝑥≤𝛼𝑥_1𝑙}, 𝑥₁⋅𝑥₂=𝑥∈ℝ∶min𝑥_1𝑙⋅𝑥_2𝑙,𝑥_1𝑙⋅𝑥_2𝑟,𝑥_1𝑟⋅𝑥_2𝑙,𝑥_1𝑟⋅𝑥_2𝑟≤𝑥≤min𝑥_1𝑙⋅𝑥_2𝑙,𝑥_1𝑙⋅𝑥_2𝑟,𝑥_1𝑟⋅𝑥_2𝑙,𝑥_1𝑟⋅𝑥_2𝑟. (1)

In [2], Moore proved that the set of all interval numbers ℝ is a complete metric space defined by 𝑑(𝑥₁,𝑥₂)=max{|𝑥_1𝑙−𝑥_2𝑙|,|𝑥_1𝑟⋅𝑥_2𝑟|}. In the special cases 𝑥₁=[𝑎,𝑎] and 𝑥₂=[𝑏,𝑏], we obtain usual metric of ℝ. Let us define transformation 𝑓∶ℕ→ℝ by 𝑘→𝑓(𝑘)=𝑥,𝑥=(𝑥_𝑘). Then 𝑥=(𝑥_𝑘) is called sequence of interval numbers. The 𝑥_𝑘 is called 𝑘th term of sequence 𝑥=(𝑥_𝑘). We denote the set of all interval numbers with real terms as 𝑤^𝑖. The algebraic properties of 𝑤^𝑖 can be found in [7]. Now we give the basic definitions used in this paper.

Definition 1 (see [7]). A sequence 𝑥=(𝑥_𝑘) of interval numbers is said to be convergent to the interval number 𝑥₀ if for each 𝜖>0 there exists a positive integer 𝑘₀ such that 𝑑(𝑥_𝑘,𝑥₀)<𝜖 for all 𝑘≥𝑘₀ and we denote it by lim_𝑘𝑥_𝑘=𝑥₀. Thus, lim_𝑘𝑥_𝑘=𝑥₀⇔lim_𝑘𝑥_𝑘𝑙=𝑥_0𝑙 and lim_𝑘𝑥_𝑘𝑟=𝑥_0𝑟.

Definition 2. A transformation 𝑓 from ℕ×ℕ to ℝ is defined by 𝑖,𝑗→𝑓(𝑖,𝑗)=𝑥,𝑥=(𝑥_𝑖𝑗). Then 𝑥=(𝑥_𝑖𝑗) is called sequence of double interval numbers. Then 𝑥_𝑖𝑗 is called 𝑖𝑗^th term of sequence 𝑥=(𝑥_𝑖𝑗).

Definition 3. An interval valued double sequence 𝑥=(𝑥_𝑖𝑗) is said to be convergent in Pringsheim’s sense or 𝑃-convergent to an interval number 𝑥₀, if, for every 𝜖>0, there exists 𝑁∈ℕ such that 𝑑𝑥_𝑖𝑗,𝑥₀<𝜖∀𝑖,𝑗>𝑁, (2) where ℕ is the set of natural numbers, and we denote it also by 𝑃−lim𝑥_𝑖𝑗=𝑥₀. The interval number 𝑥₀ is called the Pringsheim limit of 𝑥=(𝑥_𝑖𝑗).
More exactly, we say that a double sequence 𝑥=(𝑥_𝑖𝑗) converges to a finite interval number 𝑥₀ if 𝑥_𝑖𝑗 tend to 𝑥₀ as both 𝑖 and 𝑗 tend to ∞ independently of one another. We denote by 𝑐² the set of all double convergent interval numbers of double interval numbers.

Definition 4. An interval valued double sequence 𝑥=(𝑥_𝑖𝑗) is bounded if there exists a positive number 𝑀 such that 𝑑(𝑥_𝑖𝑗,𝑥₀)≤𝑀 for all 𝑖,𝑗∈ℕ. We will denote all bounded double interval number sequences by 𝑙²_∞. It should be noted that, similar to the case of double sequences, 𝑐² is not the subset of 𝑙²_∞.

Definition 5. Let 𝐴=(𝑎_{𝑚𝑛𝑖𝑗}) denote a four-dimensional summability method that maps the complex double sequences 𝑥 into the double sequence 𝐴𝑥 where the 𝑚𝑛th term to 𝐴𝑥 is as follows: (𝐴𝑥)_𝑚𝑛=^∞,∞_{𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}𝑥_𝑖𝑗. (3) Such a transformation is said to be nonnegative if 𝑎_{𝑚𝑛𝑖𝑗} is nonnegative for all 𝑚,𝑛,𝑖 and 𝑗.

The notion of difference sequence spaces was introduced by Kizmaz [18] who studied the difference sequence spaces 𝑙_∞(Δ), 𝑐(Δ), and 𝑐₀(Δ). The notion was further generalized by Et and Çolak [19] by introducing the spaces 𝑙_∞(Δ^𝑛), 𝑐(Δ^𝑛), and 𝑐₀(Δ^𝑛). Let 𝑤 denote the set of all real and complex sequences and let 𝑛 be a nonnegative integer; then for 𝑍=𝑐,𝑐₀, and 𝑙_∞, we have sequence spaces 𝑍Δ^𝑛=𝑥=𝑥_𝑘∈𝑤∶Δ^𝑛𝑥_𝑘∈𝑍, (4) where Δ^𝑛𝑥=(Δ^𝑛𝑥_𝑘)=(Δ^𝑛−1𝑥_𝑘−Δ^𝑛−1𝑥_𝑘+1) and Δ⁰𝑥_𝑘=𝑥_𝑘 for all 𝑘∈ℕ, which is equivalent to the following binomial representation: Δ^𝑛𝑥_𝑘=^𝑛_𝑣=0(−1)^𝑣𝑛𝑣𝑥_𝑘+𝑣. (5) Taking 𝑛=1, we get the spaces studied by Et and Çolak [19]. For more details about sequence spaces see [20–32] and references therein. Quite recently, Et et al. [33] defined and studied the concept of statistical convergence of order 𝛼 involving the notions of Δ and ideal 𝐼.

Definition 6. An Orlicz function 𝑀∶[0,∞)→[0,∞) is a continuous, nondecreasing, and convex such that 𝑀(0)=0, 𝑀(𝑥)>0 for 𝑥>0 and 𝑀(𝑥)→∞ as 𝑥→∞. If convexity of Orlicz function is replaced by 𝑀(𝑥+𝑦)≤𝑀(𝑥)+𝑀(𝑦), then this function is called modulus function. Lindenstrauss and Tzafriri [34] used the idea of Orlicz function to define the following sequence space: _𝑀=𝑥=𝑥_𝑘∈𝑤∶^∞_𝑘=1𝑀𝑥_𝑘𝜌<∞,forsome𝜌>0 (6) which is known as an Orlicz sequence space. The space 𝑀 is a Banach space with the norm ‖𝑥‖=inf𝜌>0∶^∞_𝑘=1𝑀𝑥_𝑘𝜌≤1. (7) Also it was shown in [34] that every Orlicz sequence space 𝑀 contains a subspace isomorphic to _𝑝(𝑝≥1). An Orlicz function 𝑀 can always be represented in the following integral form: 𝑀(𝑥)=^𝑥₀𝜂(𝑡)𝑑𝑡, (8) where 𝜂 is known as the kernel of 𝑀 and is a right differentiable for 𝑡≥0, 𝜂(0)=0, 𝜂(𝑡)>0, and 𝜂 is nondecreasing and 𝜂(𝑡)→∞ as 𝑡→∞.

Definition 7. A sequence ℳ=(𝑀_𝑘) of Orlicz functions is said to be Musielak-Orlicz function (see [35, 36]). A sequence 𝒩=(𝑁_𝑘) is defined by 𝑁_𝑘(𝑣)=sup|𝑣|𝑢−𝑀_𝑘(𝑢)∶𝑢≥0,𝑘=1,2,…, (9) and is called the complementary function of a Musielak-Orlicz function ℳ. For a given Musielak-Orlicz function ℳ, the Musielak-Orlicz sequence space 𝑡_ℳ and its subspace ℎ_ℳ are defined as follows: 𝑡_ℳ=𝑥∈𝑤∶𝐼_ℳ(),ℎ𝑐𝑥<∞forsome𝑐>0_ℳ=𝑥∈𝑤∶𝐼_ℳ(𝑐𝑥)<∞∀𝑐>0, (10) where 𝐼_ℳ is a convex modular defined by 𝐼_ℳ(𝑥)=^∞_𝑘=1𝑀_𝑘𝑥_𝑘,𝑥=𝑥_𝑘∈𝑡_ℳ. (11) We consider 𝑡_ℳ equipped with the Luxemburg norm ‖𝑥‖=inf𝑘>0∶𝐼_ℳ𝑥𝑘≤1 (12) or equipped with the Orlicz norm ‖𝑥‖⁰1=inf𝑘1+𝐼_ℳ(𝑘𝑥)∶𝑘>0. (13) A Musielak-Orlicz function ℳ=(𝑀_𝑘) is said to satisfy Δ₂-condition if there exist constants 𝑎,𝐾>0 and a sequence 𝑐=(𝑐_𝑘)^∞_𝑘=1∈𝑙¹₊ (the positive cone of 𝑙¹) such that the inequality 𝑀_𝑘(2𝑢)≤𝐾𝑀_𝑘(𝑢)+𝑐_𝑘 (14) holds for all 𝑘∈ℕ and 𝑢∈ℝ⁺, whenever 𝑀_𝑘(𝑢)≤𝑎.

Definition 8. Let 𝑋 be a linear metric space. A function 𝑝: 𝑋→ ℝ is called paranorm, if (1)𝑝(𝑥)≥0 for all 𝑥∈𝑋;(2)𝑝(−𝑥)=𝑝(𝑥) for all 𝑥∈𝑋;(3)𝑝(𝑥+𝑦)≤𝑝(𝑥)+𝑝(𝑦) for all 𝑥,𝑦∈𝑋;(4)(𝜆_𝑛) is a sequence of scalars with 𝜆_𝑛→𝜆 as 𝑛→∞ and (𝑥_𝑛) is a sequence of vectors with 𝑝(𝑥_𝑛−𝑥)→0 as 𝑛→∞, then 𝑝(𝜆_𝑛𝑥_𝑛−𝜆𝑥)→0 as 𝑛→∞.
A paranorm 𝑝 for which 𝑝(𝑥)=0 implies 𝑥=0 is called total paranorm and the pair (𝑋,𝑝) is called a total paranormed space. It is well known that the metric of any linear metric space is given by some total paranorm.

Let ℳ=(𝑀_𝑖𝑗) be a Musielak-Orlicz function and let 𝐴=(𝑎_{𝑚𝑛𝑖𝑗}) be a nonnegative four-dimensional bounded regular matrix (see [37, 38]). Let 𝑝=(𝑝_𝑖𝑗) be a bounded double sequence of positive real numbers and 𝑢=(𝑢_𝑖𝑗) be a double sequence of strictly positive real numbers. In the present paper we define the following new double sequence spaces for interval sequences: ₂𝑤ℳ,𝑝,𝑢,Δ^𝑟=,𝐴𝑥=𝑥_𝑖𝑗∶𝑃−lim_𝑚𝑛1×𝑚𝑛_{𝑚,𝑛𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}𝑀_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,𝑥₀𝜌^𝑝𝑖𝑗,=0,forsome𝜌>0₂𝑤₀ℳ,𝑝,𝑢,Δ^𝑟=,𝐴𝑥=𝑥_𝑖𝑗∶𝑃−lim_𝑚𝑛1×𝑚𝑛_{𝑚,𝑛𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}𝑀_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,0𝜌^𝑝𝑖𝑗,=0,forsome𝜌>0₂𝑤_∞ℳ,𝑝,𝑢,Δ^𝑟=,𝐴𝑥=𝑥_𝑖𝑗∶sup_𝑚𝑛1×𝑚𝑛_{𝑚,𝑛𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}𝑀_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,0𝜌^𝑝𝑖𝑗.<∞,forsome𝜌>0 (15)

Remark 9. Let us consider a few special cases of the above sequence spaces.(i)If ℳ=𝑀_𝑖𝑗(𝑥)=𝑥 for all 𝑖,𝑗∈ℕ, then we have ₂𝑤ℳ,𝑝,𝑢,Δ^𝑟,𝐴=₂𝑤𝑝,𝑢,Δ^𝑟,𝐴₂,𝑤₀ℳ,𝑝,𝑢,Δ^𝑟,𝐴=₂𝑤₀𝑝,𝑢,Δ^𝑟,𝐴₂,𝑤_∞ℳ,𝑝,𝑢,Δ^𝑟,𝐴=₂𝑤_∞𝑝,𝑢,Δ^𝑟,𝐴. (16)(ii)If 𝑝 = (𝑝_𝑖𝑗) = 1, for all 𝑖,𝑗, then we have ₂𝑤ℳ,𝑝,𝑢,Δ^𝑟,𝐴=₂𝑤ℳ,𝑢,Δ^𝑟,𝐴₂,𝑤₀ℳ,𝑝,𝑢,Δ^𝑟,𝐴=₂𝑤₀ℳ,𝑢,Δ^𝑟,𝐴₂,𝑤_∞ℳ,𝑝,𝑢,Δ^𝑟,𝐴=₂𝑤_∞ℳ,𝑢,Δ^𝑟,𝐴. (17)(iii)If 𝑢 = (𝑢_𝑖𝑗) = 1, for all 𝑖,𝑗, then we have ₂𝑤ℳ,𝑝,𝑢,Δ^𝑟,𝐴=₂𝑤ℳ,𝑝,Δ^𝑟,𝐴₂,𝑤₀ℳ,𝑝,𝑢,Δ^𝑟,𝐴=₂𝑤₀ℳ,𝑝,Δ^𝑟,𝐴₂,𝑤_∞ℳ,𝑝,𝑢,Δ^𝑟,𝐴=₂𝑤_∞ℳ,𝑝,Δ^𝑟,𝐴. (18)(iv)If 𝐴=(𝐶,1,1)=1, that is, the double Cesàro matrix, then the above classes of sequences reduce to the following sequence spaces: ₂𝑤ℳ,𝑝,𝑢,Δ^𝑟,𝐴=₂𝑤ℳ,𝑝,𝑢,Δ^𝑟,₂𝑤₀ℳ,𝑝,𝑢,Δ^𝑟,𝐴=₂𝑤₀ℳ,𝑝,𝑢,Δ^𝑟,₂𝑤_∞ℳ,𝑝,𝑢,Δ^𝑟,𝐴=₂𝑤_∞ℳ,𝑝,𝑢,Δ^𝑟. (19)(v)Let 𝐴 = (𝐶,1,1) = 1 and 𝑢_𝑖𝑗=1 for all 𝑖,𝑗. If, in addition, ℳ(𝑥)=𝑀(𝑥) and 𝑟=0, then the spaces ₂𝑤(ℳ,𝑝,𝑢,Δ^𝑟,𝐴), ₂𝑤₀(ℳ,𝑝,𝑢,Δ^𝑟,𝐴), and ₂𝑤_∞(ℳ,𝑝,𝑢,Δ^𝑟,𝐴) are reduced to ₂𝑤(𝑀,𝑝),₂𝑤₀(𝑀,𝑝), and ₂𝑤_∞(𝑀,𝑝) which were introduced and studied by Esi and Hazarika [39].

The following inequality will be used throughout the paper. If 0≤𝑝_𝑖𝑗≤sup𝑝_𝑖𝑗=𝐻, 𝐾=max(1,2^𝐻−1) then 𝑎_𝑖𝑗+𝑏_𝑖𝑗^𝑝𝑖𝑗≤𝐾𝑎_𝑖𝑗^𝑝𝑖𝑗+𝑏_𝑖𝑗^𝑝𝑖𝑗 (20) for all 𝑖,𝑗 and 𝑎_𝑖𝑗,𝑏_𝑖𝑗∈ℂ. Also |𝑎|^𝑝𝑖𝑗≤max(1,|𝑎|^𝐻) for all 𝑎∈ℂ.

The main purpose of this paper is to introduce interval valued double difference sequence spaces ₂𝑤(ℳ,𝑝,𝑢,Δ^𝑟,𝐴), ₂𝑤₀(ℳ,𝑝,𝑢,Δ^𝑟,𝐴), and ₂𝑤_∞(ℳ,𝑝,𝑢,Δ^𝑟,𝐴) and to study different properties of these spaces like linearity, paranorm, solidity, monotone, and so forth. Some inclusion relations between theses spaces are also established.

2. Main Results

Theorem 10. If 0<𝑝_𝑖𝑗<𝑞_𝑖𝑗 for each 𝑖 and 𝑗, then we have ₂𝑤_∞(ℳ,𝑝,𝑢,Δ^𝑟,𝐴)⊂₂𝑤_∞(ℳ,𝑞,𝑢,Δ^𝑟,𝐴).

Proof. Let 𝑥=(𝑥_𝑖𝑗)∈₂𝑤_∞(ℳ,𝑝,𝑢,Δ^𝑟,𝐴). Then there exists 𝜌>0 such that sup_𝑚𝑛1𝑚𝑛^𝑚,𝑛_{𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}𝑀_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,0𝜌^𝑝𝑖𝑗<∞. (21) This implies that 𝑎_{𝑚𝑛𝑖𝑗}𝑀_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,0𝜌^𝑝𝑖𝑗<1, (22) for sufficiently large values of 𝑖 and 𝑗. Since 𝑀_𝑖𝑗 is nondecreasing, we get sup_𝑚𝑛1𝑚𝑛^𝑚,𝑛_{𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}𝑀_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,0𝜌^𝑞𝑖𝑗≤sup_𝑚𝑛1𝑚𝑛^𝑚,𝑛_{𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}𝑀_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,0𝜌^𝑝𝑖𝑗<∞. (23) Thus 𝑥=(𝑥_𝑖𝑗)∈₂𝑤_∞(ℳ,𝑞,𝑢,Δ^𝑟,𝐴). This completes the proof.

Theorem 11. Suppose that ℳ=(𝑀_𝑖𝑗) is a Musielak-Orlicz function, 𝑝=(𝑝_𝑖𝑗) a bounded double sequence of positive real numbers, and 𝑢=(𝑢_𝑖𝑗) a double sequence of strictly positive real numbers. Then the following hold.(i)If 0 < inf𝑝_𝑖𝑗 < 𝑝_𝑖𝑗 ≤ 1, then ₂𝑤_∞(ℳ,𝑝,𝑢,Δ^𝑟,𝐴)⊂₂𝑤_∞(ℳ,𝑢,Δ^𝑟,𝐴).(ii)If 1 ≤ 𝑝_𝑖𝑗 ≤ sup𝑝_𝑖𝑗 < ∞, then ₂𝑤_∞(ℳ,𝑢,Δ^𝑟,𝐴)⊂₂𝑤_∞(ℳ,𝑝,𝑢,Δ^𝑟,𝐴).

Proof. (i) Let 𝑥=(𝑥_𝑖𝑗)∈₂𝑤_∞(ℳ,𝑝,𝑢,Δ^𝑟,𝐴). Since 0<inf𝑝_𝑖𝑗≤1, we obtain the following: sup_𝑚𝑛1𝑚𝑛^𝑚,𝑛_{𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}𝑀_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,0𝜌≤sup_𝑚𝑛1𝑚𝑛^𝑚,𝑛_{𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}𝑀_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,0𝜌^𝑝𝑖𝑗<∞, (24) and hence 𝑥=(𝑥_𝑖𝑗)∈₂𝑤_∞(ℳ,𝑢,Δ^𝑟,𝐴).
(ii) Let 𝑝_𝑖𝑗≥1 for each 𝑖 and 𝑗 and sup𝑝_𝑖𝑗<∞. Let 𝑥=(𝑥_𝑖𝑗)∈₂𝑤_∞(ℳ,𝑢,Δ^𝑟,𝐴). Then for each 0<𝜖<1 there exists a positive integer 𝑁 such that sup_𝑚𝑛1𝑚𝑛^𝑚,𝑛_{𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}𝑀_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,0𝜌≤𝜖<1∀𝑛,𝑚≥𝑁. (25) This implies that sup_𝑚𝑛1𝑚𝑛^𝑚,𝑛_{𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}𝑀_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,0𝜌^𝑝𝑖𝑗≤sup_𝑚𝑛1𝑚𝑛^𝑚,𝑛_{𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}𝑀_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,0𝜌<∞. (26) Therefore, 𝑥=(𝑥_𝑖𝑗)∈₂𝑤_∞(ℳ,𝑝,𝑢,Δ^𝑟,𝐴). This completes the proof.

Theorem 12. Let 0<𝑝_𝑖𝑗≤𝑞_𝑖𝑗 for all 𝑖,𝑗∈ℕ and (𝑞_𝑖𝑗/𝑝_𝑖𝑗) be bounded. Then we have ₂𝑤_∞(ℳ,𝑞,𝑢,Δ^𝑟,𝐴)⊂₂𝑤_∞(ℳ,𝑝,𝑢,Δ^𝑟,𝐴).

Proof. Let 𝑥=(𝑥_𝑖𝑗)∈₂𝑤_∞(ℳ,𝑞,𝑢,Δ^𝑟,𝐴). Then sup_𝑚𝑛1𝑚𝑛^𝑚,𝑛_{𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}𝑀_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,0𝜌^𝑞𝑖𝑗<∞,forsome𝜌>0. (27) Let 𝑠_𝑖𝑗=sup_𝑚𝑛(1/𝑚𝑛)∑^𝑚,𝑛_{𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}[𝑀_𝑖𝑗(𝑢_𝑖𝑗𝑑(Δ^𝑟𝑥_𝑖𝑗,0)/𝜌)]^𝑞𝑖𝑗 and 𝜆_𝑖𝑗=𝑝_𝑖𝑗/𝑞_𝑖𝑗. Since 𝑝_𝑖𝑗≤𝑞_𝑖𝑗, we have 0≤𝜆_𝑖𝑗≤1. Take 0<𝜆<𝜆_𝑖𝑗.
Define 𝑢_𝑖𝑗=𝑠_𝑖𝑗if𝑠_𝑖𝑗≥10if𝑠_𝑖𝑗𝑣<1,_𝑖𝑗=0if𝑠_𝑖𝑗𝑠≥1_𝑖𝑗if𝑠_𝑖𝑗<1, (28)𝑠_𝑖𝑗=𝑢_𝑖𝑗+𝑣_𝑖𝑗,𝑠^𝜆𝑖𝑗_𝑖𝑗=𝑢^𝜆𝑖𝑗_𝑖𝑗+𝑣^𝜆𝑖𝑗_𝑖𝑗. It follows that 𝑢^𝜆𝑖𝑗_𝑖𝑗≤𝑢_𝑖𝑗≤𝑠_𝑖𝑗,𝑣^𝜆𝑖𝑗_𝑖𝑗≤𝑣^𝜆_𝑖𝑗. since 𝑠^𝜆𝑖𝑗_𝑖𝑗=𝑢^𝜆𝑖𝑗_𝑖𝑗+𝑣^𝜆𝑖𝑗_𝑖𝑗, then 𝑠^𝜆𝑖𝑗_𝑖𝑗≤𝑠_𝑖𝑗+𝑣^𝜆_𝑖𝑗sup_𝑚𝑛1𝑚𝑛^𝑚,𝑛_{𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}𝑀_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,0𝜌^{𝑞𝑖𝑗𝜆𝑖𝑗}≤sup_𝑚𝑛1𝑚𝑛^𝑚,𝑛_{𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}𝑀_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,0𝜌^𝑞𝑖𝑗⟹sup_𝑚𝑛1𝑚𝑛^𝑚,𝑛_{𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}𝑀_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,0𝜌^{𝑞𝑖𝑗𝑝𝑖𝑗/𝑞𝑖𝑗}≤sup_𝑚𝑛1𝑚𝑛^𝑚,𝑛_{𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}𝑀_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,0𝜌^𝑞𝑖𝑗⟹sup_𝑚𝑛1𝑚𝑛^𝑚,𝑛_{𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}𝑀_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,0𝜌^𝑝𝑖𝑗≤sup_𝑚𝑛1𝑚𝑛^𝑚,𝑛_{𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}𝑀_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,0𝜌^𝑞𝑖𝑗, (29) but sup_𝑚𝑛1𝑚𝑛^𝑚,𝑛_{𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}𝑀_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,0𝜌^𝑞𝑖𝑗<∞forsome𝜌>0. (30) Therefore, sup_𝑚𝑛1𝑚𝑛^𝑚,𝑛_{𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}𝑀_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,0𝜌^𝑝𝑖𝑗<∞forsome𝜌>0. (31) Hence 𝑥=(𝑥_𝑖𝑗)∈₂𝑊_∞(ℳ,𝑝,𝑢,Δ^𝑟,𝐴). Thus, we get ₂𝑊_∞(ℳ,𝑞,𝑢,Δ^𝑟,𝐴)⊂₂𝑤_∞(ℳ,𝑝,𝑢,Δ^𝑟,𝐴).

Theorem 13. Let ℳ^′=(𝑀^′_𝑖𝑗) and ℳ^′′=(𝑀^′′_𝑖𝑗) be two Musielak-Orlicz functions, ₂𝑤_∞ℳ^′,𝑝,𝑢,Δ^𝑟,𝐴∩₂𝑤_∞ℳ^′′,𝑝,𝑢,Δ^𝑟⊂,𝐴₂𝑤_∞ℳ^′+ℳ^′′,𝑝,𝑢,Δ^𝑟,𝐴. (32)

Proof. Let 𝑥=(𝑥_𝑖𝑗)∈₂𝑤_∞(ℳ^′,𝑝,𝑢,Δ^𝑟,𝐴)∩₂𝑤_∞(ℳ^′′,𝑝,𝑢,Δ^𝑟,𝐴). Then sup_𝑚𝑛1𝑚𝑛^𝑚,𝑛_{𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}𝑀^′_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,0𝜌₁^𝑝𝑖𝑗<∞,forsome𝜌₁>0,sup_𝑚𝑛1𝑚𝑛^𝑚,𝑛_{𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}𝑀^′′_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,0𝜌₂^𝑝𝑖𝑗<∞,forsome𝜌₂>0. (33) Let 𝜌=max{𝜌₁,𝜌₂}. The result follows from the inequality sup_𝑚𝑛1𝑚𝑛^𝑚,𝑛_{𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}𝑀^′_𝑖𝑗+𝑀^′′_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,0𝜌^𝑝𝑖𝑗=sup_𝑚𝑛1𝑚𝑛^𝑚,𝑛_{𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}𝑀^′_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,0𝜌₁^𝑝𝑖𝑗+sup_𝑚𝑛1𝑚𝑛^𝑚,𝑛_{𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}𝑀^′′_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,0𝜌₂^𝑝𝑖𝑗≤𝐾sup_𝑚𝑛1𝑚𝑛^𝑚,𝑛_{𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}𝑀^′_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,0𝜌₁^𝑝𝑖𝑗+𝐾sup_𝑚𝑛1𝑚𝑛^𝑚,𝑛_{𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}𝑀^′′_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,0𝜌₂^𝑝𝑖𝑗<∞. (34) Thus, sup_𝑚𝑛(1/𝑚𝑛)∑^𝑚,𝑛_{𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}[(𝑀^′_𝑖𝑗+𝑀^′′_𝑖𝑗)(𝑢_𝑖𝑗𝑑(Δ^𝑟𝑦_𝑖𝑗,0)/𝜌)]^𝑝𝑖𝑗<∞. Therefore, 𝑥=(𝑥_𝑖𝑗)∈₂𝑤_∞(ℳ^′+ ℳ^′′,𝑝,𝑢,Δ^𝑟,𝐴).

Theorem 14. Let ℳ=(𝑀_𝑖𝑗) be a Musielak-Orlicz function and let 𝐴=(𝑎_{𝑛𝑚𝑖𝑗}) be a nonnegative four-dimensional regular summability method. Suppose that 𝛽=lim_𝑡→∞(𝑀_𝑖𝑗(𝑡)/𝑡)<∞. Then ₂𝑤(𝑝,𝑢,Δ^𝑟,𝐴)=₂𝑤(ℳ,𝑝,𝑢,Δ^𝑟,𝐴).

Proof. In order to prove that ₂𝑊(𝑝,𝑢,Δ^𝑟,𝐴)=₂𝑤(ℳ,𝑝,𝑢,Δ^𝑟,𝐴), it is sufficient to show that ₂𝑊(ℳ,𝑝,𝑢,Δ^𝑟,𝐴)⊂₂𝑤(𝑝,𝑢,Δ^𝑟,𝐴). Now, let 𝛽>0. By definition of 𝛽, we have 𝑀_𝑖𝑗(𝑡)≥𝛽𝑡 for all 𝑡≥0. Since 𝛽>0, we have 𝑡≤(1/𝛽)𝑀_𝑖𝑗(𝑡) for all 𝑡≥0. Let 𝑥=(𝑥_𝑖𝑗)∈₂𝑤(ℳ,𝑝,𝑢,Δ^𝑟,𝐴). Thus, we have sup_𝑚𝑛1𝑚𝑛^𝑚,𝑛_{𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,𝑥₀𝜌^𝑝𝑖𝑗≤1𝛽sup_𝑚𝑛1𝑚𝑛^𝑚,𝑛_{𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}𝑀_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,𝑥₀𝜌^𝑝𝑖𝑗<∞ (35) which implies that 𝑥=(𝑥_𝑖𝑗)∈₂𝑤(𝑝,𝑢,Δ^𝑟,𝐴). This completes the proof.

Theorem 15. Let 0<ℎ=inf𝑝_𝑖𝑗≤𝑝_𝑖𝑗≤sup𝑝_𝑖𝑗=𝐻<∞. Then for a Musielak-Orlicz function ℳ=(𝑀_𝑖𝑗) which satisfies the Δ₂-condition, we have ₂𝑤(𝑝,𝑢,Δ^𝑟,𝐴)=₂𝑤(ℳ,𝑝,𝑢,Δ^𝑟,𝐴).

Proof. Let 𝑥=(𝑥_𝑖𝑗)∈₂𝑤(𝑝,𝑢,Δ^𝑟,𝐴); that is, 1𝑚𝑛^𝑚,𝑛_{𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,𝑥₀𝜌^𝑝𝑖𝑗=0,forsome𝜌>0. (36) Let 𝜖>0 and choose 𝛿 with 0<𝛿<1 such that 𝑀_𝑖𝑗(𝑡)<𝜖 for 0≤𝑡≤𝛿. Then 1𝑚𝑛^𝑚,𝑛_{𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}𝑀_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,𝑥₀𝜌^𝑝𝑖𝑗=1𝑚𝑛^𝑚,𝑛_{𝑖,𝑗=1,1𝑑(Δ𝑟𝑥𝑖𝑗,𝑥0)≤𝛿}𝑎_{𝑚𝑛𝑖𝑗}𝑀_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,𝑥₀𝜌^𝑝𝑖𝑗+1𝑚𝑛^𝑚,𝑛_{𝑖,𝑗=1,1𝑑(Δ𝑟𝑥𝑖𝑗,𝑥0)>𝛿}𝑎_{𝑚𝑛𝑖𝑗}𝑀_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,𝑥₀𝜌^𝑝𝑖𝑗=₁+₂, (37) where ₁=1𝑚𝑛^𝑚,𝑛_{𝑖,𝑗=1,1𝑑(Δ𝑟𝑥𝑖𝑗,𝑥0)≤𝛿}𝑎_{𝑚𝑛𝑖𝑗}𝑀_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,𝑥₀𝜌^𝑝𝑖𝑗<max𝜖,𝜖^𝐻 (38) by using continuity of (𝑀_𝑖𝑗). For the second summation, we will make the following procedure. Thus we have 𝑑Δ^𝑟𝑥_𝑖𝑗,𝑥₀𝜌𝑑Δ<1+^𝑟𝑥_𝑖𝑗,𝑥₀/𝜌𝛿. (39) Since ℳ=(𝑀_𝑖𝑗) is nondecreasing and convex, so we have 𝑎_{𝑚𝑛𝑖𝑗}𝑀_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,𝑥₀𝜌<𝑎_{𝑚𝑛𝑖𝑗}𝑀_𝑖𝑗𝑢1+^𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,𝑥₀/𝜌𝛿≤12𝑎_{𝑚𝑛𝑖𝑗}𝑢_𝑖𝑗𝑀_𝑖𝑗(2+1)2𝑎_{𝑚𝑛𝑖𝑗}𝑀_𝑖𝑗2𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,𝑥₀/𝜌𝛿. (40) Again, since ℳ=(𝑀_𝑖𝑗) satisfies the Δ₂-condition, it follows that 𝑎_{𝑚𝑛𝑖𝑗}𝑀_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,𝑥₀𝜌≤12𝐾𝑑Δ^𝑟𝑥_𝑖𝑗,𝑥₀/𝜌𝛿𝑎_{𝑚𝑛𝑖𝑗}𝑢_𝑖𝑗𝑀_𝑖𝑗(2+1)2𝐾𝑑Δ^𝑟𝑥_𝑖𝑗,𝑥₀/𝜌𝛿𝑎_{𝑚𝑛𝑖𝑗}𝑢_𝑖𝑗𝑀_𝑖𝑗(2𝑑Δ)=𝐾^𝑟𝑥_𝑖𝑗,𝑥₀/𝜌𝛿𝑎_{𝑚𝑛𝑖𝑗}𝑢_𝑖𝑗𝑀_𝑖𝑗(2). (41) Thus, it follows that ₂=max1,𝐾𝑎_{𝑚𝑛𝑖𝑗}𝑢_𝑖𝑗𝑀_𝑖𝑗(2)𝛿^𝐻×1𝑚𝑛^𝑚,𝑛_{𝑖,𝑗=1,1}𝑑Δ^𝑟𝑥_𝑖𝑗,𝑥₀𝜌^𝑝𝑖𝑗. (42) Taking the limit as 𝜖→0 and 𝑚,𝑛→∞, it follows that 𝑥=(𝑥_𝑖𝑗)∈₂𝑤(ℳ,𝑝,𝑢,Δ^𝑟,𝐴).

Theorem 16. Suppose that ℳ=(𝑀_𝑖𝑗) is a Musielak-Orlicz function, 𝑝=(𝑝_𝑖𝑗) a bounded double sequence of positive real numbers, and 𝑢=(𝑢_𝑖𝑗) a double sequence of strictly positive real numbers. If sup_𝑖,𝑗(𝑀_𝑖𝑗(𝑥))^𝑝𝑖𝑗<∞ for all fixed 𝑥>0, then ₂𝑤ℳ,𝑝,𝑢,Δ^𝑟,𝐴⊂₂𝑤_∞ℳ,𝑝,𝑢,Δ^𝑟,𝐴. (43)

Proof. Let 𝑥=(𝑥_𝑖𝑗)∈₂𝑤(ℳ,𝑝,𝑢,Δ^𝑟,𝐴). Then there exists a positive number 𝜌₁>0 such that 1𝑚𝑛^𝑚,𝑛_{𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}𝑀_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,𝑥₀𝜌₁^𝑝𝑖𝑗=0,forsome𝜌₁>0. (44) Define 𝜌=2𝜌₁. Since ℳ=(𝑀_𝑖𝑗) is nondecreasing and convex, for each 𝑖,𝑗, so by using (20), we have sup_𝑚𝑛1𝑚𝑛^𝑚,𝑛_{𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}𝑀_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,0𝜌^𝑝𝑖𝑗≤sup_𝑚𝑛1𝑚𝑛^𝑚,𝑛_{𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}𝑀_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,𝑥₀+𝑑𝑥₀,0𝜌^𝑝𝑖𝑗≤𝐾sup_𝑚𝑛1𝑚𝑛^𝑚,𝑛_{𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}𝑀_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,𝑥₀𝜌₁^𝑝𝑖𝑗+sup_𝑚𝑛1𝑚𝑛^𝑚,𝑛_{𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}𝑀_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,0𝜌₁^𝑝𝑖𝑗<∞. (45) Thus 𝑥=(𝑥_𝑖𝑗)∈₂𝑤_∞(ℳ,𝑝,𝑢,Δ^𝑟,𝐴). This completes the proof of the theorem.

Theorem 17. The double sequence space ₂𝑤_∞(ℳ,𝑝,𝑢,Δ^𝑟,𝐴) is solid.

Proof. Suppose 𝑥=(𝑥_𝑖𝑗)∈₂𝑤_∞(ℳ,𝑝,𝑢,Δ^𝑟,𝐴)sup_𝑚𝑛1𝑚𝑛^𝑚,𝑛_{𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}𝑀_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,0𝜌^𝑝𝑖𝑗<∞,forsome𝜌>0. (46) Let (𝛼_𝑖𝑗) be a double sequence of scalars such that |𝛼_𝑖𝑗|≤1 for all 𝑖,𝑗∈ℕ. Then we get sup_𝑚𝑛1𝑚𝑛^𝑚,𝑛_{𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}𝑀_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝛼_𝑖𝑗𝑥_𝑖𝑗,0𝜌^𝑝𝑖𝑗≤sup_𝑚𝑛1𝑚𝑛^𝑚,𝑛_{𝑖,𝑗=1,1}𝑎_{𝑚𝑛𝑖𝑗}𝑀_𝑖𝑗𝑢_𝑖𝑗𝑑Δ^𝑟𝑥_𝑖𝑗,0𝜌^𝑝𝑖𝑗<∞. (47) This completes the proof.

Theorem 18. The double sequence space ₂𝑤_∞(ℳ,𝑝,𝑢,Δ^𝑟,𝐴) is monotone.

Proof. The proof is trivial so we omit it.

Conflict of Interests

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

Acknowledgment

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