Generalized Difference λ-Sequence Spaces Defined by Ideal Convergence and the Musielak-Orlicz Function

Abstract

We introduced the ideal convergence of generalized difference sequence spaces combining an infinite matrix of complex numbers with respect to 𝜆-sequences and the Musielak-Orlicz function over 𝑛-normed spaces. We also studied some topological properties and inclusion relations between these spaces.

1. Introduction

Throughout the paper𝜔,ℓ_∞,𝑐,𝑐₀, andℓ_𝑝denote the classes of all, bounded, convergent, null, and p-absolutely summable sequences of complex numbers. The sets of natural numbers and real numbers will be denoted by ℕ and ℝ, respectively. Many authors studied various sequence spaces using normed or seminormed linear spaces. In this paper, using an infinite matrix of complex numbers and the notion of ideal, we aimed to introduce some new sequence spaces with respect to generalized difference operatorΔ^𝑠_𝑚on𝜆-sequences and the Musielak-Orlicz function in𝑛-normed linear spaces. By an ideal we mean a family𝐼⊂2^𝑌of subsets of a nonempty set𝑌satisfying the following: (i)𝜙∈𝐼; (ii)𝐴,𝐵∈𝐼imply𝐴∪𝐵∈𝐼; (iii)𝐴∈𝐼, 𝐵⊂𝐴imply𝐵∈𝐼, while an admissible ideal𝐼of𝑌further satisfies{𝑥}∈𝐼for each𝑥∈𝑌. The notion of ideal convergence was introduced first by Kostyrko et al. [1] as a generalization of statistical convergence. The concept of 2-normed spaces was initially introduced by Gähler [2] in the 1960s, while that of𝑛-normed spaces can be found in [3]; this concept has been studied by many authors; see for instance [4–7]. The notion of ideal convergence in a 2-normed space was initially introduced by Gürdal [8]. Later on, it was extended to𝑛-normed spaces by Gürdal and Şahiner [9]. Given that𝐼⊂2^ℕis a nontrivial ideal in ℕ, the sequence(𝑥_𝑛)_𝑛∈ℕin a normed space(𝑋;‖⋅‖)is said to be𝐼-convergent to𝑥∈𝑋, if, for each𝜀>0, 𝐴(𝜀)=𝑛∈ℕ∶𝑥_𝑛−𝑥≥𝜀∈𝐼. (1)

A sequence(𝑥_𝑘)in a normed space(𝑋,‖⋅‖)is said to be𝐼-bounded if there exists𝐿>0such that 𝑘∈ℕ∶𝑥_𝑘>𝐿∈𝐼. (2)

A sequence(𝑥_𝑘)in a normed space(𝑋,‖⋅‖)is said to be𝐼-Cauchy if, for each𝜀>0, there exists a positive integer𝑚=𝑚(𝜀)such that 𝑘∈ℕ∶𝑥_𝑘−𝑥_𝑚≥𝜀∈𝐼. (3)

In paper [10], the notion of𝜆-convergent and bounded sequences is introduced as follows: let𝜆=(𝜆_𝑗)^∞_𝑗=1be a strictly increasing sequence of positive real numbers tending to infinity; that is, 0<𝜆₁<𝜆₂<⋯,𝜆_𝑗⟶∞as𝑗⟶∞. (4)

We say that a sequence 𝑥=(𝑥_𝑗)∈𝜔 is 𝜆-convergent to the number𝑙∈ℂ, called the𝜆-limit of𝑥, ifΛ_𝑗(𝑥)→𝑙as𝑗→∞, where Λ_𝑗(𝑥)=1𝜆_𝑗^𝑗_𝑟=1𝜆_𝑟−𝜆_𝑟−1𝑥_𝑟,𝑗∈ℕ. (5)

The class of all sequences(𝜆_𝑗)satisfying this property is denoted byΛ.

In particular, we say that 𝑥 is a 𝜆-null sequence ifΛ_𝑗(𝑥)→0as𝑗→∞. Further, we say that𝑥is 𝜆-bounded ifsup_𝑗|Λ_𝑗(𝑥)|<∞. Here and in the sequel, we will use the convention that any term with a zero subscript is equal to naught; for example,𝜆₀=0and𝑥₀=0. Now, it is well known [10] that iflim_𝑗𝑥_𝑗=𝑎in the ordinary sense of convergence, then lim_𝑗→∞1𝜆_𝑗^𝑗_𝑟=1𝜆_𝑟−𝜆_𝑟−1𝑥_𝑟−𝑎=0. (6)

This implies that lim_𝑗Λ_𝑗(𝑥)−𝑎=lim_𝑗→∞1𝜆_𝑗^𝑗_𝑟=1𝜆_𝑟−𝜆_𝑟−1𝑥_𝑟−𝑎=0, (7) which yields that lim_𝑗Λ_𝑗(𝑥)=𝑎 and hence 𝑥 is 𝜆-convergent to𝑎. We therefore deduce that the ordinary convergence implies the𝜆-convergence to the same limit.

An Orlicz function is a function𝑀∶[0,∞)→[0,∞)which is continuous, nondecreasing, and convex with𝑀(0)=0 and𝑀(𝑥)>0for𝑥>0and𝑀(𝑥)→∞, as𝑥→∞. If convexity of𝑀is replaced by𝑀(𝑥+𝑦)≤𝑀(𝑥)+𝑀(𝑦), then it is called a modulus function, introduced by Nakano [11]. Ruckle [12] and Maddox [13] used the idea of a modulus function to construct some spaces of complex sequences. An Orlicz function𝑀is said to satisfy the Δ₂-condition for all values of𝑥≥0, if there exists a constant𝑘>0, such that𝑀(2𝑥)≤𝑘𝑀(𝑥). TheΔ₂-condition is equivalent to𝑀(𝑙𝑥)≤𝑘𝑙𝑀(𝑥)for all values of𝑥and for𝑙>1. Lindentrauss and Tzafriri [14] used the idea of an Orlicz function to define the following sequence spaces: ℓ_𝑀=𝑥∈𝜔∶^∞_𝑘=1𝑀|𝑥(𝑘)|𝜌<∞, (8) which is a Banach space with the Luxemburg norm defined by ‖𝑥‖=inf𝜌>0∶^∞_𝑘=1𝑀|𝑥(𝑘)|𝜌≤1. (9)

The spaceℓ_𝑀is closely related to the spaceℓ_𝑝, which is an Orlicz sequence space with𝑀(𝑥)=𝑥^𝑝for1≤𝑝<∞. Recently different classes of sequences have been introduced using Orlicz functions. See [7, 9, 15–17].

A sequence𝑀=(𝑀_𝑘)of Orlicz functions𝑀_𝑘for all𝑘∈ℕ is called a Musielak-Orlicz function.

Kizmaz [18] defined the difference sequences ℓ_∞(Δ), 𝑐(Δ), and 𝑐₀(Δ) as follows.

𝑍(Δ)={𝑥=(𝑥_𝑘)∶(Δ𝑥_𝑘)∈𝑍}. For𝑍=ℓ_∞, 𝑐, and 𝑐₀, whereΔ𝑥=(𝑥_𝑘−𝑥_𝑘+1), for all𝑘∈ℕ. The above spaces are Banach spaces, normed by‖𝑥‖=|𝑥₁|+sup_𝑘|Δ𝑥_𝑘|. The notion of difference sequence spaces was generalized by Et and Colak [19] as follows:𝑍(Δ^𝑠)={𝑥=(𝑥_𝑘)∶(Δ^𝑠𝑥_𝑘)∈𝑍}. For𝑍=ℓ_∞, 𝑐and𝑐₀, where𝑠∈ℕ, (Δ^𝑠𝑥_𝑘)=(Δ^𝑠−1𝑥_𝑘−Δ^𝑠−1𝑥_𝑘+1)and so thatΔ^𝑠𝑥_𝑘=∑^𝑠_𝑛=0(−1)^𝑛𝐶^𝑠_𝑛𝑥_𝑘+𝑛. Tripathy and Esi [20] introduced the following new type of difference sequence spaces.

𝑍(Δ_𝑚)={𝑥=(𝑥_𝑘)∶(Δ_𝑚𝑥_𝑘)∈𝑍}, 𝑍=ℓ_∞,𝑐, and𝑐₀, whereΔ_𝑚𝑥_𝑘=(𝑥_𝑘−𝑥_𝑘+𝑚), for all𝑘∈ℕ. Tripathy et al. [21], generalized the previous notions and unified them as follows.

Let 𝑚 and 𝑠 be nonnegative integers, then for𝑍a given sequence space we have 𝑍Δ^𝑠_𝑚=𝑥=𝑥_𝑘∶Δ^𝑠_𝑚𝑥_𝑘Δ∈𝑍,where_𝑠𝑚𝑥_𝑘=^𝑠_𝑛=0(−1)^𝑛𝐶^𝑠_𝑛𝑥_{𝑘+𝑚𝑛}(),𝑍Δforwarddifference_𝑚(𝑠)=𝑥=𝑥_𝑘∶Δ_𝑚^(𝑠)𝑥_𝑘Δ∈𝑍,where_𝑚(𝑠)𝑥_𝑘=^𝑠_𝑛=0(−1)^𝑛𝐶^𝑠_𝑛𝑥_{𝑘−𝑚𝑛}(backwarddifference), (10) where𝑥_𝑘=0, for𝑘<0.

2. Definitions and Preliminaries

Let𝑛∈ℕ and𝑋be a linear space over the field𝐾of dimension𝑑, where𝑑≥𝑛≥2and𝐾is the field of real or complex numbers. A real valued function ‖⋅…⋅‖ on𝑋^𝑛satisfies the following four conditions:(1)‖𝑥₁,𝑥₂,…,𝑥_𝑛‖=0if and only if𝑥₁,𝑥₂,…,and𝑥_𝑛are linearly dependent in𝑋;(2)‖𝑥₁,𝑥₂,…,𝑥_𝑛‖is invariant under permutation;(3)‖𝛼𝑥₁,𝑥₂,…,𝑥_𝑛‖=|𝛼|‖𝑥₁,𝑥₂,…,𝑥_𝑛‖for any𝛼∈𝐾;(4)‖𝑥+𝑥^′,𝑥₂,…,𝑥_𝑛‖≤‖𝑥,𝑥₂,…,𝑥_𝑛‖+‖𝑥₁,𝑥₂,…,𝑥_𝑛‖, which is called an𝑛-norm on𝑋and the pair(𝑋;‖⋅…⋅‖)is called an𝑛-normed space over the field𝐾. For example, we may take𝑋=ℝ^𝑛being equipped with the𝑛-norm‖𝑥₁,𝑥₂,…,𝑥_𝑛‖_𝐸=the volume of the𝑛-dimensional parallelepiped spanned by the vectors𝑥₁,𝑥₂,…,and𝑥_𝑛which may be given explicitly by the formula 𝑥₁,𝑥₂,…,𝑥_𝑛𝐸=det𝑥_𝑖𝑗𝑥=abs¹¹𝑥₁₂⋯𝑥_1𝑛𝑥₂₁𝑥₂₂⋯𝑥_2𝑛𝑥⋮⋮⋱⋮_𝑛1𝑥_𝑛2⋯𝑥_𝑛𝑛, (11)where 𝑥_𝑖=(𝑥_𝑖1,𝑥_𝑖2,…,𝑥_𝑖𝑛)for each𝑖∈ℕ.

Let(𝑋,‖⋅…⋅‖)be an𝑛-normed space of dimension𝑑≥𝑛≥2and{𝑎₁,𝑎₂,𝑎₃,…,𝑎_𝑛}a linearly independent set in𝑋. Then, the function‖⋅…⋅‖_∞on𝑋^𝑛−1defined by 𝑥₁,𝑥₂,…,𝑥_𝑛∞=max_{1≤𝑖≤𝑛}𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑎_𝑖 (12) defines an (𝑛−1)-norm on 𝑋 with respect to 𝑎₁,𝑎₂,𝑎₃,…,and𝑎_𝑛 and this is known as the derived (𝑛−1)-norm. The standard (𝑛)-norm on 𝑋, a real inner product space of dimension 𝑑≥𝑛, is as follows: 𝑥₁,𝑥₂,…,𝑥_𝑛𝑠𝑥=abs¹,𝑥₁𝑥₁,𝑥₂⋯𝑥₁,𝑥_𝑛𝑥₂,𝑥₁𝑥₂,𝑥₂⋯𝑥₂,𝑥_𝑛𝑥⋮⋮⋱⋮_𝑛,𝑥₁𝑥_𝑛,𝑥₂⋯𝑥_𝑛,𝑥_𝑛^1/2, (13) where ⟨⋅,⋅⟩ denotes the inner product on𝑋. If we take𝑋=ℝ^𝑛, then 𝑥₁,𝑥₂,…,𝑥_𝑛𝐸=𝑥₁,𝑥₂,…,𝑥_𝑛𝑠. (14)

For𝑛=1, this𝑛-norm is the usual norm‖𝑥₁‖=⟨𝑥₁,𝑥₁⟩.

Definition 1. A sequence(𝑥_𝑘)in an𝑛-normed space is said to be convergent to𝑥∈𝑋if lim_𝑘→∞𝑧₁,𝑧₂,…,𝑧_𝑛−1,𝑥_𝑘−𝑥_𝑛=0,∀𝑧₁,𝑧₂,…,𝑧_𝑛−1∈𝑋. (15)

Definition 2. A sequence(𝑥_𝑘)in an𝑛-normed space is called Cauchy (with respect to𝑛-norm) if lim_{𝑘,𝑗→∞}𝑧₁,𝑧₂,…,𝑧_𝑛−1,𝑥_𝑘−𝑥_𝑗𝑛=0,∀𝑧₁,𝑧₂,…,𝑧_𝑛−1∈𝑋. (16)

If every Cauchy sequence in𝑋converges to an𝑥∈𝑋, then𝑋is said to be complete (with respect to the𝑛-norm). A complete𝑛-normed space is called an 𝑛-Banach space.

Definition 3. A sequence(𝑥_𝑘)in an𝑛-normed space (𝑋,‖⋅…⋅‖) is said to be𝐼-convergent to𝑥₀∈𝑋with respect to𝑛-norm, if, for each 𝜀>0, the set 𝑘∈ℕ∶𝑥_𝑘−𝑥₀,𝑧₁,𝑧₂,…,𝑧_𝑛−1≥𝜀,forevery𝑧₁,𝑧₂,…,𝑧_𝑛−1∈𝐼. (17)

Definition 4. A sequence(𝑥_𝑘)in an𝑛-normed space(𝑋,‖⋅…⋅‖)is said to be𝐼-Cauchy if, for each 𝜀>0, there exists a positive integer𝑚=𝑚(𝜀)such that the set 𝑘∈ℕ∶𝑥_𝑘−𝑥_𝑚,𝑧₁,𝑧₂,…,𝑧_𝑛−1≥𝜀,forevery𝑧₁,𝑧₂,…,𝑧_𝑛−1∈𝐼. (18)
Let𝑥=(𝑥_𝑘)be a sequence; then𝑆(𝑥)denotes the set of all permutations of the elements of(𝑥_𝑘); that is,𝑆(𝑥)=(𝑥_𝜋(𝑛))∶𝜋is a permutation of ℕ.

Definition 5. A sequence space𝐸is said to be symmetric if𝑆(𝑥)⊂𝐸for all𝑥∈𝐸.

Definition 6. A sequence space𝐸is said to be normal (or solid) if(𝛼_𝑘𝑥_𝑘)∈𝐸, whenever(𝑥_𝑘)∈𝐸and for all sequences (𝛼_𝑘)of scalars with|𝛼_𝑘|≤1for all𝑘∈ℕ.

Definition 7. A sequence space𝐸is said to be a sequence algebra if𝑥,𝑦∈𝐸; then𝑥⋅𝑦=(𝑥_𝑘𝑦_𝑘)∈𝐸.

Lemma 8. Every𝑛-normed space is an(𝑛−𝑟)-normed space for all𝑟=1,2,3,…,𝑛−1. In particular, every𝑛-normed space is a normed space.

Lemma 9. On a standard𝑛-normed space𝑋, the derived(𝑛−1)-norm‖⋅…⋅‖_∞defined with respect to the orthogonal set{𝑒₁,𝑒₂,…,𝑒_𝑛}is equivalent to the standard(𝑛−1)-norm‖⋅…⋅‖_𝑠. To be precise, one has 𝑥₁,𝑥₂,…,𝑥_𝑛−1∞≤‖⋅…⋅‖_𝑠≤√𝑛𝑥₁,𝑥₂,…,𝑥_𝑛−1∞, (19) for all𝑥₁,𝑥₂,…,𝑥_𝑛−1∈𝑋, where‖𝑥₁,𝑥₂,…,𝑥_𝑛−1‖_∞=max_{1≤𝑖≤𝑛}{‖𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑒_𝑖‖_𝑆}.
For any bounded sequence(𝑝_𝑛)of positive numbers, one has the following well known inequality: if0≤𝑝_𝑘≤sup_𝑘𝑝_𝑘=𝐺and𝐷=max(1,2^𝐺−1), then|𝑎_𝑛+𝑏_𝑛|^𝑝𝑛≤𝐷(|𝑎_𝑛|^𝑝𝑛+|𝑏_𝑛|^𝑝𝑛), for all𝑘and𝑎_𝑘,𝑏_𝑘∈ℂ.

3. Main Results

In this section, we define some new ideal convergent sequence spaces and investigate their linear topological structures. We find out some relations related to these sequence spaces. Let 𝐼 be an admissible ideal of ℕ, ℳ=(𝑀_𝑗) a Musielak-Orlicz function, and Δ^𝑠_𝑚 the forward generalized difference operator on the class of all sequences(𝜆_𝑗)satisfying the propertyΛand an𝑛-normed space(𝑋,‖⋅…⋅‖). Further, let𝑝=(𝑝_𝑘)be any bounded sequence of positive real numbers; we will define the following sequence spaces: 𝑊𝐴,ℳ,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖^𝐼=()×𝑥∈𝜔𝑛−𝑋∶∀𝜀>0𝑘∈ℕ∶^∞_𝑗=1𝑎_𝑘𝑗×𝑀_𝑗Δ^𝑠_𝑚Λ_𝑗(𝑥)−𝑙𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗≥𝜀∈𝐼,forsome𝜌>0,𝑙∈𝑋andeach𝑧₁,𝑧₂,…,𝑧_𝑛−1,𝑊∈𝑋𝐴,ℳ,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖^𝐼₀=()×𝑥∈𝜔𝑛−𝑋∶∀𝜀>0𝑘∈ℕ∶^∞_𝑗=1𝑎_𝑘𝑗×𝑀_𝑗Δ^𝑠_𝑚Λ_𝑗(𝑥)𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗≥𝜀∈𝐼,forsome𝜌>0,andeach𝑧₁,𝑧₂,…,𝑧_𝑛−1,𝑊∈𝑋𝐴,ℳ,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖_∞=𝑥∈𝜔(𝑛−𝑋)∶∃𝐾>0st.sup_𝑘^∞_𝑗=1𝑎_𝑘𝑗×𝑀_𝑗Δ^𝑠_𝑚Λ_𝑗(𝑥)𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗<∞,forsome𝜌>0andeach𝑧₁,𝑧₂,…,𝑧_𝑛−1,𝑊∈𝑋𝐴,ℳ,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖^𝐼_∞=𝑥∈𝜔(𝑛−𝑋)∶∃𝐾>0,s.t.𝑘∈ℕ∶^∞_𝑗=1𝑎_𝑘𝑗×𝑀_𝑗Δ^𝑠_𝑚Λ_𝑗(𝑥)𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗≥𝐾∈𝐼,forsome𝜌>0,andeach𝑧₁,𝑧₂,…,𝑧_𝑛−1∈𝑋. (20)

Let us consider a few special cases of the aforementioned sets.

(1) If𝑀_𝑘(𝑥)=𝑀(𝑥), for all𝑘∈ℕ then the previous classes of sequences are denoted by𝑊[𝐴,𝑀,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]^𝐼, 𝑊[𝐴,𝑀,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]^𝐼₀, 𝑊[𝐴,𝑀,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]_∞, and 𝑊[𝐴,𝑀,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]^𝐼_∞, respectively.

(2) If 𝑝_𝑘=1for all𝑘∈ℕ then the previous classes of sequences are denoted by 𝑊[𝐴,ℳ,Δ^𝑠_𝑚,Λ,‖⋅…⋅‖]^I, 𝑊[𝐴,ℳ,Δ^𝑠_𝑚,Λ,‖⋅…⋅‖]^𝐼₀, 𝑊[𝐴,ℳ,Δ^𝑠_𝑚,Λ,‖⋅…⋅‖]_∞, and 𝑊[𝐴,ℳ,Δ^𝑠_𝑚,Λ,‖⋅…⋅‖]^𝐼_∞, respectively.

(3) If𝑀_𝑘(𝑥)=𝑥, for all𝑘∈ℕ and𝑥∈[0,∞[, then the previous classes of sequences are denoted by 𝑊[𝐴,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]^𝐼, 𝑊[𝐴,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]^𝐼₀, 𝑊[𝐴,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]_∞, and 𝑊[𝐴,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]^𝐼_∞, respectively.

(4) If we take𝑀_𝑘(𝑥)=𝑀(𝑥), for all𝑘∈ℕand𝐴=(𝑎_𝑘𝑗)as 𝑎_𝑘𝑗=1𝑘,𝑘≥𝑗,0,otherwise, (21) then we denote the previous classes of sequences by𝑊[𝐶,𝑀,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]^𝐼, 𝑊[𝐶,𝑀,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]^𝐼₀, 𝑊[𝐶,𝑀,Δ^𝑠_𝑚,𝑝,‖⋅…⋅‖]_∞, and𝑊[𝐶,𝑀,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]^𝐼_∞, respectively.

(5) If we take𝑀_𝑘(𝑥)=𝑀(𝑥)and𝐴=(𝑎_𝑘𝑗)as 𝑎_𝑘𝑗=1𝜙_𝑘,𝑗∈𝐼_𝑘=𝑘−𝜙_𝑘+1,𝑘,0,otherwise, (22) where(𝜙_𝑘)is a nondecreasing sequence of positive numbers tending to ∞,𝜙₁=1, and 𝜙_𝑘+1≤𝜙_𝑘+1, then we denote the previous classes of sequences by𝑊[Φ,𝑀,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]^𝐼,𝑊[Φ,𝑀,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]^𝐼₀, 𝑊[Φ,𝑀,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]_∞, and 𝑊[Φ,𝑀,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]^𝐼_∞.

(6) If𝐴=(𝑎_𝑘𝑗)as in (22), then we denote the previous classes of sequences by𝑊[Φ,ℳ,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]^𝐼, 𝑊[Φ,ℳ,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]^𝐼₀, 𝑊[Φ,ℳ,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]_∞, and𝑊[Φ,ℳ,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]^𝐼_∞.

And if 𝜆_𝑗=𝑗for all𝑗∈ℕ, then the previous classes of sequences are denoted by𝑊[Φ,ℳ,Δ^𝑠_𝑚,𝐶,𝑝,‖⋅…⋅‖]^𝐼, 𝑊[Φ,ℳ,Δ^𝑠_𝑚,𝐶,𝑝,‖⋅…⋅‖]^𝐼₀, 𝑊[Φ,ℳ,Δ^𝑠_𝑚,𝐶,𝑝,‖⋅…⋅‖]_∞, and𝑊[Φ,ℳ,Δ^𝑠_𝑚,𝐶,𝑝,‖⋅…⋅‖]^𝐼_∞and they are a generalization of the sequence spaces defined by Bakery et al. [22].

(7) By a lacunary𝜃=(𝑗_𝑟), 𝑟=0,1,2,…, where𝑗₀=0, we will mean an increasing sequence of nonnegative integers with𝑗_𝑟−𝑗_𝑟−1→∞as𝑟→∞. The interval determined by𝜃will be denoted by𝐼_𝑟=]𝑗_𝑟−1,𝑗_𝑟]andℎ_𝑟=𝑗_𝑟−𝑗_𝑟−1and let𝐴=(𝑎_𝑘𝑗)as 𝑎_𝑘𝑗=1ℎ_𝑟,𝑗∈𝐼_𝑟=𝑗_𝑟−1,𝑗_𝑟,0,otherwise. (23)

Then we denote the previous classes of sequences by𝑊[𝜃,𝑀,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]^𝐼, 𝑊[𝜃,𝑀,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]^𝐼₀, 𝑊[𝜃,𝑀,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]_∞, and𝑊[𝜃,𝑀,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]^𝐼_∞, respectively.

(8) If𝑀_𝑘(𝑥)=𝑀(𝑥), for all𝑘∈ℕ, 𝐴=𝐼, and𝜆_𝑗=𝑗, then the previous classes of sequences are denoted by𝑊[𝑀,Δ^𝑠_𝑚,𝐶,𝑝,‖⋅…⋅‖]^𝐼, 𝑊[𝑀,Δ^𝑠_𝑚,𝐶,𝑝,‖⋅…⋅‖]^𝐼₀, 𝑊[𝑀,Δ^𝑠_𝑚,𝐶,𝑝,‖⋅…⋅‖]_∞, and𝑊[𝑀,Δ^𝑠_𝑚,𝐶,𝑝,‖⋅…⋅‖]^𝐼_∞.

(9) If𝑠=1, then the previous classes of sequences are denoted by𝑊[𝐴,ℳ,Δ_𝑚,Λ,𝑝,‖⋅…⋅‖]^𝐼, 𝑊[𝑀,Δ_𝑚,𝐶,𝑝,‖⋅…⋅‖]^𝐼₀, 𝑊[𝑀,Δ_𝑚,𝐶,𝑝,‖⋅…⋅‖]_∞, and 𝑊[𝑀,Δ_𝑚,𝐶,𝑝,‖⋅…⋅‖]^𝐼_∞.

(10) If𝑚=1, then the previous classes of sequences are denoted by[𝐴,ℳ,Δ^𝑠,Λ,𝑝,‖⋅…⋅‖]^𝐼, 𝑊[𝑀,Δ^𝑠,𝐶,𝑝,‖⋅…⋅‖]^𝐼₀, 𝑊[𝑀,Δ^𝑠,𝐶,𝑝,‖⋅…⋅‖]_∞, and𝑊[𝑀,Δ^𝑠,𝐶,𝑝,‖⋅…⋅‖]^𝐼_∞.

Theorem 10. The spaces 𝑊[𝐴,ℳ,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]^𝐼, 𝑊[𝐴,ℳ,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]^𝐼₀ and 𝑊[𝐴,ℳ,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]^𝐼_∞ are linear spaces.

Proof. We will prove the assertion for𝑊[𝐴,ℳ,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]^𝐼₀; the others can be proved similarly. Assume that𝑥=(𝑥_𝑘), 𝑦=(𝑦_𝑘)∈𝑊[𝐴,ℳ,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]^𝐼₀, and𝛼,𝛽∈ℂ. Then, there exist𝜌₁and𝜌₂such that the sets 𝑘∈ℕ∶^∞_𝑗=1𝑎_𝑘𝑗×𝑀_𝑗Δ^𝑠_𝑚Λ_𝑗(𝑥)𝜌₁,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗≥𝜀2∈𝐼,𝑘∈ℕ∶_∞𝑗=1𝑎_𝑘𝑗𝑀_𝑗Δ^𝑠_𝑚(Λ_𝑗(𝑦)𝜌₂,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗≥𝜀2∈𝐼. (24)(25)
Since(𝑋,‖⋅…⋅‖)is an𝑛-norm,Δ^𝑠_𝑚andΛ_𝑗are linear, and the Orlicz function𝑀_𝑗is convex for all𝑗∈ℕ, the following inequality holds: ^∞_𝑗=1𝑎_𝑘𝑗𝑀_𝑗Δ^𝑠_𝑚Λ_𝑗𝛼𝑥+𝛽𝑦|𝛼|𝜌_1𝐹+𝛽𝜌₂,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗≤𝐷^∞_𝑗=1𝑎_𝑘𝑗|𝛼|𝜌₁|𝛼|𝜌₁+𝛽𝜌₂×𝑀_𝑗Δ^𝑠_𝑚Λ_𝑗(𝑥)𝜌₁,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗+𝐷^∞_𝑗=1𝑎_𝑘𝑗𝛽𝜌₂|𝛼|𝜌₁+𝛽𝜌₂×𝑀_𝑗Δ^𝑠_𝑚Λ_𝑗𝑦𝜌₂,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗≤𝐷𝐿^∞_𝑗=1𝑎_𝑘𝑗×𝑀_𝑗Δ^𝑠_𝑚Λ_𝑗(𝑥)𝜌₁,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗+𝐷𝐿^∞_𝑗=1𝑎_𝑘𝑗×𝑀_𝑗Δ^𝑠_𝑚Λ_𝑗𝑦𝜌₂,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗, (26) where𝐿=max{|𝛼|𝜌₁/(|𝛼|𝜌₁+|𝛽|𝜌₂),|𝛽|𝜌₂/(|𝛼|𝜌₁+|𝛽|𝜌₂)}. On the other hand from the above inequality we get 𝑘∈ℕ∶^∞_𝑗=1𝑎_𝑘𝑗𝑀_𝑗Δ^𝑠_𝑚Λ_𝑗𝛼𝑥+𝛽𝑦|𝛼|𝜌₁+𝛽𝜌₂,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗⊆×≥𝜀𝑘∈ℕ∶𝐷𝐿_∞𝑗=1𝑎_𝑘𝑗𝑀_𝑗Δ^𝑠_𝑚Λ_𝑗(𝑥)𝜌₁,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗≥𝜀2∪×𝑘∈ℕ∶𝐷𝐿_∞𝑗=1𝑎_𝑘𝑗𝑀_𝑗Δ^𝑠_𝑚Λ_𝑗𝑦𝜌₂,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗≥𝜀2. (27)
Since the two sets on the right hand side belong to𝐼, this completes the proof.

Theorem 11. The spaces 𝑊[𝐴,ℳ,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]^𝐼, 𝑊[𝐴,ℳ,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]^𝐼₀, and 𝑊[𝐴,ℳ,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]^𝐼_∞ are paranormed spaces (not totally paranormed) with respect to the paranorm 𝑔_Δ defined by 𝑔_Δ(𝑥)=^𝑚𝑠_𝑗=1𝑥_𝑗,𝑧₁,𝑧₂,…,𝑧_𝑛−1+inf𝜌^{𝑝𝑘/𝐻}∶sup_𝑘^∞_𝑗=1𝑎_𝑘𝑗𝑀_𝑗Δ^𝑠_𝑚Λ_𝑗(𝑥)𝜌₁,𝑧₁,𝑧₂,…,𝑧_𝑛−1^{𝑝𝑗1/𝐻}≤1,𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝜌>0,𝑎𝑛𝑑𝑒𝑎𝑐ℎ𝑧₁,𝑧₂,…,𝑧_𝑛−1∈𝑋, (28) where𝐻=max{1,sup_𝑘𝑝_𝑘}.

Proof. Clearly𝑔_Δ(−𝑥)=𝑔_Δ(𝑥)and𝑔_Δ(𝜃)=0. Let𝑥=(𝑥_𝑘)and𝑦=(𝑦_𝑘)∈𝑊[𝐴,ℳ,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]^𝐼₀. Then, for𝜌>0we set 𝐴₁=𝜌∶sup_𝑘^∞_𝑗=1𝑎_𝑘𝑗×𝑀_𝑗Δ^𝑠_𝑚Λ_𝑗(𝑥)𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1^{𝑝𝑗1/𝐻}≤1,foreach𝑧₁,𝑧₂,…,𝑧_𝑛−1,𝐴∈𝑋₂=𝜌∶sup_𝑘^∞_𝑗=1𝑎_𝑘𝑗×𝑀_𝑗Δ^𝑠_𝑚Λ_𝑗𝑦𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1^{𝑝𝑗1/𝐻}≤1,foreach𝑧₁,𝑧₂,…,𝑧_𝑛−1∈𝑋. (29)
Let𝜌₁∈𝐴₁, 𝜌₂∈𝐴₂, and𝜌=𝜌₁+𝜌₂; then we have sup_𝑘^∞_𝑗=1𝑎_𝑘𝑗×𝑀_𝑗Δ^𝑠_𝑚Λ_𝑗𝑥+𝑦𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1^{𝑝𝑗1/𝐻}≤𝜌₁𝜌₁+𝜌₂×sup_𝑘^∞_𝑗=1𝑎_𝑘𝑗𝑀_𝑗Δ^𝑠_𝑚Λ_𝑗(𝑥)𝜌₁,𝑧₁,𝑧₂,…,𝑧_𝑛−1^{𝑝𝑗1/𝐻}+𝜌₂𝜌₁+𝜌₂×sup_𝑘^∞_𝑗=1𝑎_𝑘𝑗𝑀_𝑗Δ^𝑠_𝑚Λ_𝑗𝑦𝜌₂,𝑧₁,𝑧₂,…,𝑧_𝑛−1^{𝑝𝑗1/𝐻}𝑔≤1,_Δ=𝑥+𝑦_{𝑚𝑠𝑗=1}𝑥_𝑗+𝑦_𝑗,𝑧₁,𝑧₂,…,𝑧_𝑛−1+inf𝜌₁+𝜌₂^{𝑝𝑘/𝐻}∶𝜌₁∈𝐴₁,𝜌₂∈𝐴₂≤^𝑚𝑠_𝑗=1𝑥_𝑗,𝑧₁,𝑧₂,…,𝑧_𝑛−1+inf𝜌₁^{𝑝𝑘/𝐻}∶𝜌₁∈𝐴₁+^𝑚𝑠_𝑗=1𝑦_𝑗,𝑧₁,𝑧₂,…,𝑧_𝑛−1+inf𝜌₂^{𝑝𝑘/𝐻}∶𝜌₂∈𝐴₂=𝑔_Δ(𝑥)+𝑔_Δ𝑦. (30)
Let𝜆^𝑡→𝜆where𝜆^𝑡,𝜆∈ℂ, and let 𝑔_Δ(𝑥^𝑡−𝑥)→0 as 𝑡→∞. We have to show that 𝑔_Δ(𝜆^𝑡𝑥^𝑡−𝜆𝑥)→0as 𝑡→∞. We set 𝐴₃=𝜌_𝑡∶sup_𝑘^∞_𝑗=1𝑎_𝑘𝑗×𝑀_𝑗Δ^𝑠_𝑚Λ_𝑗(𝑥)𝜌_𝑡,𝑧₁,𝑧₂,…,𝑧_𝑛−1^{𝑝𝑗1/𝐻}≤1,foreach𝑧₁,𝑧₂,…,𝑧_𝑛−1,𝐴∈𝑋₄=𝜌¹_𝑡∶sup_𝑘^∞_𝑗=1𝑎_𝑘𝑗×𝑀_𝑗Δ^𝑠_𝑚Λ_𝑗𝑦𝜌¹_𝑡,𝑧₁,𝑧₂,…,𝑧_𝑛−1^{𝑝𝑗1/𝐻}≤1,foreach𝑧₁,𝑧₂,…,𝑧_𝑛−1∈𝑋. (31)
If 𝜌_𝑡∈𝐴₃ and 𝜌¹_𝑡∈𝐴₄, then by using non-decreasing and convexity of the Orlicz function𝑀_𝑗for all 𝑗∈ℕ we get sup_𝑘^∞_𝑗=1𝑎_𝑘𝑗×𝑀_𝑗Δ^𝑠_𝑚𝜆^𝑡𝑥^𝑡_𝑗−𝜆𝑥_𝑗𝜆^𝑡−𝜆𝜌_𝑡+|𝜆|𝜌¹_𝑡,𝑧₁,𝑧₂,…,𝑧_𝑛−1^{𝑝𝑗1/𝐻}≤sup_𝑘^∞_𝑗=1𝑎_𝑘𝑗×𝑀_𝑗Δ^𝑠_𝑚𝜆^𝑡𝑥^𝑡_𝑗−𝜆𝑥^𝑡_𝑗𝜆^𝑡−𝜆𝜌_𝑡+|𝜆|𝜌¹_𝑡,𝑧₁,𝑧₂,…,𝑧_𝑛−1^{𝑝𝑗1/𝐻}+sup_𝑘^∞_𝑗=1𝑎_𝑘𝑗×𝑀_𝑗Δ^𝑠_𝑚𝜆𝑥^𝑡_𝑗−𝜆𝑥_𝑗𝜆^𝑡−𝜆𝜌_𝑡+|𝜆|𝜌¹_𝑡,𝑧₁,𝑧₂,…,𝑧_𝑛−1^{𝑝𝑗1/𝐻}≤𝜆^𝑡−𝜆𝜌_𝑡𝜆^𝑡−𝜆𝜌_𝑡+|𝜆|𝜌¹_𝑡×sup_𝑘^∞_𝑗=1𝑎_𝑘𝑗𝑀_𝑗Δ^𝑠_𝑚𝑥^𝑡_𝑗𝜌_𝑡,𝑧₁,𝑧₂,…,𝑧_𝑛−1^{𝑝𝑗1/𝐻}+|𝜆|𝜌¹_𝑡𝜆^𝑡−𝜆𝜌_𝑡+|𝜆|𝜌¹_𝑡×sup_𝑘^∞_𝑗=1𝑎_𝑘𝑗𝑀_𝑗Δ^𝑠_𝑚𝑥^𝑡_𝑗−𝑥_𝑗𝜌¹_𝑡,𝑧₁,𝑧₂,…,𝑧_𝑛−1^{𝑝𝑗1/𝐻}. (32)
From the previous inequality, it follows that sup_𝑘^∞_𝑗=1𝑎_𝑘𝑗×𝑀_𝑗Δ^𝑠_𝑚𝜆^𝑡𝑥^𝑡_𝑗−𝜆𝑥_𝑗𝜆^𝑡−𝜆𝜌_𝑡+|𝜆|𝜌¹_𝑡,𝑧₁,𝑧₂,…,𝑧_𝑛−1^{𝑝𝑗1/𝐻}≤1, (33) and consequently 𝑔_Δ𝜆^𝑡𝑥^𝑡=−𝜆𝑥_{𝑚𝑠𝑗=1}𝜆^𝑡𝑥^𝑡_𝑗−𝜆𝑥_𝑗,𝑧₁,𝑧₂,…,𝑧_𝑛−1+inf𝜆^𝑡−𝜆𝜌_𝑡+|𝜆|𝜌¹_𝑡^{𝑝𝑘/𝐻}∶𝜌_𝑡∈𝐴₃,𝜌¹_𝑡∈𝐴₄≤𝜆^𝑡−𝜆^𝑚𝑠_𝑗=1𝑥^𝑡_𝑗,𝑧₁,𝑧₂,…,𝑧_𝑛−1+𝜆^𝑡−𝜆^{𝑝𝑘/𝐻}inf𝜌_𝑡^{𝑝𝑘/𝐻}∶𝜌_𝑡∈𝐴₃+|𝜆|^𝑚𝑠_𝑗=1𝜆^𝑡𝑥^𝑡_𝑗−𝜆𝑥_𝑗,𝑧₁,𝑧₂,…,𝑧_𝑛−1+|𝜆|^{𝑝𝑘/𝐻}inf𝜌¹_𝑡^{𝑝𝑘/𝐻}∶𝜌¹_𝑡∈𝐴₄≤max𝜆^𝑡−𝜆,𝜆^𝑡−𝜆^{𝑝𝑘/𝐻}𝑔_Δ𝑥^𝑡+max|𝜆|,|𝜆|^{𝑝𝑘/𝐻}𝑔_Δ𝑥^𝑡−𝑥. (34)
Note that𝑔_Δ(𝑥^𝑡)≤𝑔_Δ(𝑥)+𝑔_Δ(𝑥^𝑡−𝑥), for all𝑡∈ℕ. Hence, by our assumption, the right hand of (34) tends to 0 as𝑡→∞, and the result follows. This completes the proof of the theorem.

Theorem 12. Let ℳ=(𝑀_𝑗), ℳ^′=(𝑀^′_𝑗), and ℳ^′′=(𝑀^′′_𝑗) be the Musielak-Orlicz functions. Then, the following hold:(a)𝑊[𝐴,ℳ^′,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]^𝐼₀⊆𝑊[𝐴,ℳ.ℳ^′,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]^𝐼₀, provided𝑝=(𝑝_𝑘) such that𝐺₀=inf𝑝_𝑘>0,(b)𝑊[𝐴,ℳ^′,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]^𝐼₀⊆𝑊[𝐴,ℳ^′+ℳ^′′,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]^𝐼₀.

Proof. (a) Let𝜀>0be given. Choose𝜀₁>0such thatsup_𝑘(∑^∞_𝑗=1𝑎_𝑘𝑗)max{𝜀^𝐺₁,𝜀^𝐺0₁}<𝜀. Using the continuity of the Orlicz function𝑀, choose0<𝛿<1such that0<𝑡<𝛿implies that𝑀(𝑡)<𝜀₁.
Let 𝑥=(𝑥_𝑘) be any element in 𝑊[𝐴,ℳ^′,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]^𝐼₀ and put 𝐴_𝛿=𝑘∈ℕ∶_∞𝑗=1𝑎_𝑘𝑗𝑀^′_𝑗Δ^𝑠_𝑚Λ_𝑗(𝑥)𝜌₁,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗≥𝛿^𝐺. (35)
Then, by the definition of ideal convergent, we have the set𝐴_𝛿∈𝐼. If𝑛∉𝐴_𝛿, then we have ^∞_𝑗=1𝑎_𝑘𝑗𝑀^′_𝑗Δ^𝑠_𝑚Λ_𝑗(𝑥)𝜌₁,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗<𝛿^𝐺𝑀^′_𝑗Δ^𝑠_𝑚Λ_𝑗(𝑥)𝜌₁,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗<𝛿^𝐺⟹𝑀^′_𝑗Δ^𝑠_𝑚Λ_𝑗(𝑥)𝜌₁,𝑧₁,𝑧₂,…,𝑧_𝑛−1<𝛿. (36)
Using the continuity of the Orlicz function𝑀_𝑗for all𝑗and the relation (36), we have 𝑀_𝑗𝑀^′_𝑗Δ^𝑠_𝑚Λ_𝑗(𝑥)𝜌₁,𝑧₁,𝑧₂,…,𝑧_𝑛−1<𝜀₁. (37)
Consequently, we get ^∞_𝑗=1𝑎_𝑘𝑗𝑀_𝑗𝑀^′_𝑗Δ^𝑠_𝑚Λ_𝑗(𝑥)𝜌₁,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗<sup_𝑘^∞_𝑗=1𝑎_𝑘𝑗max𝜀^𝐺₁,𝜀^𝐺0₁⟹<𝜀_∞𝑗=1𝑎_𝑘𝑗𝑀_𝑗𝑀^′_𝑗Δ^𝑠_𝑚Λ_𝑗(𝑥)𝜌₁,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗<𝜀. (38)
This shows that 𝑘∈ℕ∶_∞𝑗=1𝑎_𝑘𝑗𝑀_𝑗𝑀^′_𝑗Δ^𝑠_𝑚Λ_𝑗(𝑥)𝜌₁,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗≥𝜀⊆𝐴_𝛿∈𝐼. (39)
This proves the assertion.
(b) Let 𝑥=(𝑥_𝑘)be any element in 𝑊[𝐴,ℳ^′,Λ,𝑝,‖⋅…⋅‖]^𝐼₀. Then, by the following inequality, the results follow: ^∞_𝑗=1𝑎_𝑘𝑗𝑀^′_𝑗+𝑀^′′_𝑗Δ^𝑠_𝑚Λ_𝑗(𝑥)𝜌₁,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗≤𝐷^∞_𝑗=1𝑎_𝑘𝑗𝑀^′_𝑗Δ^𝑠_𝑚Λ_𝑗(𝑥)𝜌₁,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗+𝐷^∞_𝑗=1𝑎_𝑘𝑗𝑀^′′_𝑗Δ^𝑠_𝑚Λ_𝑗(𝑥)𝜌₁,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗. (40)

Theorem 13. The inclusions𝑍[𝐴,ℳ,Δ_𝑚^𝑠−1,Λ,‖⋅…⋅‖]⊆𝑍[𝐴,ℳ,Δ^𝑠_𝑚,Λ,‖⋅…⋅‖] are strict for 𝑠,𝑚≥1 in general where𝑍=𝑊^𝐼,𝑊^𝐼₀, and𝑊^𝐼_∞.

Proof. We will give the proof for 𝑊[𝐴,ℳ,Δ_𝑚^𝑠−1,Λ,‖⋅…⋅‖]^𝐼₀⊆𝑊[𝐴,ℳ,Δ^𝑠_𝑚,Λ,‖⋅…⋅‖]^𝐼₀ only. The others can be proved by similar arguments. Let 𝑥=(𝑥_𝑘)∈𝑊[𝐴,ℳ,Δ_𝑚^𝑠−1,Λ,‖⋅…⋅‖]^𝐼₀. Then let 𝜀>0 be given; there exist 𝜌>0 such that 𝑘∈ℕ∶^∞_𝑗=1𝑎_𝑘𝑗𝑀_𝑗Δ_𝑚^𝑠−1Λ_𝑗(𝑥)𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1≥𝜀2∈𝐼. (41)
Since 𝑀_𝑗 for all 𝑗∈ℕ is non-decreasing and convex, it follows that ^∞_𝑗=1𝑎_𝑘𝑗𝑀_𝑗Δ^𝑠_𝑚Λ_𝑗(𝑥)2𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1=^∞_𝑗=1𝑎_𝑘𝑗×𝑀_𝑗Δ_𝑚^𝑠−1Λ_𝑗+1(𝑥)−Δ_𝑚^𝑠−1Λ_𝑗(𝑥),𝑧2𝜌₁,𝑧₂,…,𝑧_𝑛−1≤12^∞_𝑗=1𝑎_𝑘𝑗𝑀_𝑗Δ_𝑚^𝑠−1Λ_𝑗+1(𝑥)𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1+12^∞_𝑗=1𝑎_𝑘𝑗𝑀_𝑗Δ_𝑚^𝑠−1Λ_𝑗(𝑥)𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1, (42) and then we have 𝑘∈ℕ∶^∞_𝑗=1𝑎_𝑘𝑗×𝑀_𝑗Δ^𝑠_𝑚Λ_𝑗(𝑥),𝑧2𝜌₁,𝑧₂,…,𝑧_𝑛−1⊆1≥𝜀𝑘∈ℕ∶2×^∞_𝑗=1𝑎_𝑘𝑗𝑀_𝑗Δ_𝑚^𝑠−1𝑥_𝑗+1𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1≥𝜀2∪1𝑘∈ℕ∶2×^∞_𝑗=1𝑎_𝑘𝑗𝑀_𝑗Δ_𝑚^𝑠−1Λ_𝑗(𝑥)𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1≥𝜀2. (43)
Let 𝑀_𝑘(𝑥)=𝑀(𝑥)=𝑥 for all𝑥∈[0,∞[, 𝑘∈ℕ and 𝜆_𝑘=𝑘 for all 𝑘∈ℕ. Consider a sequence 𝑥=(𝑥_𝑘)=(𝑘^𝑠). Then, 𝑥∈𝑊[𝐴,ℳ,Δ^𝑠_𝑚,Λ,‖⋅…⋅‖]^𝐼₀but does not belong to 𝑊[𝐴,ℳ,Δ_𝑚^𝑠−1,Λ,‖⋅…⋅‖]^𝐼₀, for 𝑠=𝑚=1. This shows that the inclusion is strict.

Theorem 14. Let 0<𝑝_𝑘≤𝑞_𝑘 for all 𝑘∈ℕ; then 𝑊𝐴,ℳ,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖_∞⊆𝑊𝐴,ℳ,Δ^𝑠_𝑚,Λ,𝑞,‖⋅…⋅‖_∞. (44)

Proof. Let 𝑥=(𝑥_𝑗)∈𝑊[𝐴,ℳ,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]_∞; then there exists some 𝜌>0 such that sup_𝑘^∞_𝑗=1𝑎_𝑘𝑗𝑀_𝑗Δ^𝑠_𝑚Λ_𝑗(𝑥)𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗<∞. (45)
This implies that 𝑀_𝑗Δ^𝑠_𝑚Λ_𝑗(𝑥)𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1<1, (46) for a sufficiently large value of 𝑗. Since 𝑀_𝑗 for all 𝑗∈ℕ is non-decreasing, we get sup_𝑘^∞_𝑗=1𝑎_𝑘𝑗𝑀_𝑗Δ^𝑠_𝑚Λ_𝑗(𝑥)𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑞𝑗≤sup_𝑘^∞_𝑗=1𝑎_𝑘𝑗𝑀_𝑗Δ^𝑠_𝑚Λ_𝑗(𝑥)𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗<∞. (47)
Thus, 𝑥∈𝑊[𝐴,ℳ,Δ^𝑠_𝑚,Λ,𝑞,‖⋅…⋅‖]_∞. This completes the proof of the theorem.

Theorem 15. (i) If0<inf𝑝_𝑘≤𝑝_𝑘<1, then 𝑊𝐴,ℳ,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖_∞⊆𝑊𝐴,ℳ,Δ^𝑠_𝑚,Λ,‖⋅…⋅‖_∞. (48)
(ii) If 1<𝑝_𝑘≤sup_𝑘𝑝_𝑘<∞, then 𝑊[𝐴,ℳ,Δ^𝑠_𝑚,Λ,‖⋅…⋅‖]_∞⊆𝑊[𝐴,ℳ,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]_∞.

Proof. (i) Let 𝑥=(𝑥_𝑗)∈𝑊[𝐴,ℳ,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]_∞; since 0<inf_𝑘𝑝_𝑘≤𝑝_𝑘<1, then we have sup_𝑘^∞_𝑗=1𝑎_𝑘𝑗𝑀_𝑗Δ^𝑠_𝑚Λ_𝑗(𝑥)𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1≤sup_𝑘^∞_𝑗=1𝑎_𝑘𝑗𝑀_𝑗Δ^𝑠_𝑚Λ_𝑗(𝑥)𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗<∞, (49) and hence 𝑥∈𝑊[𝐴,ℳ,Δ^𝑠_𝑚,Λ,‖⋅…⋅‖]_∞.
(ii) Let 1<𝑝_𝑘≤sup_𝑘𝑝_𝑘<∞ and 𝑥=(𝑥_𝑗)∈𝑊[𝐴,ℳ,Δ^𝑠_𝑚,Λ,‖⋅…⋅‖]_∞. Then for each 0<𝜀<1 there exists a positive integer 𝑗₀ such that sup_𝑘^∞_𝑗=1𝑎_𝑘𝑗𝑀_𝑗Δ^𝑠_𝑚Λ_𝑗(𝑥)𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1≤𝜀<1, (50) for all 𝑗≥𝑗₀. This implies that sup_𝑘^∞_𝑗=1𝑎_𝑘𝑗𝑀_𝑗Δ^𝑠_𝑚Λ_𝑗(𝑥)𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗≤sup_𝑘^∞_𝑗=1𝑎_𝑘𝑗𝑀_𝑗Δ^𝑠_𝑚Λ_𝑗(𝑥)𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1<∞. (51) Thus 𝑥∈𝑊[𝐴,ℳ,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]_∞ and this completes the proof.

Theorem 16. For any sequence of the Orlicz functions ℳ=(𝑀_𝑗) which satisfies the Δ₂-condition, we have 𝑊[𝐴,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]^𝐼⊂𝑊[𝐴,ℳ,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]^𝐼.

Proof. Let 𝑥=(𝑥_𝑗)∈𝑊[𝐴,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]^𝐼 and 𝜀>0be given. Then, there exist𝜌>0such that the set 𝑘∈ℕ∶_∞𝑗=1𝑎_𝑘𝑗Δ^𝑠_𝑚Λ_𝑗(𝑥)−𝑙𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗≥𝜀∈𝐼,forsome𝑙. (52)
By taking 𝑦_𝑗=‖(Δ^𝑠_𝑚(Λ_𝑗(𝑥))−𝑙)/𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1‖, let 𝜀>0 and choose 𝛿 wit 0<𝛿<1 such that 𝑀_𝑗(𝑡)<𝜀 for all 𝑗∈ℕ; for 0≤𝑡≤𝛿, consider that ^∞_𝑗=1𝑀_𝑗𝑦_𝑗^𝑝𝑗=^∞_{𝑗=1,𝑦𝑗≤𝛿}𝑀_𝑗𝑦_𝑗^𝑝𝑗+^∞_{𝑗=1,𝑦𝑗>𝛿}𝑀_𝑗𝑦_𝑗^𝑝𝑗, (53) since𝑀_𝑗is continuous for all𝑛∈ℕ.
∑_{𝑗∈𝐼𝑘,𝑦𝑗≤𝛿}[𝑀_𝑗(𝑦_𝑗)]^𝑝𝑗<𝜀 and for𝑦_𝑗>𝛿, we use the fact that𝑦_𝑗<𝑦_𝑗/𝛿<1+𝑦_𝑗/𝛿. Since ℳ=(𝑀_𝑗)is non-decreasing and convex, it follows that 𝑀_𝑗𝑦_𝑗<𝑀_𝑗𝑦1+^𝑗𝛿<12𝑀_𝑗(2)+12𝑀_𝑗2𝑦_𝑗𝛿. (54)
Since ℳ=(𝑀_𝑗)satisfies the Δ₂-condition, then 𝑀_𝑗𝑦_𝑗<𝑦_𝑗2𝛿𝐿𝑀_𝑗(2)+𝑦_𝑗2𝛿𝐿𝑀_𝑗(2)=𝑦_𝑗𝛿𝐿𝑀_𝑗(2). (55)
Hence ^∞_{𝑗=1,𝑦𝑗>𝛿}𝑀_𝑗𝑦_𝑗^𝑝𝑗<max1,sup_𝑗𝐿𝛿⁻¹𝑀_𝑗(2)^𝑝𝑗×^∞_{𝑗=1,𝑦𝑗>𝛿}𝑦_𝑗^𝑝𝑗, (56) and then we have ^∞_𝑗=1𝑀_𝑗𝑦_𝑗^𝑝𝑗<𝜀+max1,sup_𝑗𝐿𝛿⁻¹𝑀_𝑗(2)^𝑝𝑗∞_{𝑗=1,𝑦𝑗>𝛿}𝑦_𝑗^𝑝𝑗. (57)
This proves that 𝑊[𝐴,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]^𝐼⊂𝑊[𝐴,ℳ,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]^𝐼.

Theorem 17. Let0<𝑝_𝑛≤𝑞_𝑛<1and(𝑞_𝑛/𝑝_𝑛)be bounded; then 𝑊𝐴,ℳ,Δ^𝑠_𝑚,Λ,𝑞,‖⋅…⋅‖^𝐼⊆𝑊𝐴,ℳ,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖^𝐼. (58)

Proof. Let𝑥=(𝑥_𝑗)∈𝑊[𝐴,ℳ,Λ,𝑞,‖⋅…⋅‖]_∞and we put 𝑦_𝑗=𝑀_𝑗Δ^𝑠_𝑚Λ_𝑗(𝑥)−𝑙𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑞𝑗,𝛽_𝑗=𝑝_𝑗𝑞_𝑗∀𝑗∈ℕ. (59)
Then0<𝛽_𝑗≤1, for all𝑗∈ℕ. Let it be such that0<𝛽≤𝛽_𝑗for all𝑗∈ℕ. Define the sequences(𝑎_𝑗)and(𝑏_𝑗)as follows: for𝑦_𝑗≥1, let𝑎_𝑗=𝑦_𝑗and𝑏_𝑗=0; for𝑦_𝑗<1, let𝑎_𝑗=0and𝑏_𝑗=𝑦_𝑗. Then clearly, for all𝑗∈ℕ we have𝑦_𝑗=𝑎_𝑗+𝑏_𝑗,𝑦^𝛽𝑗_𝑗=𝑎^𝛽𝑗_𝑗+𝑏^𝛽𝑗_𝑗, 𝑎^𝛽𝑗_𝑗≤𝑎_𝑗≤𝑦_𝑗, and𝑏^𝛽𝑗_𝑗≤𝑏^𝛽_𝑗. Therefore, we have ^∞_𝑗=1𝑎_𝑘𝑗𝑦^𝛽𝑗_𝑗≤^∞_𝑗=1𝑎_𝑘𝑗𝑦_𝑗≤^∞_𝑗=1𝑎_𝑘𝑗𝑦_𝑗^𝛽. (60)
Hence𝑥∈𝑊[𝐴,ℳ,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]_∞.

Theorem 18. For any two sequences𝑝=(𝑝_𝑘)and𝑞=(𝑞_𝑘)of positive real numbers and for any two n-norms‖⋅…⋅‖₁and‖⋅…⋅‖₂on𝑋, the following holds: 𝑍𝐴,ℳ,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖₁∩𝑍𝐴,ℳ,Δ^𝑠_𝑚,Λ,𝑞,‖⋅…⋅‖₂≠𝜙, (61) where𝑍=𝑊^𝐼,𝑊^𝐼₀,𝑊^𝐼_∞, and𝑊_∞.

Proof. The proof of the theorem is obvious, because the zero element belongs to each of the sequence spaces involved in the intersection.

Theorem 19. The sequence spaces 𝑊[𝐴,ℳ,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]^𝐼, 𝑊[𝐴,ℳ,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]^𝐼₀, 𝑊[𝐴,ℳ,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]_∞, and 𝑊[𝐴,ℳ,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]^𝐼_∞ are neither solid nor symmetric nor sequence algebras for𝑠,𝑚≥1in general.

Proof. The proof is obtained by using the same techniques of Et [23] and Theorems 15, 17, and 18.

Note 1. It is clear from definitions that 𝑊𝐴,ℳ,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖^𝐼₀⊆𝑊𝐴,ℳ,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖^𝐼⊆𝑊𝐴,ℳ,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖^𝐼_∞. (62)

Theorem 20. The spaces 𝑍[𝐴,ℳ,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖] and 𝑍[𝐴,ℳ,𝑝,‖⋅…⋅‖] are equivalent as topological spaces, where 𝑍=𝑊^𝐼,𝑊^𝐼₀,𝑊^𝐼_∞, and 𝑊_∞.

Proof. Consider the mapping 𝑇∶𝑍𝐴,ℳ,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖⟶𝑍𝐴,ℳ,𝑝,‖⋅…⋅‖, (63) defined by 𝑇(𝑥)=(Δ^𝑠_𝑚(Λ_𝑗𝑥)) for each 𝑥∈𝑍[𝐴,ℳ,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖]. Then, clearly 𝑇 is a linear homeomorphism and the proof follows.

Remark 21. If we replace the difference operator Δ^𝑠_𝑚 by Δ_𝑚^(𝑠), then for each 𝜀>0 we get the following sequence spaces: 𝑊𝐴,ℳ,Δ_𝑚^(𝑠),Λ,𝑝,‖⋅…⋅‖^𝐼=()∶𝜆𝑥∈𝜔𝑛−𝑋𝑘∈ℕ∶_{𝑘−1𝑗∈𝐼𝑘}𝑀_𝑗Δ_𝑚^(𝑠)𝑥_𝑗−𝑙𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗𝑧≥𝜀∈𝐼,forsome𝜌>0,𝑙∈𝑋andeach₁,𝑧₂,…,𝑧_𝑛−1,𝑊∈𝑋𝐴,ℳ,Δ_𝑚^(𝑠),Λ,𝑝,‖⋅…⋅‖^𝐼₀=()∶𝜆𝑥∈𝜔𝑛−𝑋𝑘∈ℕ∶_{𝑘−1𝑗∈𝐼𝑘}𝑀_𝑗Δ_𝑚^(𝑠)𝑥_𝑗𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗≥𝜀∈𝐼,forsome𝜌>0andeach𝑧₁,𝑧₂,…,𝑧_𝑛−1,𝑊∈𝑋𝐴,ℳ,Δ_𝑚^(𝑠),Λ,𝑝,‖⋅…⋅‖_∞=𝑥∈𝜔(𝑛−𝑋)∶sup_𝑘𝜆_𝑘⁻¹_{𝑗∈𝐼𝑘}𝑀_𝑗Δ_𝑚^(𝑠)𝑥_𝑗𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗<∞,forsome𝜌>0andeach𝑧₁,𝑧₂,…,𝑧_𝑛−1,𝑊∈𝑋𝐴,ℳ,Δ_𝑚^(𝑠),Λ,𝑝,‖⋅…⋅‖^𝐼_∞=𝑥∈𝜔(𝑛−𝑋)∶∃𝐾>0,s.t.𝑘∈ℕ∶𝜆_𝑘⁻¹_{𝑗∈𝐼𝑘}𝑀_𝑗Δ_𝑚^(𝑠)𝑥_𝑗𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗≥𝐾∈𝐼,forsome𝜌>0andeach𝑧₁,𝑧₂,…,𝑧_𝑛−1∈𝑋. (64)

Corollary 22. The sequence spaces𝑍[𝐴,ℳ,Δ^𝑠_𝑚,Λ,𝑝,‖⋅…⋅‖], where𝑍=𝑊^𝐼,𝑊^𝐼₀,𝑊^𝐼_∞, and𝑊_∞, are paranormed spaces (not totally paranormed) with respect to the paranormℎ_Δdefined by ℎ_Δ(𝑥)=^𝑚𝑠_𝑗=1𝑥_𝑗,𝑧₁,𝑧₂,…,𝑧_𝑛−1+inf𝜌^{𝑝𝑘/𝐻}∶sup_𝑘^∞_𝑗=1𝑎_𝑘𝑗×𝑀_𝑗Δ_𝑚^(𝑠)Λ_𝑗(𝑥)𝜌₁,𝑧₁,𝑧₂,…,𝑧_𝑛−1^{𝑝𝑗1/𝐻}≤1,𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝜌>0,𝑎𝑛𝑑𝑒𝑎𝑐ℎ𝑧₁,𝑧₂,…,𝑧_𝑛−1∈𝑋, (65) where𝐻=max{1,sup_𝑘𝑝_𝑘} and𝑍=𝑊^𝐼,𝑊^𝐼₀,𝑊^𝐼_∞, and𝑊_∞. Also it is clear that the paranorms𝑔_Δandℎ_Δare equivalent.

We state the following theorem in view of Lemma 9.

Theorem 23. Let 𝑋 be a standard 𝑛-normed space and {𝑒₁,𝑒₂,…,𝑒_𝑛} an orthogonal set in 𝑋. Then, the following hold:(a)𝑊[𝐴,ℳ,Δ_𝑚^(𝑠),Λ,𝑝,‖⋅…⋅‖_∞]^𝐼=𝑊[𝐴,ℳ,Δ_𝑚^(𝑠),Λ,𝑝,‖⋅…⋅‖_𝑛−1]^𝐼,(b)𝑊[𝐴,ℳ,Δ_𝑚^(𝑠),Λ,𝑝,‖⋅…⋅‖_∞]^𝐼₀=𝑊[𝐴,ℳ,Δ_𝑚^(𝑠),Λ,𝑝,‖⋅…⋅‖_𝑛−1]^𝐼₀,(c)𝑊[𝐴,ℳ,Δ_𝑚^(𝑠),Λ,𝑝,‖⋅…⋅‖_∞]_∞=𝑊[𝐴,ℳ,Δ_𝑚^(𝑠),Λ,𝑝,‖⋅…⋅‖_𝑛−1]_∞,(d)𝑊[𝐴,ℳ,Δ_𝑚^(𝑠),Λ,𝑝,‖⋅…⋅‖_∞]^𝐼_∞=𝑊[𝐴,ℳ,Δ_𝑚^(𝑠),Λ,𝑝,‖⋅…⋅‖_𝑛−1]^𝐼_∞,where ‖⋅…⋅‖_∞is the derived(𝑛−1)-norm defined with respect to the set{𝑒₁,𝑒₂,…,𝑒_𝑛}and‖⋅…⋅‖_𝑛−1is the standard(𝑛−1)-norm on𝑋.

Acknowledgment

The author is most grateful to the editor and anonymous referee for careful reading of the paper and valuable suggestions which helped in improving an earlier version of it.