Abstract

We generalize some sequence spaces from single to double, we study some topological properties of these double sequence spaces by using ideal convergence, difference sequence spaces, and an Orlicz function in 2-normed spaces, and we give some results related to these sequence spaces.

1. Introduction

The concept of 𝐼-convergent was introduced by Das et al. [1] and developed by many scholars. B. Tripathy and B. C. Tripathy [2] extended the concept of 𝐼-convergence from single sequence to double sequence. Some difference double-sequence spaces and paranormed double-sequence spaces defined by Orlicz function were introduced by Tripathy and Sarma [3, 4]. And also some new single- and double-sequence spaces defined in 2-normed spaces using ideal convergence and an Orlicz function were introduced by Savaş [5, 6].

Recall that an Orlicz Function, which was presented by Karasnoselskii and Rutishky [7] 𝑀∶[0,∞)→[0,∞) is continuous, convex, nondecreasing 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 nondecreasing, and 𝑝(𝑡)→∞ as 𝑡→∞.

Ruckle [8] and Maddox [9] presented and discussed that if convexity of Orlicz function 𝑀 is replaced by 𝑀(𝑥+𝑦)≤𝑀(𝑥)+𝑀(𝑦) then this function is called Modulus function.

An Orlicz function 𝑀 is said to satisfy Δ2-condition for all values of 𝑢, if there exists constant 𝐾>0, such that 𝑀(2𝑢)≤𝐾𝑀(𝑢),(𝑢≥0). The Δ2-condition is equivalent to the inequality 𝑀(𝐿𝑢)≤𝐾𝐿𝑀(𝑢) for all values of 𝑢 and for 𝐿>1 being satisfied [7].

The notion of difference double-sequence spaces was introduced by Tripathy and Sarma [3]. These notions are further studied by Tripathy and Sarma [4]. Let (𝑎𝑛𝑘) be a double sequence. Then the operator Δ is defined as Δ𝑎𝑛𝑘=𝑎𝑛,𝑘−𝑎𝑛,𝑘+1−𝑎𝑛+1,𝑘+𝑎𝑛+1,𝑘+1, for all 𝑘,𝑛∈ℕ.

Let us recall some well-known concepts that (𝑋,‖⋅‖) is a normed space and a sequence (𝑥𝑚𝑛)(𝑚,𝑛∈ℕ) of elements of 𝑋 is called to be statistically convergent to 𝑥∈𝑋 if the set 𝐴(𝜀)={𝑚,𝑛∈ℕ∶‖𝑥𝑚𝑛−𝑥‖≥𝜀}has zero natural density for each 𝜀>0 [2].

A real double-sequence 𝑥=(𝑥𝑘,𝑙) is said to be statistically bounded above if there exists a real number 𝐷>0 such that 𝛿{𝑘∶𝑥𝑘,𝑙>𝐷}=0. A real double-sequence 𝑥=(𝑥𝑘,𝑙) is said to be statistically bounded below if there exists a real number 𝐸>0 such that 𝛿{𝑘∶𝑥𝑘,𝑙<𝐸}=0. If a real double-sequence 𝑥=(𝑥𝑘,𝑙) is statistically bounded both above and below, then we say that 𝑥=(𝑥𝑘,𝑙) is statistically bounded. It is clear that any bounded double-sequence is also statistically bounded [10].

A family 𝐼⊂2𝑌 of subset of a nonempty set 𝑌 is said to be an ideal in 𝑌 if(i)∅∈𝑌,(ii)𝐴,𝐵∈𝐼 imply 𝐴∪𝐵∈𝐼,(iii)𝐴∈𝐼,𝐵⊂𝐴 imply 𝐵∈𝐼 [11, 12].

𝐼⊂2𝑌 is called an admissible ideal in 𝑌 if and only if it contains {{𝑦}∶𝑦∈𝑌} [13].

Let 𝐼⊂2ℕ×ℕ be a nontrivial ideal (𝑖.𝑒.,𝐼≠∅ and ℕ×ℕ∉𝐼) in N. The sequence (𝑥𝑚𝑛)(𝑚,𝑛∈ℕ) is said to be 𝐼-convergent to 𝑥∈𝑋 if for each 𝜀>0 the set 𝐴(𝜀)={𝑚,𝑛∈ℕ∶‖𝑥𝑚𝑛−𝑥‖≥𝜀}∈𝐼 [1].

A real double-sequence 𝑥=(𝑥𝑘,𝑙) is said to be 𝐼-bounded above if there exists a real number 𝐺>0 such that {𝑘∶𝑥𝑘,𝑙>𝐺}∈𝐼. A real double-sequence 𝑥=(𝑥𝑘,𝑙) is said to be 𝐼-bounded below if there exists a real number 𝐻>0 such that {𝑘∶𝑥𝑘,𝑙<𝐻}∈𝐼. If a real double-sequence 𝑥=(𝑥𝑘,𝑙) is 𝐼-bounded both above and below, then we say that 𝑥=(𝑥𝑘,𝑙) is 𝐼-bounded. One can observe easily that any bounded double sequence is 𝐼-bounded [14].

Let 𝑋 be a real vector space of dimension 𝑑, where 2≤𝑑<∞. A 2-norm on 𝑋 is a function ‖⋅,⋅‖∶𝑋×𝑋→ℝ which satisfied 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 space [15, 16].

The sequence (𝑥𝑘)𝑘∈ℕ in a 2-normed space (𝑋,‖⋅,⋅‖) is said to be convergent to 𝐿 in 𝑋 if lim𝑘→∞‖𝑥𝑘−𝐿,𝑧‖=0 for every 𝑧∈𝑋. In this instance, we write lim𝑘→∞‖𝑥𝑘,𝑧‖=‖𝐿,𝑧‖ [17].

Let 𝐼⊂2ℕ be a nontrivial ideal in ℕ. The sequence (𝑥𝑘)𝑘∈ℕ in a2-normed space 𝑋 is said to be 𝐼-convergent to 𝑥, if for every 𝜀>0 and 𝑧 in 𝑋 the set 𝐴(𝜀)={𝑘∈ℕ∶‖𝑥𝑘−𝑥,𝑧‖≥𝜀} belongs to 𝐼 [18].

The following inequalities which will be used throughout the paper can be introduced like in Maddox [19]. If 0≤𝑝𝑘,𝑙≤sup𝑘,𝑙𝑝𝑘,𝑙=𝐻,𝐷=max(1,2𝐻−1) then |𝑎𝑘,𝑙+𝑏𝑘,𝑙|𝑝𝑘,𝑙≤𝐷{|𝑎𝑘,𝑙|𝑝𝑘,𝑙+|𝑏𝑘,𝑙|𝑝𝑘,𝑙} for all 𝑘,𝑙 and 𝑎𝑘,𝑙,𝑏𝑘,𝑙∈ℂ. And also |𝑎|𝑝𝑘,𝑙≤max(1,2𝐻) for all 𝑎∈ℂ.

2. Main Results

Let 𝜆=(𝜆𝑛) and 𝜇=(𝜇𝑚) be two nondecreasing sequences of positive real numbers both of which tend to ∞ as 𝑛 and 𝑚 approach ∞, respectively. Also let 𝜆𝑛+1≤𝜆𝑛+1,𝜆1=0 and 𝜇𝑚+1≤𝜇𝑚+1,𝜇1=0. Let 𝐼 be an admissible ideal of ℕ×ℕ, 𝑀 an Orlicz function, and (𝑋,‖⋅,⋅‖) a 2-normed spaces. In addition let 𝑝=𝑝𝑘,𝑙 be a bounded sequence of positive real numbers. With 𝑆(2−𝑋), we symbolize the space of all double sequences defined over (𝑋,‖⋅,⋅‖). Now we define the following double sequence spaces:𝑊𝐼2)=⎧⎪⎨⎪⎩(𝜆,𝑀,Δ,𝑝,‖⋅,⋅‖𝑥∈𝑆⎧⎪⎨⎪⎩1(2−𝑋)∶∀𝜀>0𝑚,𝑛∈ℕ∶𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚𝑀‖‖‖Δ𝑥𝑘,𝑙−𝐿𝜌‖‖‖,𝑧𝑝𝑘,𝑙⎫⎪⎬⎪⎭⎫⎪⎬⎪⎭,𝑊≥𝜀∈𝐼forsome𝜌>0,𝐿∈𝑋andeach𝑧∈𝑋𝐼02)=⎧⎪⎨⎪⎩(𝜆,𝑀,Δ,𝑝,‖⋅,⋅‖𝑥∈𝑆⎧⎪⎨⎪⎩1(2−𝑋)∶∀𝜀>0𝑚,𝑛∈ℕ∶𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚𝑀‖‖‖Δ𝑥𝑘,𝑙𝜌‖‖‖,𝑧𝑝𝑘,𝑙⎫⎪⎬⎪⎭⎫⎪⎬⎪⎭,𝑊≥𝜀∈𝐼forsome𝜌>0andeach𝑧∈𝑋∞2)=⎧⎪⎨⎪⎩(𝜆,𝑀,Δ,𝑝,‖⋅,⋅‖𝑥∈𝑆(2−𝑋)∶∃𝐾>0s.t.sup𝑚,𝑛1𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚𝑀‖‖‖Δ𝑥𝑘,𝑙𝜌‖‖‖,𝑧𝑝𝑘,𝑙⎫⎪⎬⎪⎭,𝑊≤𝐾forsome𝜌>0andeach𝑧∈𝑋𝐼∞2)=⎧⎪⎨⎪⎩(𝜆,𝑀,Δ,𝑝,‖⋅,⋅‖𝑥∈𝑆⎧⎪⎨⎪⎩1(2−𝑋)∶∃𝐾>0s.t.𝑚,𝑛∈ℕ∶𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚𝑀‖‖‖Δ𝑥𝑘,𝑙𝜌‖‖‖,𝑧𝑝𝑘,𝑙⎫⎪⎬⎪⎭⎫⎪⎬⎪⎭,≥𝐾∈𝐼forsome𝜌>0andeach𝑧∈𝑋(2.1)

where 𝐼𝑛=[𝑛−𝜆𝑛+1,𝑛], 𝐼𝑚=[𝑚−𝜇𝑚+1,𝑚], and 𝜆𝑛,𝑚=𝜆𝑛𝜇𝑚. Throughout this paper we will denote 𝜆𝑛𝜇𝑚 by 𝜆𝑛,𝑚 and (𝑘∈𝐼𝑛,𝑙∈𝐼𝑚) by (𝑘,𝑙)∈𝐼𝑛,𝑚.

Theorem 2.1. (𝑊𝐼)2(𝜆,𝑀,Δ,𝑝,‖⋅,⋅‖),(𝑊𝐼0)2(𝜆,𝑀,Δ,𝑝,‖⋅,⋅‖),(𝑊𝐼∞)2(𝜆,𝑀,Δ,𝑝,‖⋅,⋅‖) are linear spaces.

Proof. Because all the statements can be proved in a similar way, we will prove the assertion for (𝑊𝐼0)2(𝜆,𝑀,Δ,𝑝,‖⋅,⋅‖). Suppose that 𝑥,𝑦∈(𝑊𝐼0)2(𝜆,𝑀,Δ,𝑝,‖⋅,⋅‖) and 𝛼,𝛽∈ℝ. So ⎧⎪⎨⎪⎩1𝑚,𝑛∈ℕ∶𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚𝑀‖‖‖Δ𝑥𝑘,𝑙𝜌1‖‖‖,𝑧𝑝𝑘,𝑙≥𝜀forsome𝜌1⎫⎪⎬⎪⎭⎧⎪⎨⎪⎩1>0∈𝐼,𝑚,𝑛∈ℕ∶𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚𝑀‖‖‖Δ𝑦𝑘,𝑙𝜌2‖‖‖,𝑧𝑝𝑘,𝑙≥𝜀forsome𝜌2⎫⎪⎬⎪⎭>0∈𝐼.(2.2) Since we study (𝑋,‖⋅,⋅‖)  2-normed, Δ is linear and 𝑀 is an Orlicz function, we have the following inequality: 1𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚𝑀‖‖‖‖Δ𝛼𝑥𝑘,𝑙+𝛽𝑦𝑘,𝑙|𝛼|𝜌1+||𝛽||𝜌2‖‖‖‖,𝑧𝑝𝑘,𝑙1≤𝐷𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚|𝛼||𝛼|𝜌1+||𝛽||𝜌2𝑀‖‖‖Δ𝑥𝑘,𝑙𝜌1‖‖‖,𝑧𝑝𝑘,𝑙1+𝐷𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚||𝛽|||𝛼|𝜌1+||𝛽||𝜌2𝑀‖‖‖Δ𝑦𝑘,𝑙𝜌2‖‖‖,𝑧𝑝𝑘,𝑙1≤𝐷𝐹𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚𝑀‖‖‖Δ𝑥𝑘,𝑙𝜌1‖‖‖,𝑧𝑝𝑘,𝑙1+𝐷𝐹𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚𝑀‖‖‖Δ𝑦𝑘,𝑙𝜌2‖‖‖,𝑧𝑝𝑘,𝑙,(2.3) where 𝐹=max[1,(|𝛼|/(|𝛼|𝜌1+|𝛽|𝜌2))𝐻,(|𝛽|/(|𝛼|𝜌1+|𝛽|𝜌2))𝐻].
From the above inequality we get ⎧⎪⎨⎪⎩1𝑚,𝑛∈ℕ∶𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚𝑀‖‖‖‖Δ𝛼𝑥𝑘,𝑙+𝛽𝑦𝑘,𝑙|𝛼|𝜌1+||𝛽||𝜌2‖‖‖‖,𝑧𝑝𝑘,𝑙⎫⎪⎬⎪⎭⊆⎧⎪⎨⎪⎩1≥𝜀𝑚,𝑛∈ℕ∶𝐷𝐹𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚𝑀‖‖‖Δ𝑥𝑘,𝑙𝜌1‖‖‖,𝑧𝑝𝑘,𝑙≥𝜀2⎫⎪⎬⎪⎭∪⎧⎪⎨⎪⎩1𝑚,𝑛∈ℕ∶𝐷𝐹𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚𝑀‖‖‖Δ𝑦𝑘,𝑙𝜌2‖‖‖,𝑧𝑝𝑘,𝑙≥𝜀2⎫⎪⎬⎪⎭(2.4) the two set on the right side belonging to 𝐼 and this completes the proof.

Theorem 2.2. For any fixed 𝑚,𝑛∈ℕ,(𝑊∞)2(𝜆,𝑀,Δ,𝑝,‖⋅,⋅‖) is a paranormed space with the paranorm defined by 𝑔𝑚,𝑛(𝑥)=inf𝑧∈𝑋1𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚‖‖𝑥𝑘,𝑙‖‖⎧⎪⎨⎪⎩𝜌,𝑧+inf𝑝𝑚𝑛/𝐻∶s.t.sup𝑘,𝑙𝑀‖‖‖Δ𝑥𝑘,𝑙𝜌‖‖‖,𝑧𝑝𝑘,𝑙⎫⎪⎬⎪⎭≤1𝜌>0,∀𝑧∈𝑋.(2.5)

Proof. (i) 𝑔𝑚,𝑛(0)=0 and (ii) 𝑔𝑚,𝑛(−𝑥)=𝑔𝑚,𝑛(𝑥) are easy to prove. So we leave them out. (iii) Let us take 𝑥,𝑦∈(𝑊∞)2(𝜆,𝑀,Δ,𝑝,‖⋅,⋅‖). Let 𝐴(𝑥)=𝜌>0∶sup𝑘,𝑙𝑀‖‖‖Δ𝑥𝑘,𝑙𝜌‖‖‖,𝑧𝑝𝑘,𝑙,≤1,∀𝑧∈𝑋𝐴(𝑦)=𝜌>0∶sup𝑘,𝑙𝑀‖‖‖Δ𝑦𝑘,𝑙𝜌‖‖‖,𝑧𝑝𝑘,𝑙.≤1,∀𝑧∈𝑋(2.6)
Let 𝜌1∈𝐴(𝑥) and 𝜌2∈𝐴(𝑦). Then if 𝜌=𝜌1+𝜌2 then we have 𝑀‖‖‖‖Δ𝑥𝑘,𝑙+𝑦𝑘,𝑙𝜌‖‖‖‖≤𝜌,𝑧1𝜌1+𝜌2𝑀‖‖‖Δ𝑥𝑘,𝑙𝜌1‖‖‖+𝜌,𝑧2𝜌1+𝜌2𝑀‖‖‖Δ𝑦𝑘,𝑙𝜌2‖‖‖,𝑧.(2.7) Thus sup𝑘,𝑙𝑀‖‖‖‖Δ𝑥𝑘,𝑙+𝑦𝑘,𝑙𝜌1+𝜌2‖‖‖‖𝑔,𝑧≤1,𝑚,𝑛(𝑥+𝑦)=inf𝑧∈𝑋1𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚‖‖𝑥𝑘,𝑙+𝑦𝑘,𝑙‖‖𝜌,𝑧+inf1+𝜌2𝑝𝑚𝑛/𝐻∶𝜌1∈𝐴(𝑥),𝜌2∈𝐴(𝑦)≤inf𝑧∈𝑋1𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚‖‖𝑥𝑘,𝑙‖‖𝜌,𝑧+inf1𝑝𝑚𝑛/𝐻∶𝜌1∈𝐴(𝑥)+inf𝑧∈𝑋1𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚‖‖𝑦𝑘,𝑙‖‖𝜌,𝑧+inf2𝑝𝑚𝑛/𝐻∶𝜌2∈𝐴(𝑦)=𝑔𝑚,𝑛(𝑥)+𝑔𝑚,𝑛(𝑦).(2.8)   (iv) Let 𝜎𝑖→𝜎, where 𝜎,𝜎𝑖∈ℂ and 𝑔𝑚,𝑛(𝑥𝑖𝑘,𝑙−𝑥)→0 as 𝑖→∞. We have to show that 𝑔𝑚,𝑛(𝜎𝑖𝑥𝑖𝑘,𝑙−𝜎𝑥)→0 as 𝑖→∞. Let 𝐴𝑥𝑖=𝜌𝑖>0∶sup𝑘,𝑙𝑀‖‖‖‖Δ𝑥𝑖𝑘,𝑙𝜌𝑖‖‖‖‖,𝑧𝑝𝑘,𝑙,𝐴𝑥≤1,∀𝑧∈𝑋𝑖=⎧⎪⎨⎪⎩𝜌−𝑥𝑖>0∶sup𝑘,𝑙⎡⎢⎢⎣𝑀⎛⎜⎜⎝‖‖‖‖‖Δ𝑥𝑖𝑘,𝑙−𝑥𝑘,𝑙𝜌𝑖‖‖‖‖‖⎞⎟⎟⎠⎤⎥⎥⎦,𝑧𝑝𝑘,𝑙⎫⎪⎬⎪⎭.≤1,∀𝑧∈𝑋(2.9) If 𝜌𝑖∈𝐴(𝑥𝑖) and 𝜌𝑖∈𝐴(𝑥𝑖−𝑥) then we observe that 𝑀⎛⎜⎜⎝‖‖‖‖‖Δ𝜎𝑖𝑥𝑖𝑘,𝑙−𝜎𝑥𝑘,𝑙𝜌𝑖||𝜎𝑖||−𝜎+𝜌𝑖‖‖‖‖‖⎞⎟⎟⎠⎛⎜⎜⎝‖‖‖‖‖Δ𝜎|𝜎|,𝑧≤𝑀𝑖𝑥𝑖𝑘,𝑙−𝜎𝑥𝑖𝑘,𝑙𝜌𝑖||𝜎𝑖||−𝜎+𝜌𝑖‖‖‖‖‖+‖‖‖‖‖Δ|𝜎|,𝑧𝜎𝑥𝑖𝑘,𝑙−𝜎𝑥𝑖𝑘,𝑙𝜌𝑖||𝜎𝑖||−𝜎+𝜌𝑖‖‖‖‖‖⎞⎟⎟⎠≤𝜌|𝜎|,𝑧𝑖||𝜎𝑖||−𝜎𝜌𝑖||𝜎𝑖||−𝜎+𝜌𝑖|𝑀‖‖‖‖𝜎|Δ𝑥𝑖𝑘,𝑙𝜌𝑖‖‖‖‖+𝜌,𝑧𝑖|𝜎|𝜌𝑖||𝜎𝑖||−𝜎+𝜌𝑖𝑀⎛⎜⎜⎝‖‖‖‖‖Δ𝑥|𝜎|𝑖𝑘,𝑙−𝑥𝑘,𝑙𝜌𝑖‖‖‖‖‖⎞⎟⎟⎠.,𝑧(2.10) From above inequality it obviously follows that ⎛⎜⎜⎝𝑀⎛⎜⎜⎝‖‖‖‖‖Δ𝜎𝑖𝑥𝑖𝑘,𝑙−𝜎𝑥𝑘,𝑙𝜌𝑖||𝜎𝑖||−𝜎+𝜌𝑖‖‖‖‖‖⎞⎟⎟⎠⎞⎟⎟⎠|𝜎|,𝑧𝑝𝑘,𝑙≤1(2.11) and consequently 𝑔𝑚,𝑛𝜎𝑖𝑥𝑖−𝜎𝑥≤inf𝑧∈𝑋1𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚‖‖𝜎𝑖𝑥𝑖𝑘,𝑙−𝜎𝑥𝑘,𝑙‖‖𝜌,𝑧+inf𝑖||𝜎𝑖||−𝜎+𝜌𝑖|𝜎|𝑝𝑚𝑛/𝐻∶𝜌𝑖𝑥∈𝐴𝑖,𝜌𝑖𝑥∈𝐴𝑖≤||𝜎−𝑥𝑖||−𝜎inf𝑧∈𝑋1𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚‖‖𝑥𝑖𝑘,𝑙‖‖,𝑧+|𝜎|inf𝑧∈𝑋1𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚‖‖𝑥𝑖𝑘,𝑙−𝑥𝑘,𝑙‖‖+||𝜎,𝑧𝑖||−𝜎𝑝𝑚𝑛/𝐻𝜌inf𝑖𝑝𝑚𝑛/𝐻∶𝜌𝑖𝑥∈𝐴𝑖+(|𝜎|)𝑝𝑚𝑛/𝐻𝜌inf𝑖𝑝𝑚𝑛/𝐻∶𝜌𝑖𝑥∈𝐴𝑖||𝜎−𝑥≤max𝑖||,||𝜎−𝜎𝑖||−𝜎𝑝𝑚𝑛/𝐻𝑔𝑚,𝑛𝑥𝑖+max|𝜎|,(|𝜎|)𝑝𝑚𝑛/𝐻𝑔𝑚,𝑛𝑥𝑖.−𝑥(2.12)
Note that 𝑔𝑚,𝑛(𝑥𝑖)<𝑔𝑚,𝑛(𝑥)+𝑔𝑚,𝑛(𝑥𝑖−𝑥) for all 𝑖∈ℕ. Hence by our assumption the right-hand side tends to 0 as 𝑖→∞ and the results follow. This completes the proof of the theorem.

Theorem 2.3. Let 𝑀 be an Orlicz function which satisfies the Δ2-condition. Then (𝑊𝐼0)2(𝜆,𝑀,Δ,𝑝,‖⋅,⋅‖)⊂(𝑊𝐼)2(𝜆,𝑀,Δ,𝑝,‖⋅,⋅‖)⊂(𝑊𝐼∞)2(𝜆,𝑀,Δ,𝑝,‖⋅,⋅‖) and the inclusions are strict.

Proof. Because it can be proven in a similar way, we introduce the proof for (𝑊𝐼)2(𝜆,𝑀,Δ,𝑝,‖⋅,⋅‖)⊂(𝑊𝐼∞)2(𝜆,𝑀,Δ,𝑝,‖⋅,⋅‖) only. Let 𝑥∈(𝑊𝐼)2(𝜆,𝑀,Δ,𝑝,‖⋅,⋅‖). Then 1𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚𝑀‖‖‖Δ𝑥𝑘,𝑙‖‖‖2𝜌,𝑧𝑝𝑘,𝑙=1𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚𝑀‖‖‖Δ𝑥𝑘,𝑙+𝐿−𝐿‖‖‖2𝜌,𝑧𝑝𝑘,𝑙≤1𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚𝑀‖‖‖Δ𝑥𝑘,𝑙−𝐿‖‖‖+‖‖‖𝐿2𝜌,𝑧‖‖‖2𝜌,𝑧𝑝𝑘,𝑙1≤𝐷𝐺𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚𝑀‖‖‖Δ𝑥𝑘,𝑙−𝐿𝜌‖‖‖,𝑧𝑝𝑘,𝑙1+𝐷𝐺𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚𝑀‖‖‖𝐿𝜌‖‖‖,,𝑧𝑝𝑘,𝑙(2.13) where 𝐺=max{1,(1/2)𝐻} and we get from Δ2-condition 1𝐷𝐺𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚𝑀‖‖‖Δ𝑥𝑘,𝑙−𝐿𝜌‖‖‖,𝑧𝑝𝑘,𝑙+𝐾‖𝜌𝛿𝐿,𝑧‖𝑀(2)𝐻,(2.14) so 𝑥∈(𝑊𝐼∞)2(𝜆,𝑀,Δ,𝑝,‖⋅,⋅‖) and this completes the proof.

Theorem 2.4. Let 𝑋(Δ) stand for (𝑊𝐼)2(𝜆,𝑀,Δ,𝑝,‖⋅,⋅‖),(𝑊𝐼0)2(𝜆,𝑀,Δ,𝑝,‖⋅,⋅‖) or (𝑊𝐼∞)2(𝜆,𝑀,Δ,𝑝,‖⋅,⋅‖). Then the inclusion 𝑋(Δ)⊂𝑋(Δ2) is strict.

Proof. Because this can be proven in a similar way, we will give the proof for (𝑊𝐼0)2(𝜆,𝑀,Δ,𝑝,‖⋅,⋅‖) only. Let 𝑥∈(𝑊𝐼0)2(𝜆,𝑀,Δ,𝑝,‖⋅,⋅‖). Then given 𝜀>0 we have ⎧⎪⎨⎪⎩1𝑚,𝑛∈ℕ∶𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚𝑀‖‖‖Δ𝑥𝑘,𝑙𝜌‖‖‖,𝑧𝑝𝑘,𝑙⎫⎪⎬⎪⎭≥𝜀forsome𝜌>0∈𝐼.(2.15) Since 𝑀 is nondecreasing and convex it follows that 1𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚𝑀‖‖‖‖Δ2𝑥𝑘,𝑙‖‖‖‖4𝜌,𝑧𝑝𝑘,𝑙=1𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚𝑀‖‖‖Δ𝑥𝑘,𝑙−Δ𝑥𝑘+1,𝑙−Δ𝑥𝑘,𝑙+1+Δ𝑥𝑘+1,𝑙+1‖‖‖4𝜌,𝑧𝑝𝑘,𝑙≤1𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚𝑀‖‖‖Δ𝑥𝑘,𝑙−Δ𝑥𝑘+1,𝑙‖‖‖+‖‖‖4𝜌,𝑧Δ𝑥𝑘+1,𝑙+1−Δ𝑥𝑘,𝑙+1‖‖‖4𝜌,𝑧𝑝𝑘,𝑙1≤𝐷𝐺𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚𝑀‖‖‖Δ𝑥𝑘,𝑙−Δ𝑥𝑘+1,𝑙‖‖‖2𝜌,𝑧𝑝𝑘,𝑙1+𝐷𝐺𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚𝑀‖‖‖Δ𝑥𝑘+1,𝑙+1−Δ𝑥𝑘,𝑙+1‖‖‖2𝜌,𝑧𝑝𝑘,𝑙,(2.16) where 𝐺=max{1,(1/2)𝐻}. Similarly from the convexity and nondecreasing properties of 𝑀1𝐷𝐺𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚𝑀‖‖‖Δ𝑥𝑘,𝑙−Δ𝑥𝑘+1,𝑙‖‖‖2𝜌,𝑧𝑝𝑘,𝑙+𝑀‖‖‖Δ𝑥𝑘+1,𝑙+1−Δ𝑥𝑘,𝑙+1‖‖‖2𝜌,𝑧𝑝𝑘,𝑙1≤𝐷𝐺𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚𝑀‖‖‖Δ𝑥𝑘,𝑙‖‖‖+‖‖‖2𝜌,𝑧Δ𝑥𝑘+1,𝑙‖‖‖2𝜌,𝑧𝑝𝑘,𝑙1+𝐷𝐺𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚𝑀‖‖‖Δ𝑥𝑘+1,𝑙+1‖‖‖+‖‖‖2𝜌,𝑧Δ𝑥𝑘,𝑙+1‖‖‖2𝜌,𝑧𝑝𝑘,𝑙≤𝐷2𝐺21𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚𝑀‖‖‖Δ𝑥𝑘,𝑙𝜌‖‖‖,𝑧𝑝𝑘,𝑙+𝑀‖‖‖Δ𝑥𝑘+1,𝑙𝜌‖‖‖,𝑧𝑝𝑘,𝑙+𝐷2𝐺21𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚𝑀‖‖‖Δ𝑥𝑘,𝑙+1𝜌‖‖‖,𝑧𝑝𝑘,𝑙+𝑀‖‖‖Δ𝑥𝑘+1,𝑙+1𝜌‖‖‖,𝑧𝑝𝑘,𝑙,⎧⎪⎨⎪⎩1𝑚,𝑛∈ℕ∶𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚𝑀‖‖‖‖Δ2𝑥𝑘,𝑙𝜌‖‖‖‖,𝑧𝑝𝑘,𝑙⎫⎪⎬⎪⎭⊆⎧⎪⎨⎪⎩≥𝜀𝑚,𝑛∈ℕ∶𝐷2𝐺21𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚𝑀‖‖‖Δ𝑥𝑘,𝑙𝜌‖‖‖,𝑧𝑝𝑘,𝑙≥𝜀4⎫⎪⎬⎪⎭∪⎧⎪⎨⎪⎩𝑚,𝑛∈ℕ∶𝐷2𝐺21𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚𝑀‖‖‖Δ𝑥𝑘+1,𝑙𝜌‖‖‖,𝑧𝑝𝑘,𝑙≥𝜀4⎫⎪⎬⎪⎭∪⎧⎪⎨⎪⎩𝑚,𝑛∈ℕ∶𝐷2𝐺21𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚𝑀‖‖‖Δ𝑥𝑘,𝑙+1𝜌‖‖‖,𝑧𝑝𝑘,𝑙≥𝜀4⎫⎪⎬⎪⎭∪⎧⎪⎨⎪⎩𝑚,𝑛∈ℕ∶𝐷2𝐺21𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚𝑀‖‖‖Δ𝑥𝑘+1,𝑙+1𝜌‖‖‖,𝑧𝑝𝑘,𝑙≥𝜀4⎫⎪⎬⎪⎭.(2.17)
Since the right-hand side belongs to 𝐼 so does the left hand side.

Theorem 2.5. Let 𝑀,𝑀1,𝑀2 be Orlicz function. Then one has(i)(𝑊𝐼0)2(𝜆,𝑀1,Δ,𝑝,‖⋅,⋅‖)⊆(𝑊𝐼0)2(𝜆,𝑀𝑜𝑀1,Δ,𝑝,‖⋅,⋅‖) provided (𝑝𝑘,𝑙) is such that 𝐻0=inf𝑝𝑘,𝑙>0,(ii)(𝑊𝐼0)2(𝜆,𝑀1,Δ,𝑝,‖⋅,⋅‖)∩(𝑊𝐼0)2(𝜆,𝑀2,Δ,𝑝,‖⋅,⋅‖)⊆(𝑊𝐼0)2(𝜆,𝑀1+𝑀2,Δ,𝑝,‖⋅,⋅‖).

Proof. (i) For given 𝜀>0, we can choose 𝜀0>0 such that max{𝜀𝐻0,𝜀𝐻00}<𝜀. Because of the continuity of 𝑀, we can choose 0<𝛿<1 such that 0<𝑡<𝛿⇒𝑀(𝑡)<𝜀0. Let 𝑥∈(𝑊𝐼0)2(𝜆,𝑀1,Δ,𝑝,‖⋅,⋅‖). We can write from the definition ⎧⎪⎨⎪⎩1𝐴(𝛿)=𝑚,𝑛∈ℕ∶𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚𝑀1‖‖‖Δ𝑥𝑘,𝑙𝜌‖‖‖,𝑧𝑝𝑘,𝑙≥𝛿𝐻⎫⎪⎬⎪⎭∈𝐼.(2.18)
Hence if 𝑚,𝑛∉𝐴(𝛿), then 1𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚𝑀1‖‖‖Δ𝑥𝑘,𝑙𝜌‖‖‖,𝑧𝑝𝑘,𝑙<𝛿𝐻,(2.19) that is, 𝑘,𝑙∈𝐼𝑛,𝑚𝑀1‖‖‖Δ𝑥𝑘,𝑙𝜌‖‖‖,𝑧𝑝𝑘,𝑙<𝜆𝑛,𝑚𝛿𝐻,(2.20) that is, 𝑀1‖‖‖Δ𝑥𝑘,𝑙𝜌‖‖‖,𝑧𝑝𝑘,𝑙<𝛿𝐻,∀𝑘,𝑙∈𝐼𝑛,𝑚,(2.21) that is, 𝑀1‖‖‖Δ𝑥𝑘,𝑙𝜌‖‖‖,𝑧<𝛿,∀𝑘,𝑙∈𝐼𝑛,𝑚.(2.22) From this inequality using continuity of 𝑀 we must have 𝑀𝑀1‖‖‖Δ𝑥𝑘,𝑙𝜌‖‖‖,𝑧<𝜀0,∀𝑘,𝑙∈𝐼𝑛,𝑚,(2.23) which consequently implies that 𝑘,𝑙∈𝐼𝑛,𝑚𝑀𝑀1‖‖‖Δ𝑥𝑘,𝑙𝜌‖‖‖,𝑧𝑝𝑘,𝑙<𝜆𝑛,𝑚𝜀max𝐻0,𝜀𝐻00<𝜆𝑛,𝑚𝜀,(2.24) that is, 1𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚𝑀𝑀1‖‖‖Δ𝑥𝑘,𝑙𝜌‖‖‖,𝑧𝑝𝑘,𝑙<𝜀.(2.25) This show that ⎧⎪⎨⎪⎩1𝑚,𝑛∈ℕ∶𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚𝑀𝑀1‖‖‖Δ𝑥𝑘,𝑙𝜌‖‖‖,𝑧𝑝𝑘,𝑙⎫⎪⎬⎪⎭≥𝜀⊂𝐴(𝛿),(2.26) and from the definition ideal the left side belongs to 𝐼. This proves the result.
 (ii)  Let (𝑥𝑘,𝑙)∈(𝑊𝐼0)2(𝜆,𝑀1,Δ,𝑝,‖⋅,⋅‖)∩(𝑊𝐼0)2(𝜆,𝑀2,Δ,𝑝,‖⋅,⋅‖). Then the fact 𝑀1+𝑀2‖‖‖Δ𝑥𝑘,𝑙𝜌‖‖‖,𝑧𝑝𝑘,𝑙𝑀≤𝐷1‖‖‖Δ𝑥𝑘,𝑙𝜌‖‖‖,𝑧𝑝𝑘,𝑙+𝑀2‖‖‖Δ𝑥𝑘,𝑙𝜌‖‖‖,𝑧𝑝𝑘,𝑙(2.27) give us the result. Thus (𝑥𝑘,𝑙)∈(𝑊𝐼0)2(𝜆,𝑀1+𝑀2,Δ,𝑝,‖⋅,⋅‖), hence the proof.

Definition 2.6. Let 𝑋 be sequence space. Then 𝑋 is called solid if (𝛼𝑘𝑥𝑘)∈𝑋 whenever (𝑥𝑘)∈𝑋 for all sequence (𝛼𝑘) of scalar with |𝛼𝑘|≤1 for all 𝑘∈ℕ.

Theorem 2.7. The sequence spaces (𝑊𝐼0)2(𝜆,𝑀,Δ,𝑝,‖⋅,⋅‖),(𝑊𝐼∞)2(𝜆,𝑀,Δ,𝑝,‖⋅,⋅‖) are solid.

Proof. We give the proof for (𝑊𝐼0)2(𝜆,𝑀,Δ,𝑝,‖⋅,⋅‖) only. Let (𝑥𝑘,𝑙)∈(𝑊𝐼0)2(𝜆,𝑀,Δ,𝑝,‖⋅,⋅‖) and (𝛼𝑘,𝑙) be a sequence of scalars such that |𝛼𝑘,𝑙|≤1 for all 𝑘,𝑙∈ℕ. Then we have ⎧⎪⎨⎪⎩1𝑚,𝑛∈ℕ∶𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚𝑀‖‖‖‖Δ𝛼𝑘,𝑙𝑥𝑘,𝑙𝜌‖‖‖‖,𝑧𝑝𝑘,𝑙⎫⎪⎬⎪⎭⊆⎧⎪⎨⎪⎩𝐶≥𝜀𝑚,𝑛∈ℕ∶𝜆𝑛,𝑚𝑘,𝑙∈𝐼𝑛,𝑚𝑀‖‖‖Δ𝑥𝑘,𝑙𝜌‖‖‖,z𝑝𝑘,𝑙⎫⎪⎬⎪⎭≥𝜀∈𝐼,(2.28) where 𝐶=max{1,(|𝛼𝑘,𝑙|)𝐻}. Hence (𝛼𝑘,𝑙𝑥𝑘,𝑙)∈(𝑊𝐼0)2(𝜆,𝑀,Δ,𝑝,‖⋅,⋅‖) for all sequences of scalars (𝛼𝑘,𝑙) with |𝛼𝑘,𝑙|≤1 for all 𝑘,𝑙∈ℕ whenever (𝑥𝑘,𝑙)∈(𝑊𝐼0)2(𝜆,𝑀,Δ,𝑝,‖⋅,⋅‖).