On Paranorm Zweier 𝐼-Convergent Sequence Spaces

Abstract

In this paper, we introduce the paranorm Zweier 𝐼-convergent sequence spaces 𝒵^𝐼(𝑞), 𝒵⁰_𝐼(𝑞), and 𝒵^𝐼_∞(𝑞)𝒵^𝐼_∞(𝑞)for𝑞=(𝑞_𝑘), a sequence of positive real numbers. We study some topological properties, prove the decomposition theorem, and study some inclusion relations on these spaces.

1. Introduction

Let ℕ,ℝ, and ℂ be the sets of all natural, real, and complex numbers, respectively. We write 𝜔=𝑥=𝑥_𝑘∶𝑥_𝑘∈ℝorℂ, (1) the space of all real or complex sequences.

Let 𝑙_∞, 𝑐, and 𝑐₀ denote the Banach spaces of bounded, convergent, and null sequences, respectively, normed by ||𝑥||_∞=sup_𝑘|𝑥_𝑘|.

The following subspaces of 𝜔 were first introduced and discussed by Maddox [1]: 𝑙(𝑝)∶={𝑥∈𝜔∶∑_𝑘|𝑥_𝑘|^𝑝𝑘<∞},  𝑙_∞(𝑝)∶={𝑥∈𝜔∶sup_𝑘|𝑥_𝑘|^𝑝𝑘<∞},  𝑐(𝑝)∶={𝑥∈𝜔∶lim_𝑘|𝑥_𝑘−𝑙|^𝑝𝑘=0,forsome𝑙∈ℂ},  𝑐₀(𝑝)∶={𝑥∈𝜔∶lim_𝑘|𝑥_𝑘|^𝑝𝑘=0}, where 𝑝=(𝑝_𝑘) is a sequence of strictly positive real numbers.

After that Lascarides [2, 3] defined the following sequence spaces: 𝑙_∞𝑝=𝑥∈𝜔∶thereexists𝑟>0suchthatsup_𝑘𝑥_𝑘𝑟^𝑝𝑘𝑡_𝑘,𝑐<∞₀𝑝=𝑥∈𝜔∶thereexists𝑟>0suchthatlim_𝑘𝑥_𝑘𝑟^𝑝𝑘𝑡_𝑘,𝑙𝑝==0𝑥∈𝜔∶thereexists𝑟>0suchthat^∞_𝑘=1𝑥_𝑘𝑟^𝑝𝑘𝑡_𝑘<∞, (2)  where 𝑡_𝑘=𝑝_𝑘⁻¹, for all 𝑘∈ℕ.

Each linear subspace of 𝜔, for example, 𝜆,𝜇⊂𝜔, is called a sequence space.

A sequence space 𝜆 with linear topology is called a 𝐾-space provided each map 𝑝_𝑖→ℂ defined by 𝑝_𝑖(𝑥)=𝑥_𝑖 is continuous for all 𝑖∈ℕ.

A 𝐾-space 𝜆 is called an FK-space provided 𝜆 is a complete linear metric space.

An FK-space whose topology is normable is called a BK-space.

Let 𝜆 and 𝜇 be two sequence spaces and 𝐴=(𝑎_𝑛𝑘) an infinite matrix of real or complex numbers 𝑎_𝑛𝑘, where 𝑛,𝑘∈ℕ. Then we say that 𝐴 defines a matrix mapping from 𝜆 to 𝜇, and we denote it by writing 𝐴∶𝜆→𝜇.

If for every sequence 𝑥=(𝑥_𝑘)∈𝜆 the sequence 𝐴𝑥={(𝐴𝑥)_𝑛}, the 𝐴 transform of 𝑥 is in 𝜇, where (𝐴𝑥)_𝑛=_𝑘𝑎_𝑛𝑘𝑥_𝑘,(𝑛∈ℕ). (3) By (𝜆∶𝜇), we denote the class of matrices 𝐴 such that 𝐴∶𝜆→𝜇.

Thus, 𝐴∈(𝜆∶𝜇) if and only if series on the right side of (3) converges for each 𝑛∈ℕ and every 𝑥∈𝜆.

The approach of constructing the new sequence spaces by means of the matrix domain of a particular limitation method has been recently employed by Altay et al. [4], Başar and Altay [5], Malkowsky [6], Ng and Lee [7], and Wang [8].

Şengönül[9] defined the sequence 𝑦=(𝑦_𝑖) which is frequently used as the 𝑍^𝑝 transform of the sequence 𝑥=(𝑥_𝑖), that is, 𝑦_𝑖=𝑝𝑥_𝑖+1−𝑝𝑥_𝑖−1, (4) where 𝑥₋₁=0,𝑝≠1, 1<𝑝<∞ and 𝑍^𝑝 denotes the matrix 𝑍^𝑝=(𝑧_𝑖𝑘) defined by 𝒵_𝑖𝑘=(),();(),𝑝,𝑖=𝑘1−𝑝,𝑖−1=𝑘𝑖,𝑘∈ℕ0,otherwise. (5) Following Başar and Altay [5], Şengönül[9] introduced the Zweier sequence spaces 𝒵 and 𝒵₀ as follows: 𝒵=𝑥=𝑥_𝑘∈𝜔∶𝑍^𝑝,𝒵𝑥∈𝑐₀=𝑥=𝑥_𝑘∈𝜔∶𝑍^𝑝𝑥∈𝑐₀. (6) Here we quote below some of the results due to Şengönül [9] which we will need in order to establish the results of this paper.

Theorem 1 (see [9, Theorem  2.1]). The sets 𝒵 and 𝒵₀ are the linear spaces with the coordinate wise addition and scalar multiplication which are the BK-spaces with the norm ‖𝑥‖_𝒵=‖𝑥‖_𝒵0=𝑍^𝑝𝑥_𝑐. (7)

Theorem 2 (see [9, Theorem  2.2]). The sequence spaces 𝒵 and 𝒵₀ are linearly isomorphic to the spaces 𝑐 and 𝑐₀, respectively, that is, 𝒵≅𝑐 and 𝒵₀≅𝑐₀.

Theorem 3 (see [9, Theorem  2.3]). The inclusions 𝒵₀⊂𝒵 strictly hold for 𝑝≠1.

Theorem 4 (see [9, Theorem  2.6]). 𝒵₀ is solid.

Theorem 5 (see [9, Theorem  3.6]). 𝒵 is not a solid sequence space.

The concept of statistical convergence was first introduced by Fast [10] and also independently by Buck [11] and Schoenberg [12] for real and complex sequences. Further this concept was studied by Connor [13, 14], Connor et al. [15], and many others. Statistical convergence is a generalization of the usual notion of convergence that parallels the usual theory of convergence. A sequence 𝑥=(𝑥_𝑘) is said to be statistically convergent to 𝐿 if for a given 𝜖>0 as lim_𝑘1𝑘𝑖∶𝑥_𝑖−𝐿≥𝜖,𝑖≤𝑘=0. (8) The notion of 𝐼-convergence is a generalization of the statistical convergence. At the initial stage, it was studied by Kostyrko et al. [16]. Later on, it was studied by Šalát et al. [17, 18], Demirci [19], Tripathy and Hazarika [20, 21], and Khan et al. [22–24].

Here we give some preliminaries about the notion of 𝐼-convergence.

Let X be a nonempty set. Then a family of sets 𝐼⊆2^𝑋 (denoting the power set of 𝑋) is said to be an ideal if 𝐼 is additive, that is, 𝐴,𝐵∈𝐼⇒𝐴∪𝐵∈𝐼, and hereditary, that is, 𝐴∈𝐼, 𝐵⊆𝐴⇒𝐵∈𝐼.

A nonempty family of sets £⊆2^𝑋 is said to be a filter on 𝑋 if and only if 𝜙∉£, for 𝐴,𝐵∈£ we have 𝐴∩𝐵∈£ and for each 𝐴∈£ and 𝐴⊆𝐵 implies 𝐵∈£.

An ideal 𝐼⊆2^𝑋 is called nontrivial if 𝐼≠2^𝑋.

A non-trivial ideal 𝐼⊆2^𝑋 is called admissible if {{𝑥}∶𝑥∈𝑋}⊆𝐼.

A non-trivial ideal 𝐼 is maximal if there cannot exist any non-trivial ideal 𝐽≠𝐼 containing 𝐼 as a subset.

For each ideal 𝐼, there is a filter £(𝐼) corresponding to I. that is, £(𝐼) = {𝐾⊆𝑁∶𝐾^𝑐∈𝐼}, where 𝐾^𝑐=𝑁−𝐾.

Definition 6. A sequence (𝑥_𝑘)∈𝜔 is said to be 𝐼-convergent to a number 𝐿 if {𝑘∈𝑁∶|𝑥_𝑘−𝐿|≥𝜖}∈𝐼 for every 𝜀>0. In this case we write 𝐼−lim𝑥_𝑘=𝐿.
The space 𝑐^𝐼 of all 𝐼-convergent sequences converging to 𝐿 is given by 𝑐^𝐼=𝑥_𝑘∈𝜔∶𝑘∈ℕ∶𝑥_𝑘−𝐿≥𝜖∈𝐼,forsome𝐿∈ℂ. (9)

Definition 7. A sequence (𝑥_𝑘)∈𝜔 is said to be 𝐼-null if 𝐿=0. In this case we write 𝐼−lim𝑥_𝑘=0.

Definition 8. A sequence (𝑥_𝑘)∈𝜔 is said to be 𝐼-Cauchy if for every 𝜖>0 there exists a number 𝑘₀∈ℕ such that {𝑘∈𝑁∶|𝑥_𝑘−𝑥_𝑚|≥𝜖}∈𝐼 for all 𝑘,𝑚≥𝑘₀.

Definition 9. A sequence (𝑥_𝑘)∈𝜔 is said to be 𝐼-bounded if there exists 𝑀>0 such that {𝑘∈𝑁∶|𝑥_𝑘|>𝑀}∈𝐼.

Definition 10. Let (𝑥_𝑘),(𝑦_𝑘) be two sequences. We say that (𝑥_𝑘)=(𝑦_𝑘) for almost all k relative to 𝐼 (a.a.k.r.I), if {𝑘∈ℕ∶𝑥_𝑘≠𝑦_𝑘}∈𝐼.

The following lemma will be used for establishing some results of this paper.

Lemma 11. If  𝐼⊂2^𝑁 and 𝑀⊆𝑁. If 𝑀∉𝐼, then 𝑀∩𝑁∉𝐼 (see [20, 21]) cf. ([17, 18, 20–24]).

Recently Khan and Ebadullah [25] introduced the following classes of sequence spaces: 𝒵^𝐼=𝑘∈ℕ∶𝑥=𝑥_𝑘∈𝜔∶𝐼−lim𝑍^𝑝𝑥=𝐿forsome𝐿𝒵∈𝐼,_𝐼0=𝑘∈ℕ∶𝑥=𝑥_𝑘∈𝜔∶𝐼−lim𝑍^𝑝𝒵𝑥=0∈𝐼,_𝐼∞=𝑘∈ℕ∶𝑥=𝑥_𝑘∈𝜔∶sup_𝑘𝑍^𝑝𝑥<∞∈𝐼, (10) We also denote by 𝑚^𝐼_𝒵=𝒵^𝐼_∞∩𝒵^𝐼,𝑚^𝐼_𝒵0=𝒵^𝐼_∞∩𝒵^𝐼₀. (11)

In this paper we introduce the following classes of sequence spaces: 𝒵^𝐼𝑞=𝑥=𝑥_𝑘∈𝜔∶𝑘∈ℕ∶𝑍^𝑝𝑥−𝐿^𝑞𝑘≥𝜖∈𝐼,forsome;𝒵𝐿∈ℂ_𝐼0𝑞=𝑥=𝑥_𝑘∈𝜔∶𝑘∈ℕ∶𝑍^𝑝𝑥^𝑞𝑘;𝒵≥𝜖∈𝐼_𝐼∞𝑞=𝑥=𝑥_𝑘∈𝜔∶sup_𝑘𝑍^𝑝𝑥^𝑞𝑘<∞. (12) We also denote by 𝑚^𝐼_𝒵𝑞=𝒵^𝐼_∞𝑞∩𝒵^𝐼𝑞,𝑚^𝐼_𝒵0𝑞=𝒵^𝐼_∞𝑞∩𝒵^𝐼₀𝑞, (13) where 𝑞=(𝑞_𝑘), is a sequence of positive real numbers.

Throughout the paper, for the sake of convenience now we will denote by  𝑍^𝑝𝑥=𝑥^/,𝑍^𝑝𝑦=𝑦^/,𝑍^𝑝𝑧=𝑧^/ for all 𝑥,𝑦,𝑧∈𝜔.

2. Main Results

Theorem 12. The classes of sequences 𝒵^𝐼(𝑞),𝒵^𝐼₀(𝑞),𝑚^𝐼_𝒵(𝑞) and 𝑚^𝐼_𝒵0(𝑞) are linear spaces.

Proof. We shall prove the result for the space 𝒵^𝐼(𝑞).
The proof for the other spaces will follow similarly.
Let (𝑥_𝑘),(𝑦_𝑘)∈𝒵^𝐼(𝑞), and let 𝛼,𝛽 be scalars. Then for a given 𝜖>0: we have 𝑘∈ℕ∶𝑥^/_𝑘−𝐿₁^𝑞𝑘≥𝜖2𝑀₁,forsome𝐿₁𝑦∈ℂ∈𝐼𝑘∈ℕ∶^/_𝑘−𝐿₂^𝑞𝑘≥𝜖2𝑀₂,forsome𝐿₂∈ℂ∈𝐼, (14) where 𝑀₁=𝐷⋅max1,sup_𝑘|𝛼|^𝑞𝑘,𝑀₂=𝐷⋅max1,sup_𝑘𝛽^𝑞𝑘,𝐷=max1,2^𝐻−1where𝐻=sup_𝑘𝑞_𝑘≥0. (15) Let 𝐴₁=𝑘∈ℕ∶𝑥^/_𝑘−𝐿₁^𝑞𝑘<𝜖2𝑀₁,forsome𝐿₁𝐴∈ℂ∈𝐼₂=𝑘∈ℕ∶𝑦^/_𝑘−𝐿₂^𝑞𝑘<𝜖2𝑀₂,forsome𝐿₂∈ℂ∈𝐼 (16) be such that 𝐴^𝑐₁,𝐴^𝑐₂∈𝐼.
Then 𝐴₃=𝑘∈ℕ∶𝛼𝑥^/_𝑘+𝛽𝑦^/_𝑘−𝛼𝐿₁+𝛽𝐿₂^𝑞𝑘⊇|𝛼|<𝜖𝑘∈ℕ∶^𝑞𝑘𝑥^/_𝑘−𝐿₁^𝑞𝑘<𝜖2𝑀₁|𝛼|^𝑞𝑘∩𝛽⋅𝐷𝑘∈ℕ∶^𝑞𝑘𝑦^/_𝑘−𝐿₂^𝑞𝑘<𝜖2𝑀₂𝛽^𝑞𝑘⋅𝐷. (17) Thus 𝐴^𝑐₃=𝐴^𝑐₁∩𝐴^𝑐₂∈𝐼. Hence (𝛼𝑥_𝑘+𝛽𝑦_𝑘)∈𝒵^𝐼(𝑞). Therefore 𝒵^𝐼(𝑞) is a linear space. The rest of the result follows similarly.

Theorem 13. Let (𝑞_𝑘)∈𝑙_∞. Then 𝑚^𝐼_𝒵(𝑞) and 𝑚^𝐼_𝒵0(𝑞) are paranormed spaces, paranormed by 𝑔(𝑥)=sup_𝑘|𝑥_𝑘|^{𝑞𝑘/𝑀} where 𝑀=max{1,sup_𝑘𝑞_𝑘}.

Proof. Let 𝑥=(𝑥_𝑘),𝑦=(𝑦_𝑘)∈𝑚^𝐼_𝒵(𝑞).(1)Clearly, 𝑔(𝑥)=0 if and only if 𝑥=0.(2)𝑔(𝑥)=𝑔(−𝑥) is obvious.(3)Since 𝑞_𝑘/𝑀≤1 and 𝑀>1, using Minkowski’s inequality, we have sup_𝑘𝑥_𝑘+𝑦_𝑘^{𝑞𝑘/𝑀}≤sup_𝑘𝑥_𝑘^{𝑞𝑘/𝑀}+sup_𝑘𝑦_𝑘^{𝑞𝑘/𝑀} (18)(4)Now for any complex 𝜆, we have (𝜆_𝑘) such that 𝜆_𝑘→𝜆, (𝑘→∞).
Let 𝑥_𝑘∈𝑚^𝐼_𝒵(𝑞) such that |𝑥_𝑘−𝐿|^𝑞𝑘≥𝜖.
Therefore,  𝑔(𝑥−𝐿𝑒)=sup_𝑘|𝑥_𝑘−𝐿|^{𝑞𝑘/𝑀}≤sup_𝑘|𝑥_𝑘|^{𝑞𝑘/𝑀}+sup_𝑘|𝐿|^{𝑞𝑘/𝑀},  where 𝑒=(1,1,1,…).
Hence 𝑔((𝜆_𝑛𝑥_𝑘−𝜆𝐿))≤𝑔((𝜆_𝑛𝑥_𝑘))+𝑔(𝜆𝐿)=𝜆_𝑛𝑔(𝑥)+𝜆𝑔(𝐿) as (𝑘→∞).
Hence 𝑚^𝐼_𝒵(𝑞) is a paranormed space.
The rest of the result follows similarly.

Theorem 14. 𝑚^𝐼_𝒵(𝑞) is a closed subspace of 𝑙_∞(𝑞).

Proof. Let (𝑥_𝑘^(𝑛)) be a Cauchy sequence in 𝑚^𝐼_𝒵(𝑞) such that 𝑥^(𝑛)→𝑥.
We show that 𝑥∈𝑚^𝐼_𝒵(𝑞).
Since (𝑥_𝑘^(𝑛))∈𝑚^𝐼_𝒵(𝑞), then there exists 𝑎_𝑛 such that 𝑘∈ℕ∶𝑥^(𝑛)−𝑎_𝑛≥𝜖∈𝐼. (19) We need to show that (1)(𝑎_𝑛) converges to 𝑎,(2)if 𝑈={𝑘∈ℕ∶|𝑥_𝑘−𝑎|<𝜖}, then 𝑈^𝑐∈𝐼.
(1) Since (𝑥_𝑘^(𝑛)) is a Cauchy sequence in 𝑚^𝐼_𝒵(𝑞) then for a given 𝜖>0, there exists 𝑘₀∈ℕsuch that sup_𝑘𝑥_𝑘^(𝑛)−𝑥_𝑘^(𝑖)<𝜖3,∀𝑛,𝑖≥𝑘₀. (20) For a given 𝜀>0, we have 𝐵_𝑛𝑖=𝑘∈ℕ∶𝑥_𝑘^(𝑛)−𝑥_𝑘^(𝑖)<𝜖3,𝐵_𝑖=𝑘∈ℕ∶𝑥_𝑘^(𝑖)−𝑎_𝑖<𝜖3,𝐵_𝑛=𝑘∈ℕ∶𝑥_𝑘^(𝑛)−𝑎_𝑛<𝜖3. (21) Then 𝐵^𝑐_𝑛𝑖,𝐵^𝑐_𝑖,𝐵^𝑐_𝑛∈𝐼.
Let 𝐵^𝑐=𝐵^𝑐_𝑛𝑖∩𝐵^𝑐_𝑖∩𝐵^𝑐_𝑛, where 𝐵={𝑘∈ℕ∶|𝑎_𝑖−𝑎_𝑛|<𝑥𝜖}.
Then 𝐵^𝑐∈𝐼.
We choose 𝑘₀∈𝐵^𝑐, then for each 𝑛,𝑖≥𝑘₀, we have 𝑘∈ℕ∶𝑎_𝑖−𝑎_𝑛<𝜖⊇𝑘∈ℕ∶𝑥_𝑘^(𝑖)−𝑎_𝑖<𝜖3∩𝑘∈ℕ∶𝑥_𝑘^(𝑛)−𝑥_𝑘^(𝑖)<𝜖3∩𝑘∈ℕ∶𝑥_𝑘^(𝑛)−𝑎_𝑛<𝜖3. (22) Then (𝑎_𝑛) is a Cauchy sequence of scalars in ℂ, so there exists a scalar 𝑎∈ℂ such that 𝑎_𝑛→𝑎, as 𝑛→∞.
(2) Let 0<𝛿<1 be given. Then we show that if 𝑈={𝑘∈ℕ∶|𝑥_𝑘−𝑎|^𝑞𝑘<𝛿}, then 𝑈^𝑐∈𝐼.
Since 𝑥^(𝑛)→𝑥, then there exists 𝑞₀∈ℕsuch that 𝑃=𝑘∈ℕ∶𝑥^(𝑞0)<𝛿−𝑥3𝐷^𝑀 (23) which implies that 𝑃^𝑐∈𝐼.
The number 𝑞₀ can be so chosen that together with (23), we have 𝑄=𝑘∈ℕ∶𝑎_𝑞0−𝑎^𝑞𝑘<𝛿3𝐷^𝑀 (24) such that 𝑄^𝑐∈𝐼.
Since {𝑘∈ℕ∶|𝑥^(𝑞0)−𝑎_𝑞0|^𝑞𝑘≥𝛿}∈𝐼. Then we have a subset 𝑆 of ℕsuch that 𝑆^𝑐∈𝐼, where 𝑆=𝑘∈ℕ∶𝑥^(𝑞0)−𝑎_𝑞0^𝑞𝑘<𝛿3𝐷^𝑀. (25) Let 𝑈^𝑐=𝑃^𝑐∩𝑄^𝑐∩𝑆^𝑐, where 𝑈={𝑘∈ℕ∶|𝑥_𝑘−𝑎|^𝑞𝑘<𝛿}.
Therefore for each 𝑘∈𝑈^𝑐, we have 𝑘∈ℕ∶𝑥_𝑘−𝑎^𝑞𝑘⊇𝑥<𝛿𝑘∈ℕ∶^(𝑞0)−𝑥^𝑞𝑘<𝛿3𝐷^𝑀∩𝑘∈ℕ∶𝑥^(𝑞0)−𝑎_𝑞0^𝑞𝑘<𝛿3𝐷^𝑀∩𝑘∈ℕ∶𝑎_𝑞0−𝑎^𝑞𝑘<𝛿3𝐷^𝑀. (26) Then the result follows.

Since the inclusions 𝑚^𝐼_𝒵(𝑞)⊂𝑙_∞(𝑞) and 𝑚^𝐼_𝒵0(𝑞)⊂𝑙_∞(𝑞) are strict, so in view of Theorem 14 we have the following result.

Theorem 15. The spaces 𝑚^𝐼_𝒵(𝑞) and 𝑚^𝐼_𝒵0(𝑞) are nowhere dense subsets of 𝑙_∞(𝑞).

Theorem 16. The spaces 𝑚^𝐼_𝒵(𝑞) and 𝑚^𝐼_𝒵0(𝑞) are not separable.

Proof. We shall prove the result for the space 𝑚^𝐼_𝒵(𝑞).
The proof for the other spaces will follow similarly.
Let 𝑀 be an infinite subset of ℕ of increasing natural numbers such that 𝑀∈𝐼.
Let 𝑞_𝑘=1,if𝑘∈𝑀,2,otherwise. (27) Let 𝑃₀={(𝑥_𝑘)∶𝑥_𝑘=0𝑜𝑟1,for𝑘∈𝑀and𝑥_𝑘=0,otherwise}.
Clearly 𝑃₀ is uncountable.
Consider the class of open balls 𝐵₁={𝐵(𝑧,1/2)∶𝑧∈𝑃₀}.
Let 𝐶₁ be an open cover of 𝑚^𝐼_𝒵(𝑞) containing 𝐵₁.
Since 𝐵₁ is uncountable, so 𝐶₁ cannot be reduced to a countable subcover for 𝑚^𝐼_𝒵(𝑞).
Thus 𝑚^𝐼_𝒵(𝑞) is not separable.

Theorem 17. Let 𝐺=sup_𝑘𝑞_𝑘<∞ and 𝐼 an admissible ideal. Then the following is equivalent.(a)(𝑥_𝑘)∈𝒵^𝐼(𝑞); (b)there exists (𝑦_𝑘)∈𝒵(𝑞) such that 𝑥_𝑘=𝑦_𝑘, for a.a.k.r.I;(c)there exists (𝑦_𝑘)∈𝒵(𝑞) and (𝑥_𝑘)∈𝒵^𝐼₀(𝑞) such that 𝑥_𝑘=𝑦_𝑘+𝑧_𝑘 for all 𝑘∈ℕand {𝑘∈ℕ∶|𝑦_𝑘−𝐿|^𝑞𝑘)≥𝜖}∈𝐼;(d)there exists a subset 𝐾={𝑘₁<𝑘₂<⋯} of ℕsuch that 𝐾∈£(𝐼) and lim_𝑛→∞|𝑥_𝑘𝑛−𝐿|^𝑞𝑘𝑛=0.

Proof. (a) implies (b).
Let (𝑥_𝑘)∈𝒵^𝐼(𝑞). Then there exists 𝐿∈ℂsuch that 𝑘∈ℕ∶𝑥^/_𝑘−𝐿^𝑞𝑘≥𝜖∈𝐼. (28) Let (𝑚_𝑡) be an increasing sequence with 𝑚_𝑡∈ℕ such that 𝑘≤𝑚_𝑡∶𝑥^/_𝑘−𝐿^𝑞𝑘≥𝑡⁻¹∈𝐼. (29) Define a sequence (𝑦_𝑘) as 𝑦_𝑘=𝑥_𝑘,∀𝑘≤𝑚₁. (30) For 𝑚_𝑡<𝑘≤𝑚_𝑡+1,𝑡∈ℕ. 𝑦_𝑘=𝑥_𝑘,if𝑥^/_𝑘−𝐿^𝑞𝑘<𝑡⁻¹,𝐿,otherwise. (31) Then (𝑦_𝑘)∈𝒵(𝑞) and form the following inclusion: 𝑘≤𝑚_𝑡∶𝑥_𝑘≠𝑦_𝑘⊆𝑘≤𝑚_𝑡∶𝑥^/_𝑘−𝐿^𝑞𝑘≥𝜖∈𝐼, (32) we get 𝑥_𝑘=𝑦_𝑘, for a.a.k.r.I.
(b) implies (c).
For (𝑥_𝑘)∈𝒵^𝐼(𝑞). Then there exists (𝑦_𝑘)∈𝒵(𝑞) such that 𝑥_𝑘=𝑦_𝑘, for a.a.k.r.I.
Let 𝐾={𝑘∈ℕ∶𝑥_𝑘≠𝑦_𝑘}, then 𝑘∈𝐼.
Define a sequence (𝑧_𝑘) as 𝑧_𝑘=𝑥_𝑘−𝑦_𝑘,if𝑘∈𝐾,0,otherwise. (33) Then 𝑧_𝑘∈𝒵^𝐼₀(𝑞) and 𝑦_𝑘∈𝒵(𝑞).
(c) implies (d).
Suppose (c) holds.
Let 𝜖>0 be given.
Let 𝑃₁={𝑘∈ℕ∶|𝑧^/_𝑘|^𝑞𝑘≥𝜖}∈𝐼 and 𝐾=𝑃^𝑐₁=𝑘₁<𝑘₂<𝑘₃<⋯∈£(𝐼). (34) Then we have lim_𝑛→∞|𝑥^/_𝑘𝑛−𝐿|^𝑞𝑘𝑛=0.
(d) implies (a).
Let 𝐾={𝑘₁<𝑘₂<𝑘₃<⋯}∈£(𝐼) and lim_𝑛→∞|𝑥^/_𝑘𝑛−𝐿|^𝑞𝑘𝑛=0.
Then for any 𝜖>0, and Lemma 11, we have 𝑘∈ℕ∶𝑥^/_𝑘−𝐿^𝑞𝑘≥𝜖⊆𝐾^𝑐∪𝑘∈𝐾∶𝑥^/_𝑘−𝐿^𝑞𝑘≥𝜖. (35) Thus (𝑥_𝑘)∈𝒵^𝐼(𝑞).

Theorem 18. Let ℎ=inf_𝑘𝑞_𝑘 and 𝐺=sup_𝑘𝑞_𝑘. Then the following results are equivalent.(a)𝐺<∞ and ℎ>0.(b) 𝒵^𝐼₀(𝑞)=𝒵^𝐼₀.

Proof. Suppose that 𝐺<∞ and ℎ>0, then the inequalities min{1,𝑠^ℎ}≤𝑠^𝑞𝑘≤max{1,𝑠^𝐺} hold for any 𝑠>0 and for all 𝑘∈ℕ.
Therefore the equivalence of (a) and (b) is obvious.

Theorem 19. Let (𝑞_𝑘) and (𝑟_𝑘) be two sequences of positive real numbers. Then 𝑚^𝐼_𝒵0(𝑞)⊇𝑚^𝐼_𝒵0(𝑟) if and only if lim_𝑘∈𝐾inf(𝑞_𝑘/𝑟_𝑘)>0, where 𝐾^𝑐⊆ℕsuch that 𝐾∈𝐼.

Proof. Let lim_𝑘∈𝐾inf(𝑞_𝑘/𝑟_𝑘)>0 and (𝑥_𝑘)∈𝑚^𝐼_𝒵0(𝑟). Then there exists 𝛽>0 such that 𝑞_𝑘>𝛽𝑟_𝑘, for all sufficiently large 𝑘∈𝐾.
Since (𝑥_𝑘)∈𝑚^𝐼_𝒵0(𝑟) for a given 𝜖>0, we have 𝐵₀=𝑘∈ℕ∶𝑥_𝑘^𝑟𝑘≥𝜖∈𝐼. (36) Let 𝐺₀=𝐾^𝑐∪𝐵₀ Then 𝐺₀∈𝐼.
Then for all sufficiently large 𝑘∈𝐺₀, 𝑘∈ℕ∶𝑥_𝑘^𝑞𝑘≥𝜖⊆𝑘∈ℕ∶𝑥_𝑘^𝛽𝑟𝑘≥𝜖∈𝐼. (37) Therefore (𝑥_𝑘)∈𝑚^𝐼_𝒵0(𝑞).
The converse part of the result follows obviously.

Theorem 20. Let (𝑞_𝑘) and (𝑟_𝑘) be two sequences of positive real numbers. Then 𝑚^𝐼_𝒵0(𝑟)⊇𝑚^𝐼_𝒵0(𝑞) if and only if lim_𝑘∈𝐾inf(𝑟_𝑘/𝑞_𝑘)>0, where 𝐾^𝑐⊆ℕ such that 𝐾∈𝐼.

Proof. The proof follows similarly as the proof of Theorem 19.

Theorem 21. Let (𝑞_𝑘) and (𝑟_𝑘) be two sequences of positive real numbers. Then 𝑚^𝐼₀(𝑟)=𝑚^𝐼₀(𝑞) if and only if lim_𝑘∈𝐾inf(𝑞_𝑘/𝑟_𝑘)>0, and lim_𝑘∈𝐾inf(𝑟_𝑘/𝑞_𝑘)>0, where 𝐾⊆ℕ such that 𝐾^𝑐∈𝐼.

Proof. By combining Theorems 19 and 20, we get the required result.

Acknowledgment

The authors would like to record their gratitude to the reviewer for his careful reading and making some useful corrections which improved the presentation of this paper.