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

Abstract

In the present paper we introduced the ideal convergence of generalized difference sequence spaces combining de La Vallée-Poussin mean and Musielak-Orlicz function over n-normed spaces. We also study some topological properties and inclusion relation between these spaces.

1. Introduction

Throughout the paper 𝜔, ℓ_∞, 𝑐, 𝑐₀, and ℓ_𝑝 denote 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, ℝ, respectively. Many authors studied various sequence spaces using normed or seminormed linear spaces. In this paper, using de La Vallée-Poussin mean and the notion of ideal, we aimed to introduce some new sequence spaces with respect to generalized difference operator Δ^𝑠_𝑚 and Musielak-Orlicz function in 𝑛-normed linear spaces. By an ideal we mean a family 𝐼⊂2^𝑌 of subsets of a nonempty set 𝑌 satisfying (i) 𝜙∈𝐼; (ii) 𝐴,𝐵∈𝐼 imply 𝐴∪𝐵∈𝐼; (iii) 𝐴∈𝐼,  𝐵⊂𝐴 imply 𝐵∈𝐼, while an admissible ideal 𝐼 of 𝑌 further satisfies {𝑥}∈𝐼 for each 𝑥∈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 of 𝑛-normed spaces can be found in [3], and this concept has been studied by many authors; see for instance [4–7]. The notion of ideal convergence in 2-normed space was initially introduced by Gürdal [8]. Later on, it was extended to 𝑛-normed spaces by Gürdal and Şahiner [9]. Given 𝐼⊂2^ℕ is a nontrivial ideal in ℕ, the sequence (𝑥_𝑛)_𝑛∈ℕ in a normed space (𝑋;‖⋅‖) is said to be 𝐼-convergent to 𝑥∈𝑋, if for each 𝜀>0, 𝐴(𝜀)=𝑛∈ℕ∶𝑥_𝑛−𝑥≥𝜀∈𝐼. (1)

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

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

An Orlicz function is a function 𝑀∶[0,∞)→[0,∞) which is continuous, nondecreasing, and convex with 𝑀(0)=0, 𝑀(𝑥)>0 for 𝑥>0 and 𝑀(𝑥)→∞, as 𝑥→∞. If convexity of 𝑀 is replaced by 𝑀(𝑥+𝑦)≤𝑀(𝑥)+𝑀(𝑦), then it is called a modulus function, introduced by Nakano [10]. Ruckle [11] and Maddox [12] used the idea of a modulus function to construct some spaces of complex sequences. An Orlicz function 𝑀 is said to satisfy Δ₂-condition for all values of 𝑥≥0, if there exists a constant 𝑘>0, such that 𝑀(2𝑥)≤𝑘𝑀(𝑥). The Δ₂-condition is equivalent to 𝑀(𝑙𝑥)≤𝑘𝑙𝑀(𝑥) for all values of 𝑥 and for 𝑙>1. Lindenstrauss and Tzafriri [13] used the idea of an Orlicz function to define the following sequence spaces: ℓ_𝑀=𝑥∈𝜔∶^∞_𝑘=1𝑀|𝑥(𝑘)|𝜌<∞ (4) which is a Banach space with the Luxemburg norm defined by ‖𝑥‖=inf𝜌>0∶^∞_𝑘=1𝑀|𝑥(𝑘)|𝜌≤1. (5)

The space ℓ_𝑀 is closely related to the space ℓ_𝑝, which is an Orlicz sequence space with 𝑀(𝑥)=𝑥^𝑝 for 1≤𝑝<∞.

Recently different classes of sequences have been introduced using Orlicz functions. See [7, 9, 14–16].

A sequence 𝑀=(𝑀_𝑘) of Orlicz functions 𝑀_𝑘 for all 𝑘∈ℕ is called a Musielak-Orlicz function, for a given Musielak-Orlicz function 𝑀. Kızmaz [17] defined the difference sequence spaces ℓ_∞(Δ), 𝑐(Δ), and 𝑐₀(Δ) as follows: 𝑍(Δ)  =  {𝑥=(𝑥_𝑘)∶(Δ𝑥_𝑘)∈𝑍}, for 𝑍=ℓ_∞, 𝑐, and 𝑐₀, where Δ𝑥=(𝑥_𝑘−𝑥_𝑘+1), for all 𝑘∈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 [18] as follows: 𝑍(Δ^𝑠)={𝑥=(𝑥_𝑘)∶(Δ^𝑠𝑥_𝑘)∈𝑍}, for 𝑍=ℓ_∞,  𝑐, and 𝑐₀, where 𝑠∈ℕ, (Δ^𝑠𝑥_𝑘)=(Δ^𝑠−1𝑥_𝑘−Δ^𝑠−1𝑥_𝑘+1) and so that Δ^𝑠𝑥_𝑘=∑^𝑠_𝑛=0(−1)^𝑛𝐶^𝑠_𝑛𝑥_𝑘+𝑛. Tripathy and Esi [19] introduced the following new type of difference sequence spaces.

𝑍(Δ_𝑚)={𝑥=(𝑥_𝑘)∶(Δ_𝑚𝑥_𝑘)∈𝑍}, 𝑍=ℓ_∞, 𝑐, and 𝑐₀, where Δ_𝑚𝑥_𝑘=(𝑥_𝑘−𝑥_𝑘+𝑚), for all 𝑘∈ℕ. Tripathy et al. [20] generalized the above notions and unified them as follows. Let 𝑚, 𝑠 be nonnegative integers, then for 𝑍 a given sequence space we have 𝑍Δ^𝑠_𝑚=𝑥=𝑥_𝑘∶Δ^𝑠_𝑚𝑥_𝑘∈𝑍, (6) where Δ^𝑠_𝑚𝑥_𝑘=^𝑠_𝑛=0(−1)^𝑛𝐶^𝑠_𝑛𝑥_{𝑘+𝑚𝑛}. (7)

Also let 𝑚,𝑠 be nonnegative integers, then for 𝑍 a given sequence space we have 𝑍Δ_𝑚^(𝑠)=𝑥=𝑥_𝑘∶Δ_𝑚^(𝑠)𝑥_𝑘∈𝑍, (8) where Δ_𝑚^(𝑠)𝑥_𝑘=^𝑠_𝑛=0(−1)^𝑛𝐶^𝑠_𝑛𝑥_{𝑘−𝑚𝑛}, (9) where 𝑥_𝑘=0, for 𝑘<0

2. Definitions and Preliminaries

Let 𝑛∈ℕ and 𝑋 be a linear space over the field 𝐾 of dimension 𝑑, where 𝑑≥𝑛≥2 and 𝐾 is the field of real or complex numbers. A real valued function ‖⋅,…,⋅‖ on 𝑋^𝑛 satisfies the following four conditions:(1)‖𝑥₁,𝑥₂,…,𝑥_𝑛‖=0 if and only if 𝑥₁,𝑥₂,…,𝑥_𝑛 are linearly dependent in 𝑋;(2)‖𝑥₁,𝑥₂,…,𝑥_𝑛‖ is invariant under permutation;(3)‖𝛼𝑥₁,𝑥₂,…,𝑥_𝑛‖=|𝛼|‖𝑥₁,𝑥₂,…,𝑥_𝑛‖ for any 𝛼∈𝐾;(4)‖𝑥+𝑥^′,𝑥₂,…,𝑥_𝑛‖≤‖𝑥,𝑥₂,…,𝑥_𝑛‖+‖𝑥^′,𝑥₂,…,𝑥_𝑛‖; is called an 𝑛-norm on 𝑋 and the pair (𝑋;‖⋅,…,⋅‖) is called an 𝑛-normed space over the field 𝐾. For example, we may take 𝑋=ℝ^𝑛 being equipped with the 𝑛-norm ‖𝑥₁,𝑥₂,…,𝑥_𝑛‖_𝐸= the volume of the 𝑛-dimensional parallelepiped spanned by the vectors 𝑥₁,𝑥₂,…,𝑥_𝑛 which may be given explicitly by the formula 𝑥₁,𝑥₂,…,𝑥_𝑛𝐸=det𝑥_𝑖𝑗𝑥=abs¹¹𝑥₁₂⋯𝑥_1𝑛𝑥₂₁𝑥₂₂⋯𝑥_2𝑛𝑥⋮⋮⋱⋮_𝑛1𝑥_𝑛2⋯𝑥_𝑛𝑛, (10)

where 𝑥_𝑖=(𝑥_𝑖1,𝑥_𝑖2,…,𝑥_𝑖𝑛) for each 𝑖∈ℕ.

Let (𝑋,‖⋅,…,⋅‖) be an 𝑛-normed space of dimension 𝑑≥𝑛≥2 and {𝑎₁,𝑎₂,𝑎₃,…,𝑎_𝑛} a linearly independent set in 𝑋. Then, the function ‖⋅,…,⋅‖_∞ on 𝑋^𝑛−1 defined by 𝑥₁,𝑥₂,…,𝑥_𝑛∞=max_{1≤𝑖≤𝑛}𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑎_𝑖, (11)

defines an (𝑛−1)-norm on 𝑋 with respect to 𝑎₁,𝑎₂,𝑎₃,…,𝑎_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, (12)

where ⟨⋅,⋅⟩ denotes the inner product on 𝑋. If we take 𝑋=ℝ^𝑛, then 𝑥₁,𝑥₂,…,𝑥_𝑛𝐸=𝑥₁,𝑥₂,…,𝑥_𝑛𝑆. (13)

For 𝑛=1, this 𝑛-norm is the usual norm ‖𝑥₁‖=⟨𝑥₁,𝑥₁⟩.

Definition 1. A sequence (𝑥_𝑘) in an 𝑛-normed space is said to be convergent to 𝑥∈𝑋 if, lim_𝑘→∞𝑧₁,𝑧₂,…,𝑧_𝑛−1,𝑥_𝑘−𝑥_𝑛=0,∀𝑧₁,𝑧₂,…,𝑧_𝑛−1∈𝑋. (14)

Definition 2. A sequence (𝑥_𝑘) in an 𝑛-normed space is called Cauchy (with respect to 𝑛-norm) if, lim_{𝑘,𝑗→∞}𝑧₁,𝑧₂,…,𝑧_𝑛−1,𝑥_𝑘−𝑥_𝑗𝑛=0,∀𝑧₁,𝑧₂,…,𝑧_𝑛−1∈𝑋. (15) If every Cauchy sequence in 𝑋 converges to an 𝑥∈𝑋, then 𝑋 is said to be complete (with respect to the 𝑛-norm). A complete 𝑛-normed space is called 𝑛-Banach space.

Definition 3. A sequence (𝑥_𝑘) in an 𝑛-normed space (𝑋,‖⋅,…,⋅‖) is said to be 𝐼-convergent to 𝑥₀∈𝑋 with respect to 𝑛-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. (16)

Definition 4. A sequence (𝑥_𝑘) in an 𝑛-normed space (𝑋,‖⋅,…,⋅‖) is said to be 𝐼-Cauchy if for each 𝜀>0, there exists a positive integer 𝑚=𝑚(𝜀) such that the set 𝑘∈ℕ∶𝑥_𝑘−𝑥_𝑚,𝑧₁,𝑧₂,…,𝑧_𝑛−1≥𝜀,forevery𝑧₁,𝑧₂,…,𝑧_𝑛−1∈𝐼. (17) Let 𝑥=(𝑥_𝑘) be a sequence; then 𝑆(𝑥) denotes the set of all permutations of the elements of (𝑥_𝑘); that is, 𝑆(𝑥)=(𝑥_𝜋(𝑛))∶𝜋 is a permutation of ℕ.

Definition 5. A sequence space 𝐸 is said to be symmetric if 𝑆(𝑥)⊂𝐸 for all 𝑥∈𝐸.

Definition 6. A sequence space 𝐸 is said to be normal (or solid) if (𝛼_𝑘𝑥_𝑘)∈𝐸, whenever (𝑥_𝑘)∈𝐸 and for all sequence (𝛼_𝑘) of scalars with |𝛼_𝑘|≤1 for all 𝑘∈ℕ.

Definition 7. A sequence space 𝐸 is said to be a sequence algebra if 𝑥,𝑦∈𝐸 then 𝑥⋅𝑦=(𝑥_𝑘𝑦_𝑘)∈𝐸.

Lemma 8. Every 𝑛-normed space is an (𝑛−𝑟)-normed space for all 𝑟=1,2,3,…,𝑛−1. In particular, every 𝑛-normed space is a normed space.

Lemma 9. On a standard 𝑛-normed space 𝑋, 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∞, (18)
for all 𝑥₁,𝑥₂,…,𝑥_𝑛−1∈𝑋, where ‖𝑥₁,𝑥₂,…,𝑥_𝑛−1‖_∞=max_{1≤𝑖≤𝑛}{‖𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑒_𝑖‖_𝑆}.
Let Λ=(𝜆_𝑘) be a nondecreasing sequence of positive real numbers tending to infinity and let 𝜆₁=1 and 𝜆_𝑘+1≤𝜆_𝑘+1. In summability theory, de La Vallée-Poussin mean was first used to define the (𝑉,𝜆)-summability by Leindler [21]. Also the (𝑉,𝜆)-summable sequence spaces have been studied by many authors including [22, 23]. The generalized de La Vallée-Poussin’s mean of a sequence 𝑥=(𝑥_𝑘) is defined as follows: 𝑡_𝑘(𝑥)=(1/𝜆_𝑘)∑_{𝑗∈𝐼𝑘}|𝑥_𝑗|, where 𝐼_𝑘=[𝑘−𝜆_𝑘+1,𝑘] for 𝑘∈ℕ. We write
[𝑉,𝜆]₀={𝑥∈𝜔∶lim_𝑘→∞(1/𝜆_𝑘)∑_{𝑗∈𝐼𝑘}|𝑥_𝑗|=0},
[𝑉,𝜆]={𝑥∈𝜔∶lim_𝑘→∞(1/𝜆_𝑘)∑_{𝑗∈𝐼𝑘}|𝑥_𝑗−𝑙|=0 for some 𝑙∈ℂ},
[𝑉,𝜆]_∞={𝑥∈𝜔∶sup_𝑘(1/𝜆_𝑘)∑_{𝑗∈𝐼𝑘}|𝑥_𝑗|<∞}.
For the sequence spaces that are strongly summable to zero, strongly summable and strongly bounded by the de La Vallée-Poussin's method, respectively. In the special case where 𝜆_𝑘=𝑘 for 𝑘∈ℕ the spaces [𝑉,𝜆]₀, [𝑉,𝜆], and [𝑉,𝜆]_∞ reduce to the spaces 𝑣₀, 𝑣, and 𝑣_∞ introduced by Maddox [24]. The following new paranormed sequence space is defined in [22].
𝑉(𝜆,𝑝)={𝑥∈𝜔∶∑^∞_𝑘=1((1/𝜆_𝑘)∑_{𝑗∈𝐼𝑘}|𝑥_𝑗|)^𝑝𝑘<∞}. If one takes 𝑝_𝑘=𝑝 for all 𝑘∈ℕ; the space 𝑉(𝜆,𝑝) reduced to normed space 𝑉_𝑝(𝜆) defined by 𝑉_𝑝(𝜆)={𝑥∈𝜔∶∑^∞_𝑘=1((1/𝜆_𝑘)∑_{𝑗∈𝐼𝑘}|𝑥_𝑗|)^𝑝<∞}. The details of the sequence spaces mentioned above can be found in [23].
For any bounded sequence (𝑝_𝑛) of positive numbers, one has the following well-known inequality.
If 0≤𝑝_𝑘≤sup_𝑘𝑝_𝑘=𝐺 and 𝐷=max(1,2^𝐺−1), then |𝑎_𝑛+𝑏_𝑛|^𝑝𝑛≤𝐷(|𝑎_𝑛|^𝑝𝑛+|𝑏_𝑛|^𝑝𝑛), for all 𝑘 and 𝑎_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) be a Musielak-Orlicz function, and (𝑋,‖⋅,…,⋅‖) an 𝑛-normed space. Further, let 𝑝=(𝑝_𝑘) be any bounded sequence of positive real numbers, 𝑉𝜆,ℳ,‖⋅,…,⋅‖,𝑝,Δ^𝑠_𝑚^𝐼=𝑥∈𝜔(𝑛−𝑋)∶𝑘∈ℕ∶𝜆_𝑘⁻¹×_{𝑗∈𝐼𝑘}𝑀_𝑗Δ^𝑠_𝑚𝑥_𝑗−𝑙𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗≥𝜀∈𝐼,forsome𝜌>0,𝑙∈𝑋andeach𝑧₁,𝑧₂,…,𝑧_𝑛−1,𝑉‖‖∈𝑋𝜆,ℳ,⋅,…,⋅,𝑝,Δ^𝑠_𝑚^𝐼₀=𝑥∈𝜔(𝑛−𝑋)∶𝑘∈ℕ∶𝜆_𝑘⁻¹×_{𝑗∈𝐼𝑘}𝑀_𝑗Δ^𝑠_𝑚𝑥_𝑗𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗≥𝜀∈𝐼,forsome𝜌>0,andeach𝑧₁,𝑧₂,…,𝑧_𝑛−1,𝑉‖‖∈𝑋𝜆,ℳ,⋅,…,⋅,𝑝,Δ^𝑠_𝑚∞=𝑥∈𝜔(𝑛−𝑋)∶×sup_𝑘𝜆_𝑘⁻¹_{𝑗∈𝐼𝑘}𝑀_𝑗Δ^𝑠_𝑚𝑥_𝑗𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗<∞,forsome𝜌>0,andeach𝑧₁,𝑧₂,…,𝑧_𝑛−1,𝑉‖‖∈𝑋𝜆,ℳ,⋅,…,⋅,𝑝,Δ^𝑠_𝑚^𝐼_∞=𝑥∈𝜔(𝑛−𝑋)∶∃𝐾>0,s.t.𝑘∈ℕ∶𝜆_𝑘⁻¹×_{𝑗∈𝐼𝑘}𝑀_𝑗Δ^𝑠_𝑚𝑥_𝑗𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗≥𝐾∈𝐼,forsome𝜌>0,andeach𝑧₁,𝑧₂,…,𝑧_𝑛−1∈𝑋. (19)

The above sequence spaces contain some unbounded sequences for 𝑠≥1. If 𝑀_𝑘(𝑥)=𝑥, 𝑚=1, 𝜆_𝑘=𝑘 for all 𝑘∈ℕ and 𝑝_𝑘=1 for all 𝑘∈ℕ, then (𝑘^𝑠)∈𝑉[𝜆,𝑀,‖⋅,…,⋅‖,𝑝,Δ^𝑠_𝑚]_∞but (𝑘^𝑠)∉ℓ_∞.

Let us consider a few special cases of the above sets.(1)If 𝑛 = 2,  𝑚 = 1, and 𝑀_𝑘(𝑥) = 𝑀(𝑥), then the above classes of sequences are denoted by 𝑉[𝜆, 𝑀, ‖⋅,…,⋅‖, 𝑝, Δ^𝑠]^𝐼, 𝑉[𝜆, 𝑀, ‖,⋅,‖, 𝑝, Δ^𝑠]^𝐼₀ , 𝑉[𝜆, 𝑀, ‖,⋅,‖, 𝑝, Δ^𝑠]_∞, and 𝑉[𝜆, 𝑀, ‖,⋅,‖, 𝑝, Δ^𝑠]^𝐼_∞, respectively, which were defined and studied by Savaş [25].(2)If 𝑀_𝑘(𝑥) = 𝑀(𝑥), then the above classes of sequences are denoted by 𝑉[𝜆, 𝑀, ‖⋅,…,⋅‖, 𝑝, Δ^𝑠_𝑚]^𝐼, 𝑉[𝜆, 𝑀, ‖⋅,…,⋅‖, 𝑝, Δ^𝑠_𝑚]^𝐼₀, 𝑉[𝜆, 𝑀, ‖⋅,…,⋅‖, 𝑝, Δ^𝑠_𝑚]_∞, and 𝑉[𝜆, 𝑀, ‖⋅,…,⋅‖, 𝑝, Δ^𝑠_𝑚]^𝐼_∞, respectively.(3)If 𝑀_𝑘(𝑥)=𝑥, for all 𝑘∈ℕ, then the above classes of sequences are denoted by 𝑉[𝜆,‖⋅,…,⋅‖,𝑝,Δ^𝑠_𝑚]^𝐼, 𝑉[𝜆,‖⋅,…,⋅‖,𝑝,Δ^𝑠_𝑚]^𝐼₀, 𝑉[𝜆,‖⋅,…,⋅‖,𝑝,Δ^𝑠_𝑚]_∞, and 𝑉[𝜆,‖⋅,…,⋅‖,𝑝,Δ^𝑠_𝑚]^𝐼_∞, respectively.(4)If 𝑝_𝑘=1, for all 𝑘∈ℕ, then we denote the above classes of sequences by 𝑉[𝜆, ‖⋅,…,⋅‖, Δ^𝑠_𝑚]^𝐼, 𝑉[𝜆, ‖⋅,…,⋅‖, Δ^𝑠_𝑚]^𝐼₀, 𝑉[𝜆, ‖⋅,…,⋅‖, Δ^𝑠_𝑚]_∞, and 𝑉[𝜆, ‖⋅,…,⋅‖, Δ^𝑠_𝑚]^𝐼_∞, respectively.(5)If 𝑀_𝑘(𝑥)=𝑀(𝑥), 𝑚=1, and 𝜆_𝑘=𝑘 for all 𝑘∈ℕ, then the above classes of sequences are denoted by 𝑉[𝑀,‖⋅,…,⋅‖,𝑝,Δ^𝑠]^𝐼, 𝑉[𝑀,‖⋅,…,⋅‖,𝑝,Δ^𝑠]^𝐼₀, 𝑉[𝑀,‖⋅,…,⋅‖,𝑝,Δ^𝑠]_∞, and 𝑉[𝑀,‖⋅,…,⋅‖,𝑝,Δ^𝑠]^𝐼_∞, respectively, which were defined and studied by Savaş [7].

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

Theorem 11. The spaces 𝑉[𝜆,ℳ,‖⋅,…,⋅‖,𝑝,Δ^𝑠_𝑚]^𝐼, 𝑉[𝜆,ℳ,‖⋅,…,⋅‖,𝑝,Δ^𝑠_𝑚]^𝐼₀, and 𝑉[𝜆,ℳ,‖⋅,…,⋅‖,𝑝,Δ^𝑠_𝑚]^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_𝑗𝑀_𝑗Δ^𝑠_𝑚𝑥_𝑗𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1≤1,𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝜌>0,𝑎𝑛𝑑𝑒𝑎𝑐ℎz₁,z₂,…,z_n-1∈𝑋, (20) where 𝐻=max{1,sup_𝑘𝑝_𝑘}.

Proof. Clearly 𝑔_Δ(−𝑥)=𝑔_Δ(𝑥) and 𝑔_Δ(𝜃)=0. Let 𝑥=(𝑥_𝑘) and  𝑦=(𝑦_𝑘)∈𝑉[𝜆,ℳ,‖⋅,…,⋅‖,𝑝,Δ^𝑠_𝑚]^𝐼₀. Then, for 𝜌>0 we set 𝐴₁=𝜌∶sup_𝑗𝑀_𝑗Δ^𝑠_𝑚𝑥_𝑗𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1≤1,foreach𝑧₁,𝑧₂,…,𝑧_𝑛−1,𝐴∈𝑋₂=𝜌∶sup_𝑗𝑀_𝑗Δ^𝑠_𝑚𝑦_𝑗𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1≤1,foreach𝑧₁,𝑧₂,…,𝑧_𝑛−1∈𝑋. (21) Let 𝜌₁∈𝐴₁, 𝜌₂∈𝐴₂ and 𝜌=𝜌₁+𝜌₂, then we have 𝑀_𝑗Δ^𝑠_𝑚𝑥_𝑗+𝑦_𝑗𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1≤𝜌₁𝜌₁+𝜌₂𝑀_𝑗Δ^𝑠_𝑚𝑥_𝑗𝜌₁,𝑧₁,𝑧₂,…,𝑧_𝑛−1+𝜌₂𝜌₁+𝜌₂𝑀_𝑗Δ^𝑠_𝑚𝑦_𝑗𝜌₂,𝑧₁,𝑧₂,…,𝑧_𝑛−1𝑔≤1,_Δ=𝑥+𝑦_{𝑚𝑠𝑗=1}𝑥_𝑗+𝑦_𝑗,𝑧₁,𝑧₂,…,𝑧_𝑛−1+inf𝜌₁+𝜌₂^{𝑝𝑗/𝐻}∶𝜌₁∈𝐴₁,𝜌₂∈𝐴₂≤^𝑚𝑠_𝑗=1𝑥_𝑗,𝑧₁,𝑧₂,…,𝑧_𝑛−1+inf𝜌₁^{𝑝𝑗/𝐻}∶𝜌₁∈𝐴₁+^𝑚𝑠_𝑗=1𝑦_𝑗,𝑧₁,𝑧₂,…,𝑧_𝑛−1+inf𝜌₂^{𝑝𝑗/𝐻}∶𝜌₂∈𝐴₂=𝑔_Δ(𝑥)+𝑔_Δ𝑦. (22) Let 𝜆^𝑡→𝜆 where 𝜆^𝑡,𝜆∈ℂ, and let 𝑔_Δ(𝑥^𝑡−𝑥)→0 as 𝑡→∞. We have to show that 𝑔_Δ(𝜆^𝑡𝑥^𝑡−𝜆𝑥)→0 as 𝑡→∞. We set 𝐴₃=𝜌_𝑡∶sup_𝑗𝑀_𝑗Δ^𝑠_𝑚𝑥_𝑗𝜌_𝑡,𝑧₁,𝑧₂,…,𝑧_𝑛−1≤1,foreach𝑧₁,𝑧₂,…,𝑧_𝑛−1,𝐴∈𝑋₄=𝜌¹_𝑡∶sup_𝑗𝑀_𝑗Δ^𝑠_𝑚𝑦_𝑗𝜌¹_𝑡,𝑧₁,𝑧₂,…,𝑧_𝑛−1≤1,foreach𝑧₁,𝑧₂,…,𝑧_𝑛−1∈𝑋. (23) If 𝜌_𝑡∈𝐴₃ and 𝜌¹_𝑡∈𝐴₄, by using nondecreasing and convexity of the Orlicz function 𝑀_𝑗 for all 𝑗∈ℕ that 𝑀_𝑗Δ^𝑠_𝑚𝜆^𝑡𝑥^𝑡_𝑗−𝜆𝑥_𝑗𝜆^𝑡−𝜆𝜌_𝑡+|𝜆|𝜌¹_𝑡,𝑧₁,𝑧₂,…,𝑧_𝑛−1≤𝑀_𝑗Δ^𝑠_𝑚𝜆^𝑡𝑥^𝑡_𝑗−𝜆𝑥^𝑡_𝑗𝜆^𝑡−𝜆𝜌_𝑡+|𝜆|𝜌¹_𝑡,𝑧₁,𝑧₂,…,𝑧_𝑛−1+Δ^𝑠_𝑚𝜆𝑥^𝑡_𝑗−𝜆𝑥_𝑗𝜆^𝑡−𝜆𝜌_𝑡+|𝜆|𝜌¹_𝑡,𝑧₁,𝑧₂,…,𝑧_𝑛−1≤𝜆^𝑡−𝜆𝜌_𝑡𝜆^𝑡−𝜆𝜌_𝑡+|𝜆|𝜌¹_𝑡𝑀_𝑗Δ^𝑠_𝑚𝑥^𝑡_𝑗𝜌_𝑡,𝑧₁,𝑧₂,…,𝑧_𝑛−1+|𝜆|𝜌¹_𝑡𝜆^𝑡−𝜆𝜌_𝑡+|𝜆|𝜌¹_𝑡𝑀_𝑗×Δ^𝑠_𝑚𝑥^𝑡_𝑗−𝑥_𝑗𝜌¹_𝑡,𝑧₁,𝑧₂,…,𝑧_𝑛−1. (24) From the above inequality, it follows that sup_𝑗𝑀_𝑗Δ^𝑠_𝑚𝜆^𝑡𝑥^𝑡_𝑗−𝜆𝑥_𝑗𝜆^𝑡−𝜆𝜌_𝑡+|𝜆|𝜌¹_𝑡,𝑧₁,𝑧₂,…,𝑧_𝑛−1≤1, (25) 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^{𝑝𝑗/𝐻}∶𝜌_𝑡∈𝐴₃,𝜌¹_𝑡∈𝐴₄≤𝜆^𝑡−𝜆^𝑚𝑠_𝑗=1𝑥^𝑡_𝑗,𝑧₁,𝑧₂,…,𝑧_𝑛−1+𝜆^𝑡−𝜆^{𝑝𝑗/𝐻}inf𝜌_𝑡^{𝑝𝑗/𝐻}∶𝜌_𝑡∈𝐴₃+|𝜆|^𝑚𝑠_𝑗=1𝜆^𝑡𝑥^𝑡_𝑗−𝜆𝑥_𝑗,𝑧₁,𝑧₂,…,𝑧_𝑛−1+|𝜆|^{𝑝𝑗/𝐻}inf𝜌¹_𝑡^{𝑝𝑗/𝐻}∶𝜌¹_𝑡∈𝐴₄≤max𝜆^𝑡−𝜆,𝜆^𝑡−𝜆^{𝑝𝑗/𝐻}𝑔_Δ𝑥^𝑡+max|𝜆|,|𝜆|^{𝑝𝑗/𝐻}𝑔_Δ𝑥^𝑡−𝑥. (26) Note that 𝑔_Δ(𝑥^𝑡)≤𝑔_Δ(𝑥)+𝑔_Δ(𝑥^𝑡−𝑥), for all 𝑡∈ℕ. Hence, by our assumption, the right hand of (26) tends to 0 as 𝑡→∞, 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 Musielak-Orlicz functions. Then, the following hold:
(a)  𝑉[𝜆, ℳ^′, ‖⋅,…,⋅‖, 𝑝, Δ^𝑠_𝑚]^𝐼₀⊆𝑉[𝜆, ℳ⋅ℳ^′, ‖⋅,…,⋅‖, 𝑝, Δ^𝑠_𝑚]^𝐼₀, provided 𝑝 = (𝑝_𝑘) be such that 𝐺₀ = inf𝑝_𝑘 > 0,
(b)  𝑉[𝜆, ℳ^′, ‖⋅,…,⋅‖, 𝑝, Δ^𝑠_𝑚]^𝐼₀∩𝑉[𝜆, ℳ^′′, ‖⋅,…,⋅‖, 𝑝, Δ^𝑠_𝑚]^𝐼₀⊆𝑉[𝜆, ℳ^′ + ℳ^′′, ‖⋅,…,⋅‖, 𝑝, Δ^𝑠_𝑚]^𝐼₀.

Proof. (a) Let 𝜀>0 be given. Choose 𝜀₁>0 such that max{𝜀^𝐺₁,𝜀^𝐺0₁}<𝜀. Using the continuity of the Orlicz function 𝑀, choose 0<𝛿<1 such that 0<𝑡<𝛿 implies that 𝑀(𝑡)<𝜀₁. Let 𝑥=(𝑥_𝑘) be any element in 𝑉[𝜆,ℳ^′,‖⋅,…,⋅‖,𝑝,Δ^𝑛_𝑚]^𝐼₀, put 𝐴_𝛿=𝑘∈ℕ∶𝜆_𝑘⁻¹_{𝑗∈𝐼𝑘}𝑀^′_𝑗Δ^𝑠_𝑚𝑥_𝑗𝜌₁,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗≥𝛿^𝐺. (27) Then, by definition of ideal convergent, we have the set 𝐴_𝛿∈𝐼. If 𝑛∉𝐴_𝛿 then we have 𝜆_𝑘⁻¹_{𝑗∈𝐼𝑘}𝑀^′_𝑗Δ^𝑠_𝑚𝑥_𝑗𝜌₁,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗<𝛿^𝐺⟹𝑀^′_𝑗Δ^𝑠_𝑚𝑥_𝑗𝜌₁,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗<𝜆_𝑘𝛿^𝐺,∀𝑗∈𝐼_𝑘⟹𝑀^′_𝑗Δ^𝑠_𝑚𝑥_𝑗𝜌₁,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗<𝛿^𝐺⟹𝑀^′_𝑗Δ^𝑠_𝑚𝑥_𝑗𝜌₁,𝑧₁,𝑧₂,…,𝑧_𝑛−1<𝛿. (28) Using the continuity of the Orlicz function 𝑀_𝑗 for all 𝑗 and the relation (28), we have 𝑀_𝑗𝑀^′_𝑗Δ^𝑠_𝑚𝑥_𝑗𝜌₁,𝑧₁,𝑧₂,…,𝑧_𝑛−1<𝜀₁,∀𝑗∈𝐼_𝑘. (29) Consequently, we get _{𝑗∈𝐼𝑘}𝑀_𝑗𝑀^′_𝑗Δ^𝑠_𝑚𝑥_𝑗𝜌₁,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗<𝜆_𝑗max𝜀^𝐺₁,𝜀^𝐺0₁<𝜆_𝑗𝜀⟹𝜆_𝑗⁻¹_{𝑗∈𝐼𝑘}𝑀_𝑗𝑀^′_𝑗Δ^𝑠_𝑚𝑥_𝑗𝜌₁,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗<𝜀. (30) This shows that 𝑘∈ℕ∶_{𝑗∈𝐼𝑘}𝑀_𝑗𝑀^′_𝑗Δ^𝑠_𝑚𝑥_𝑗𝜌₁,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗≥𝜀⊆𝐴_𝛿∈𝐼. (31)
This proves the assertion.
(b) Let 𝑥=(𝑥_𝑘) be any element in 𝑉[𝜆,ℳ^′,‖⋅,…,⋅‖,𝑝,Δ^𝑠_𝑚]^𝐼₀∩𝑉[𝜆,ℳ^′′,‖⋅,…,⋅‖,𝑝,Δ^𝑠_𝑚]^𝐼₀. Then, by the following inequality, the results follow: 𝜆_𝑘⁻¹_{𝑗∈𝐼𝑘}𝑀^′_𝑗+𝑀_𝑗^′′Δ^𝑠_𝑚𝑥_𝑗𝜌₁,z₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗≤𝐷𝜆_𝑘⁻¹_{𝑗∈𝐼𝑘}𝑀^′_𝑗Δ^𝑠_𝑚𝑥_𝑗𝜌₁,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗+𝐷𝜆_𝑘⁻¹_{𝑗∈𝐼𝑘}𝑀_𝑗^′′Δ^𝑠_𝑚𝑥_𝑗𝜌₁,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗. (32)

Theorem 13. The inclusions 𝑍[𝜆,ℳ,‖⋅,…,⋅‖,Δ_𝑚^𝑠−1]⊆𝑍[𝜆,ℳ,‖⋅,…,⋅‖,Δ^𝑠_𝑚] are strict for 𝑠,𝑚≥1 in general where 𝑍=𝑉^𝐼, 𝑉^𝐼₀, and 𝑉^𝐼_∞.

Proof. We will give the proof for 𝑉[𝜆,ℳ,‖⋅,…,⋅‖,Δ_𝑚^𝑠−1]^𝐼₀⊆𝑉[𝜆,ℳ,‖⋅,…,⋅‖,Δ^𝑠_𝑚]^𝐼₀ only. The others can be proved by similar arguments. Let 𝑥=(𝑥_𝑘)∈𝑉[𝜆,ℳ,‖⋅,…,⋅‖,Δ_𝑚^𝑠−1]^𝐼₀. Then let 𝜀>0 be given; there exists 𝜌>0 such that 𝑘∈ℕ∶𝜆_𝑘⁻¹_{𝑗∈𝐼𝑘}𝑀_𝑗Δ_𝑚^𝑠−1𝑥_𝑗𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1≥𝜀2∈𝐼. (33) Since 𝑀_𝑗 for all 𝑗∈ℕ is nondecreasing and convex, it follows that 𝜆_𝑘⁻¹_{𝑗∈𝐼𝑘}𝑀_𝑗Δ^𝑠_𝑚𝑥_𝑗2𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1=𝜆_𝑘⁻¹_{𝑗∈𝐼𝑘}𝑀_𝑗Δ_𝑚^𝑠−1𝑥_𝑗+1−Δ_𝑚^𝑠−1𝑥_𝑗2𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1≤𝜆_𝑘⁻¹_{𝑗∈𝐼𝑘}12𝑀_𝑗Δ_𝑚^𝑠−1𝑥_𝑗+1𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1+𝜆_𝑘⁻¹_{𝑗∈𝐼𝑘}12𝑀_𝑗Δ_𝑚^𝑠−1𝑥_𝑗𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1≤12𝜆_𝑘⁻¹_{𝑗∈𝐼𝑘}𝑀_𝑗Δ_𝑚^𝑠−1𝑥_𝑗+1𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1+12𝜆_𝑘⁻¹_{𝑗∈𝐼𝑘}𝑀_𝑗Δ_𝑚^𝑠−1𝑥_𝑗𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1, (34) then we have 𝑘∈ℕ∶𝜆_𝑘⁻¹_{𝑗∈𝐼𝑘}𝑀_𝑗Δ^𝑠_𝑚𝑥_𝑗2𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1⊆1≥𝜀𝑘∈ℕ∶2𝜆_𝑘⁻¹_{𝑗∈𝐼𝑘}𝑀_𝑗Δ_𝑚^𝑠−1𝑥_𝑗+1𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1≥𝜀2∪1𝑘∈ℕ∶2𝜆_𝑘⁻¹_{𝑗∈𝐼𝑘}𝑀_𝑗Δ_𝑚^𝑠−1𝑥_𝑗𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1≥𝜀2. (35) Let 𝑀_𝑘(𝑥)=𝑀(𝑥)=𝑥 for all 𝑥∈[0,∞[, 𝑘∈ℕ and 𝜆_𝑘=𝑘 for all 𝑘∈ℕ. Consider a sequence 𝑥=(𝑥_𝑘)=(𝑘^𝑠). Then, 𝑥∈𝑉[𝜆,ℳ,‖⋅,…,⋅‖,Δ^𝑠_𝑚]^𝐼₀ but does not belong to 𝑉[𝜆,ℳ,‖⋅,…,⋅‖,Δ_𝑚^𝑠−1]^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 𝑉[𝜆,ℳ,‖⋅,…,⋅‖,𝑝,Δ^𝑠_𝑚]_∞⊆𝑉[𝜆,ℳ,‖⋅,…,⋅‖,𝑞,Δ^𝑠_𝑚]_∞.

Proof. Let 𝑥=(𝑥_𝑗)∈𝑉[𝜆,ℳ,‖⋅,…,⋅‖,𝑝,Δ^𝑠_𝑚]_∞, then there exists some 𝜌>0 such that sup_𝑘𝜆_𝑘⁻¹_{𝑗∈𝐼𝑘}𝑀_𝑗Δ^𝑠_𝑚𝑥_𝑗𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗<∞. (36) This implies that 𝑀_𝑗Δ^𝑠_𝑚𝑥_𝑗𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1<1, (37) for sufficiently large value of 𝑗. Since 𝑀_𝑗 for all 𝑗∈ℕ is nondecreasing, we get sup_𝑘𝜆_𝑘⁻¹_{𝑗∈𝐼𝑘}𝑀_𝑗Δ^𝑠_𝑚𝑥_𝑗𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑞𝑗≤sup_𝑘𝜆_𝑘⁻¹_{𝑗∈𝐼𝑘}𝑀_𝑗Δ^𝑠_𝑚𝑥_𝑗𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗<∞. (38) Thus, 𝑥∈[𝜆,ℳ,‖⋅,…,⋅‖,𝑞,Δ^𝑠_𝑚]_∞. This completes the proof of the theorem.

Theorem 15. (i) If 0<inf𝑝_𝑘≤𝑝_𝑘 < 1, then  𝑉[𝜆, ℳ, ‖⋅,…,⋅‖, 𝑝, Δ^𝑠_𝑚]_∞⊆𝑉[𝜆, ℳ, ‖⋅,…,⋅‖, Δ^𝑠_𝑚]_∞.
(ii) If 0<𝑝_𝑘≤sup_𝑘𝑝_𝑘<∞, then 𝑉[𝜆, ℳ, ‖⋅,…,⋅‖, Δ^𝑠_𝑚]_∞⊆𝑉[𝜆, ℳ, 𝑝, ‖⋅,…,⋅‖, Δ^𝑠_𝑚]_∞.

Theorem 16. For any sequence of Orlicz functions ℳ=(𝑀_𝑗) which satisfies Δ₂-condition, one has 𝑉[𝜆,‖⋅,…,⋅‖,𝑝,Δ^𝑠_𝑚]^𝐼⊂𝑉[𝜆,ℳ,‖⋅,…,⋅‖,𝑝,Δ^𝑠_𝑚]^𝐼.

Theorem 17. Let 0<𝑝_𝑛≤𝑞_𝑛<1 and (𝑞_𝑛/𝑝_𝑛) be bounded; then 𝑉𝜆,ℳ,‖⋅,…,⋅‖,𝑞,Δ^𝑠_𝑚^𝐼⊆𝑉𝜆,ℳ,‖⋅,…,⋅‖,𝑝,Δ^𝑠_𝑚^𝐼. (39)

Theorem 18. For any two sequences 𝑝=(𝑝_𝑘) and 𝑞=(𝑞_𝑘) of positive real numbers and for any two n-norms ‖⋅,…,⋅‖₁ and ‖⋅,…,⋅‖₂ on 𝑋, the following holds: 𝑍𝜆,ℳ,‖⋅,…,⋅‖₁,𝑝,Δ^𝑠_𝑚∩𝑍𝜆,ℳ,‖⋅,…,⋅‖₂,𝑞,Δ^𝑠_𝑚≠𝜙, (40) where 𝑍=𝑉^𝐼,  𝑉^𝐼₀,  𝑉^𝐼_∞, and 𝑉_∞.

Proof. 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 𝑉[𝜆,ℳ,‖⋅,…,⋅‖,𝑝,Δ^𝑠_𝑚]^𝐼, 𝑉[𝜆,ℳ,‖⋅,…,⋅‖,𝑝,Δ^𝑠_𝑚]^𝐼₀, 𝑉[𝜆,ℳ,‖⋅,…,⋅‖,𝑝,Δ^𝑠_𝑚]_∞, and 𝑉[𝜆,ℳ,‖⋅,…,⋅‖,𝑝,Δ^𝑠_𝑚]^𝐼_∞ are neither solid nor symmetric, nor sequence algebras for 𝑠,𝑚≥1 in general.

Proof. The proof is obtained by using the same techniques of Et [26, Theorems 3.6, 3.8, and 3.9].

Remark 20. If we replace the difference operator Δ^𝑠_𝑚 by Δ_𝑚^(𝑠), then for each 𝜀>0 we get the following sequence spaces: 𝑉𝜆,ℳ,‖⋅,…,⋅‖,𝑝,Δ_𝑚^(𝑠)𝐼=𝑥∈𝜔(𝑛−𝑋)∶𝑘∈ℕ∶𝜆_𝑘⁻¹×_{𝑗∈𝐼𝑘}𝑀_𝑗Δ_𝑚^(𝑠)𝑥_𝑗−𝑙𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗≥𝜀∈𝐼,forsome𝜌>0,𝑙∈𝑋andeach𝑧₁,𝑧₂,…,𝑧_𝑛−1,𝑉‖‖∈𝑋𝜆,ℳ,⋅,…,⋅,𝑝,Δ_𝑚^(𝑠)𝐼₀=𝑥∈𝜔(𝑛−𝑋)∶𝑘∈ℕ∶𝜆_𝑘⁻¹_{𝑗∈𝐼𝑘}𝑀_𝑗Δ_𝑚^(𝑠)𝑥_𝑗𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1^𝑝𝑗≥𝜀∈𝐼,forsome𝜌>0andeach𝑧₁,𝑧₂,…,𝑧_𝑛−1,𝑉‖‖∈X𝜆,ℳ,⋅,…,⋅,𝑝,Δ_𝑚^(𝑠)_∞=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,𝑉‖‖∈𝑋𝜆,ℳ,⋅,…,⋅,𝑝,Δ_𝑚^(𝑠)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. (41)

Note. It is clear from definitions that 𝑉[𝜆,ℳ,‖⋅,…,⋅‖,𝑝,Δ_𝑚^(𝑠)]^𝐼₀⊆𝑉[𝜆,ℳ,‖⋅,…,⋅‖,𝑝,Δ_𝑚^(𝑠)]^𝐼⊆𝑉[𝜆,ℳ,‖⋅,…,⋅‖,𝑝,Δ_𝑚^(𝑠)]^𝐼_∞.

Corollary 21. The sequence spaces 𝑍[𝜆,ℳ,‖⋅,…,⋅‖,𝑝,Δ_𝑚^(𝑠)], where 𝑍=𝑉^𝐼,𝑉^𝐼₀, 𝑉^𝐼_∞, and 𝑉_∞ are paranormed spaces (not totally paranormed) with respect to the paranorm ℎ_Δ defined byℎ_Δ(𝑥)=inf𝜌^{𝑝𝑘/𝐻}∶sup_𝑗𝑀_𝑗Δ_𝑚^(𝑠)𝑥_𝑗𝜌,𝑧₁,𝑧₂,…,𝑧_𝑛−1≤1,forsome𝜌>0,andeach𝑧₁,𝑧₂,…,𝑧_𝑛−1∈𝑋, (42)
where 𝐻=max{1,sup_𝑘𝑝_𝑘} and 𝑍=𝑉^𝐼,𝑉^𝐼₀, 𝑉^𝐼_∞, and 𝑉_∞. Also it is clear that the paranorm 𝑔_Δ and ℎ_Δ are equivalent. We state the following theorem in view of Lemma 9. Let 𝑋 be a standard 𝑛-normed space and {𝑒₁,𝑒₂,…,𝑒_𝑛} an orthogonal set in 𝑋. Then, the following hold:(a)𝑉[𝜆, ℳ, ‖⋅,…,⋅‖_∞, 𝑝, Δ_𝑚^(𝑠)]^𝐼 = 𝑉[𝜆, ℳ, ‖⋅,…,⋅‖_𝑛−1, 𝑝, Δ_𝑚^(𝑠)]^𝐼; (b)𝑉[𝜆, ℳ, ‖⋅,…,⋅‖_∞, 𝑝, Δ_𝑚^(𝑠)]^𝐼₀ = 𝑉[𝜆, ℳ, ‖⋅,…,⋅‖_𝑛−1, 𝑝, Δ_𝑚^(𝑠)]^𝐼₀;(c)𝑉[𝜆, ℳ, ‖⋅,…,⋅‖_∞, 𝑝, Δ_𝑚^(𝑠)]_∞ = 𝑉[𝜆, ℳ, ‖⋅,…,⋅‖_𝑛−1, 𝑝, Δ_𝑚^(𝑠)]_∞;(d)𝑉[𝜆, ℳ, ‖⋅,…,⋅‖_∞, 𝑝, Δ_𝑚^(𝑠)]^𝐼_∞ = 𝑉[𝜆, ℳ, ‖⋅,…,⋅‖_𝑛−1, 𝑝, Δ_𝑚^(𝑠)]^𝐼_∞,
where ‖⋅,…,⋅‖_∞ is the derived (𝑛−1)-norm defined with respect to the set {𝑒₁,𝑒₂,…,𝑒_𝑛} and ‖⋅,…,⋅‖_𝑛−1 is the standard (𝑛−1)-norm on 𝑋.

Theorem 22. The spaces 𝑍[𝜆,ℳ,‖⋅,…,⋅‖_∞,𝑝,Δ_𝑚^(𝑠)] and 𝑍[𝜆,ℳ,‖⋅,…,⋅‖_∞,𝑝] are equivalent as topological spaces, where 𝑍=𝑉^𝐼,𝑉^𝐼₀, 𝑉^𝐼_∞, and 𝑉_∞.

Proof. Consider the mapping 𝑇∶𝑍[𝜆,ℳ, ‖⋅,…,⋅‖_∞, 𝑝, Δ_𝑚^(𝑠)]→𝑍[𝜆, ℳ, ‖⋅,…,⋅‖_∞, 𝑝] defined by 𝑇(𝑥) = (Δ_𝑚^(𝑠)𝑥_𝑘) for each 𝑥 = (𝑥_𝑘)∈𝑍[𝜆, ℳ, ‖⋅,…,⋅‖_∞, 𝑝, Δ_𝑚^(𝑠)]. Then, clearly 𝑇 is a linear homeomorphism and the proof follows.

Acknowledgments

The authors are 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 this paper.