Wijsman regularly ideal convergence of double sequences of sets

Abstract

In this paper, we introduce the notions of regularly $(I_{W_{2}}, I_{W})$ -convergence, regularly $(I_{W_{2}}^{*}, I_{W}^{*})$ -convergence, regularly $(I_{W_{2}}, I_{W})$ -Cauchy and regularly $(I_{W_{2}}^{*}, I_{W}^{*})$ -Cauchy double sequence of sets and investigate the relationship among them.

Keywords

Regularly ideal convergence,double sequence of sets,Wijsman convergence

1 Introduction and background

Throughout the paper, $ℕ$ and $ℝ$ denote the set of all positive integers and the set of all real numbers, respectively. The concept of convergence of real sequences has been extended to statistical convergence independently by Fast [15] and Schoenberg [30]. This concept was extended to the double sequences by Mursaleen and Edely [20].

The idea of $I$ -convergence was introduced by Kostyrko et al. [18] as a generalization of statistical convergence. Das et al. [6] introduced the concept of $I$ -convergence of double sequences in a metric space and studied some properties of this convergence. Tripathy and Tripathy [32] studied on $I$ -convergent and regularly $I$ -convergent double sequences. Dündar and Altay [7] introduced $I_{2}$ -convergence and regularly $I$ -convergence of double sequences. Also, Dündar [12] introduced regularly $I$ -convergence and regularly $I$ -Cauchy double sequences of functions. A lot of development have been made in this area after the works of [8–11 , 31].

The concept of convergence of sequences of numbers has been extended by several authors to convergence of sequences of sets (see, [3–5 , 37]). Nuray and Rhoades [22] extended the notion of convergence of sequences of sets to statistical convergence and gave some basic theorems. Nuray et. al. [24] studied Wijsman statistical convergence, Hausdorff statistical convergence and Wijsman statistical Cauchy double sequences of sets and investigate the relationship between them. Kişi and Nuray [17] introduced a new convergence notion, for sequences of sets, which is called Wijsman $I$ -convergence. Nuray et al. [25] studied $I_{2}$ -convergence and $I_{2}$ -Cauchy double sequences of sets.

Now, we recall the basic definitions and concepts (See [1–7 , 32–37]).

Let X≠ ∅. A class $I$ of subsets of X is said to be an ideal in X provided: (i) $\emptyset \in I$ , (ii) $A, B \in I$ implies $A \cup B \in I$ , (iii) $A \in I$ , B ⊂ A implies $B \in I$ .

$I$ is called a nontrivial ideal if $X \notin I$ . A nontrivial ideal $I$ in X is called admissible if ${x} \in I$ for each x ∈ X.

Let X≠ ∅. A non empty class $F$ of subsets of X is said to be a filter in X provided: (i) $\emptyset \notin F$ , (ii) $A, B \in F$ implies $A \cap B \in F$ , (iii) $A \in F$ , A ⊂ B implies $B \in F$ .

If $I$ is a nontrivial ideal in X, X≠ ∅, then the class $F (I) = {M \subset X : (\exists A \in I) (M = X ∖ A)}$ is a filter on X, called the filter associated with $I$ .

An admissible ideal $I \subset 2^{N}$ is said to satisfy the property (AP), if for every countable family of mutually disjoint sets {A₁, A₂, . . .} belonging to $I$ , there exists a countable family of sets {B₁, B₂, . . .} such that A_jΔB_j is a finite set for $j \in ℕ$ and $B = ⋃_{j = 1}^{\infty} B_{j} \in I$ .

A nontrivial ideal $I_{2}$ of $ℕ \times ℕ$ is called strongly admissible if ${i} \times ℕ$ and $ℕ \times {i}$ belong to $I_{2}$ for each i ∈ N. It is evident that a strongly admissible ideal is admissible also.

$I_{2}^{0} = {A \subset N \times N : (\exists m (A) \in N) (i, j \geq m (A) \Rightarrow$ (i, j) ∉ A)}. Then $I_{2}^{0}$ is a nontrivial strongly admissible ideal and clearly an ideal $I_{2}$ is strongly admissible if and only if $I_{2}^{0} \subset I_{2}$ .

We say that an admissible ideal $I_{2} \subset 2^{N \times N}$ satisfies the property (AP2) if for every countable family of mutually disjoint sets {A₁, A₂, . . .} belonging to $I_{2}$ , there exists a countable family of sets {B₁, B₂, . . .} such that $A_{j} Δ B_{j} \in I_{2}^{0}$ , i.e., A_jΔB_j is included in the finite union of rows and columns in $ℕ \times ℕ$ for each $j \in ℕ$ and $B = ⋃_{j = 1}^{\infty} B_{j} \in I_{2}$ (hence $B_{j} \in I_{2}$ for each $j \in ℕ) .$

Throughout the paper, we let $I \subseteq 2^{N}$ be an admissible ideal, (X, ρ) be a separable metric space and A, A_m be any non-empty closed subsets of X.

For any point x ∈ X and any non-empty subset A of X, we define the distance from x to A by $\begin{matrix} d (x, A) = inf_{a \in A} ρ (x, a) . \end{matrix}$ A sequence {A_m} is Wijsman convergent to A if for each x ∈ X, $\begin{matrix} lim_{m \to \infty} d (x, A_{m}) = d (x, A) . \end{matrix}$ A sequence {A_m} is Wijsman Cauchy sequence if for every ɛ > 0 and each x ∈ X, there is a positive integer k₀ = k₀ (ɛ, x) such that for all m, n > k₀, $\begin{matrix} | d (x, A_{m}) - d (x, A_{n}) | < ɛ . \end{matrix}$

A sequence {A_m} is Wijsman $I$ -convergent to A if for every ɛ > 0 and each x ∈ X, $A (x, ε) = {m \in N : | d (x, A_{m}) - d (x, A) | \geq ε} \in I .$ A sequence {A_m} is Wijsman $I^{*}$ -convergent to A if and only if there exists a set $M \in F (I)$ , $M = {m_{1}, m_{2}, \dots, m_{k}, \dots} \subset ℕ$ such that for each x ∈ X, $\begin{matrix} lim_{m \to \infty} d (x, A_{m_{k}}) = d (x, A) . \end{matrix}$

A sequence {A_m} is Wijsman $I$ -Cauchy sequence if for every ɛ and each x ∈ X, there exists a number N = N (ɛ) such that $\begin{matrix} {m \in ℕ : | d (x, A_{m}) - d (x, A_{N}) | \geq ɛ} \in I . \end{matrix}$ A sequence {A_m} is Wijsman $I^{*}$ -Cauchy sequence if there exists a set $M \in F (I)$ , $M = {m_{1}, m_{2}, \dots, m_{k}, \dots} \subset ℕ$ such that the subsequence A_M = {A_{m
_k}} is Wijsman Cauchy in X, that is, $\begin{matrix} lim_{k, p \to \infty} | d (x, A_{m_{k}}) - d (x, A_{m_{p}}) | = 0 . \end{matrix}$ Throughout the paper, we let $I_{2} \subseteq 2^{N \times N}$ be a strongly admissible ideal, A_mn be any non-empty closed subsets of X.

A double sequence {A_mn} is Wijsman convergent to A if for each x ∈ X, $\begin{matrix} lim_{m, n \to \infty} d (x, A_{mn}) = d (x, A) . \end{matrix}$

A double sequence {A_mn} is $I_{W_{2}}$ -convergent to A if for every ɛ > 0 and each x ∈ X, $\begin{matrix} {(m, n) \in ℕ \times ℕ : | d (x, A_{mn}) - d (x, A) | \geq ɛ} \in I_{2} . \end{matrix}$ It is denoted by $I_{W_{2}} - lim_{m, n \to \infty} d (x, A_{mn}) = d (x, A) .$

A double sequence {A_mn} is $I_{W_{2}}^{*}$ -convergent to A if there exists a set $M_{2} \in F (I_{2})$ (i.e., $ℕ \times ℕ ∖ M_{2} = H \in I_{2}$ ) such that for each x ∈ X $\begin{matrix} \underset{(m, n) \in M_{2}}{lim_{m, n \to \infty}} d (x, A_{mn}) = d (x, A) . \end{matrix}$ It is denoted by $I_{W_{2}}^{*} - lim_{m, n \to \infty} d (x, A_{mn}) = d (x, A) .$

A double sequence {A_mn} is $I_{W_{2}}$ -Cauchy sequence if for every ɛ > 0 and each x ∈ X, there exists p = p (ɛ) and q = q (ɛ) in $ℕ$ such that ${(m, n) \in ℕ \times ℕ : | d (x, A_{mn}) - d (x, A_{pq}) | \geq ɛ} \in I_{2} .$

A double sequence {A_mn} is $I_{W_{2}}^{*}$ -Cauchy sequence if there exists a set $M_{2} \in F (I_{2})$ (i.e., $ℕ \times ℕ ∖ M_{2} = H \in I_{2}$ ) such that for each x ∈ X and (m, n) , (p, q) ∈ M₂, $\begin{matrix} lim_{m, n, p, q \to \infty} | d (x, A_{mn}) - d (x, A_{pq}) | = 0 . \end{matrix}$ A double sequence x = (x_mn) is said to be regularly $(I_{2}, I)$ -convergent ( $r (I_{2}, I)$ -convergent) if it is $I_{2}$ -convergent in Pringsheim’s sense and for every ɛ > 0, the followings hold: $\begin{matrix} {m \in ℕ : | x_{mn} - K_{n} | \geq ɛ} \in I, \end{matrix}$ for some K_n ∈ X and each $n \in ℕ$ , $\begin{matrix} {n \in ℕ : | x_{mn} - L_{m} | \geq ɛ} \in I, \end{matrix}$ for some L_m ∈ X and each $m \in ℕ$ .

A double sequence x = (x_mn) is said to be $r (I_{2}^{*}, I^{*})$ -convergent if there exist the sets $M \in F (I_{2})$ , $M_{1} \in F (I)$ and $M_{2} \in F (I)$ (i.e., $ℕ \times ℕ \ M \in I_{2}$ , $ℕ \ M_{1} \in I$ and $ℕ \ M_{2} \in I$ ) such that the limits $\begin{matrix} \underset{(m, n) \in M}{lim_{m, n \to \infty}} x_{mn}, \underset{m \in M_{1}}{lim_{m \to \infty}} x_{mn}, (n \in ℕ) and \\ \underset{n \in M_{2}}{lim_{n \to \infty}} x_{mn} (m \in ℕ) \end{matrix}$ exist. Note that if x = (x_mn) is regularly convergent to L then the limits $lim_{m \to \infty} lim_{n \to \infty} x_{mn}$ and $lim_{n \to \infty} lim_{m \to \infty} x_{mn}$ exist and are equal to L.

A double sequence x = (x_mn) is said to be regularly $(I_{2}, I)$ -Cauchy ( $r (I_{2}, I)$ -Cauchy) if it is $I_{2}$ -Cauchy in Pringsheim’s sense and for every ɛ > 0, there exist k_n = k_n (ɛ) , l_m = l_m (ɛ) $\in ℕ$ such that the followings hold: $\begin{matrix} A_{1} (ɛ) = {m \in ℕ : ρ (x_{mn}, x_{k_{n} n}) \geq ɛ} \in I, (n \in ℕ), \\ A_{2} (ɛ) = {n \in ℕ : ρ (x_{mn}, x_{{ml}_{m}}) \geq ɛ} \in I, (m \in ℕ) . \end{matrix}$ A double sequence x = (x_mn) is said to be regularly $(I_{2}^{*}, I^{*})$ -Cauchy ( $r (I_{2}^{*}, I^{*})$ -Cauchy) if there exist the sets $M \in F (I_{2})$ , $M_{1} \in F (I)$ and $M_{2} \in F (I)$ (i.e., $ℕ \times ℕ \ M \in I_{2}$ , $ℕ \ M_{1} \in I$ and $ℕ \ M_{2} \in I$ ) and for every ɛ > 0, there exist $N = N (ɛ) \in ℕ$ , s = s (ɛ), t = t (ɛ), k_n = k_n (ɛ), $l_{m} = l_{m} (ɛ) \in ℕ$ such that whenever m, n, k_n, l_m > N, we have $\begin{matrix} ρ (x_{mn}, x_{st}) < ɛ, ((m,n),(s,t) ∈ M, m,n,s,t > N), \end{matrix}$ $\begin{matrix} ρ (x_{mn}, x_{k_{n} n}) < ɛ, (each m \in M_{1} and each n \in ℕ), \end{matrix}$ $\begin{matrix} ρ (x_{mn}, x_{{ml}_{m}}) < ɛ, (each n \in M_{2} and each m \in ℕ) . \end{matrix}$

1.1 Lemma [25, Theorem 3.1]

If a double sequence {A_mn} is $I_{W_{2}}^{*}$ -convergent, then it is $I_{W_{2}}$ -convergent.

1.2 Lemma [25, Theorem 3.2]

If the ideal $I_{2}$ has the property (AP2), then $I_{W_{2}}$ -convergence implies $I_{W_{2}}^{*}$ -convergence of double sequence of sets.

1.3 Lemma [25, Theorem 3.3]

If a double sequence {A_mn} is $I_{W_{2}}$ -convergent, then it is $I_{W_{2}}$ -Cauchy.

1.4 Lemma [25, Theorem 3.4]

If a double sequence {A_mn} is $I_{W_{2}}^{*}$ -Cauchy, then it is $I_{W_{2}}$ -Cauchy.

2 Main results

In this section, we introduce the notions of regularly $(I_{W_{2}}, I_{W})$ -convergence, regularly $(I_{W_{2}}^{*}, I_{W}^{*})$ -convergence, regularly $(I_{W_{2}}, I_{W})$ -Cauchy and regularly $(I_{W_{2}}^{*}, I_{W}^{*})$ -Cauchy double sequence of sets and investigate the relationship among them.

2.1 Lemma

Let {A_mn} be a double sequence of sets. Then, for each x ∈ X $\begin{matrix} lim_{m, n \to \infty} d (x, A_{mn}) = d (x, A) implies \\ I_{2} - lim_{m, n \to \infty} d (x, A_{mn}) = d (x, A) . \end{matrix}$

Proof. Let $lim_{m, n \to \infty} d (x, A_{mn}) = d (x, A) .$ Then, for every ɛ > 0 and each x ∈ X there exists $k_{0} = k_{0} (ɛ, x) \in ℕ$ such that $| d (x, A_{mn}) - d (x, A) | < ɛ,$ for all m, n > k₀. Therefore, for each x ∈ X we have $\begin{matrix} A (ɛ) & = & {(m, n) \in ℕ \times ℕ : | d (x, A_{mn}) - d (x, A) | \geq ɛ} \\ \subset & (ℕ \times {1, 2, \dots, k_{0}} \cup {1, 2, \dots, k_{0}} \times ℕ) . \end{matrix}$ Since $I_{2}$ is a strongly admissible ideal, we have $\begin{matrix} (ℕ \times {1, 2, \dots, k_{0}} \cup {1, 2, \dots, k_{0}} \times ℕ) \in I_{2} \end{matrix}$ and so, $A (ɛ) \in I_{2}$ . Hence, this completes the proof.

2.2 Definition

A double sequence {A_mn} is said to be Wijsman regularly convergent (R (W₂, W)-convergent) if it is convergent in Pringsheim’s sense and for each x ∈ X the limits

$\begin{matrix} lim_{m \to \infty} d (x, A_{mn}), (n \in ℕ) and lim_{n \to \infty} d (x, A_{mn}), (m \in ℕ) \end{matrix}$

exist. Note that if {A_mn} is Wijsman regularly convergent to A, then the limits $\begin{matrix} lim_{n \to \infty} lim_{m \to \infty} d (x, A_{mn}) = d (x, A) and \\ lim_{m \to \infty} lim_{n \to \infty} d (x, A_{mn}) = d (x, A) \end{matrix}$ exist. In this case, we write $\begin{matrix} R (W_{2}, W) - lim_{m, n \to \infty} d (x, A_{mn}) = d (x, A) or \\ A_{mn} \overset{R (W_{2}, W)}{⟶} A . \end{matrix}$

2.3 Definition

A double sequence {A_mn} is said to be regularly $(I_{W_{2}}, I_{W})$ -convergent ( $R (I_{W_{2}}, I_{W})$ -convergent) if it is $I_{W_{2}}$ -convergent in Pringsheim’s sense and for every ɛ > 0 and each x ∈ X, the followings hold: $\begin{matrix} {m \in ℕ : | d (x, A_{mn}) - d (x, K_{n}) | \geq ɛ} \in I, \end{matrix}$ for some K_n ∈ X and each $n \in ℕ$ , $\begin{matrix} {n \in ℕ : | d (x, A_{mn}) - d (x, L_{m}) | \geq ɛ} \in I, \end{matrix}$ for some L_m ∈ X and each $m \in ℕ .$

Note that if {A_mn} is $R (I_{W_{2}}, I_{W})$ -convergent to A, then we write $\begin{matrix} R (I_{W_{2}}, I_{W}) - lim_{m, n \to \infty} d (x, A_{mn}) = d (x, A) or \\ A_{mn} \overset{R (I_{W_{2}}, I_{W})}{⟶} A . \end{matrix}$

2.4 Theorem

If a double sequence {A_mn} is R (W₂, W)-convergent, then {A_mn} is $R (I_{W_{2}}, I_{W})$ -convergent.

Proof. Let {A_mn} is R (W₂, W)-convergent to A. Then, {A_mn} is convergent to A in Pringsheim’s sense and for each x ∈ X the limits $\begin{matrix} lim_{m \to \infty} d (x, A_{mn}) = d (x, A) (n \in ℕ) and \\ lim_{n \to \infty} d (x, A_{mn}) = d (x, A) (m \in ℕ) \end{matrix}$ exist. By Lemma 2.1, we have {A_mn} is $I_{W_{2}}$ -convergent. Also, for every ɛ > 0 and each x ∈ X there exist m = m₀ (ɛ, x) and n = n₀ (ɛ, x) such that $\begin{matrix} | d (x, A_{mn}) - d (x, A) | < ɛ, \end{matrix}$ for each fixed $n \in ℕ$ and every m > m₀, $\begin{matrix} | d (x, A_{mn}) - d (x, B) | < ɛ, \end{matrix}$ for each fixed $m \in ℕ$ and every n > n₀ .

Since $I$ is an admissible ideal, for every ɛ > 0 and each x ∈ X we have $\begin{matrix} {m \in ℕ : d (x, A_{mn}) - d (x, A) \geq ɛ} \subset {1, \dots, m_{0}} \in I \end{matrix}$ and $\begin{matrix} {n \in ℕ : d (x, A_{mn}) - d (x, B) \geq ɛ} \subset {1, \dots, n_{0}} \in I . \end{matrix}$ Hence, {A_mn} is $R (I_{W_{2}}, I_{W})$ -convergent.

2.5 Definition

A double sequence {A_mn} is said to be regularly $(I_{W_{2}}^{*}, I_{W}^{*})$ -convergent ( $R (I_{W_{2}}^{*}, I_{W}^{*})$ -convergent) if there exist the sets $M \in F (I_{2})$ , $M_{1} \in F (I)$ and $M_{2} \in F (I)$ (i.e., $ℕ \times ℕ \ M \in I_{2}$ , $ℕ \ M_{1} \in I$ and $ℕ \ M_{2} \in I$ ) such that the limits $\begin{matrix} \underset{(m, n) \in M}{lim_{m, n \to \infty}} d (x, A_{mn}), \underset{m \in M_{1}}{lim_{m \to \infty}} d (x, A_{mn}) and \\ \underset{n \in M_{2}}{lim_{n \to \infty}} d (x, A_{mn}) \end{matrix}$ exist for each fixed $n \in ℕ$ and each fixed $m \in ℕ$ , respectively.

Note that if {A_mn} is $R (I_{W_{2}}^{*}, I_{W}^{*})$ -convergent to A, then for each x ∈ X the limits $\begin{matrix} lim_{n \to \infty} lim_{m \to \infty} d (x, A_{mn}) and lim_{m \to \infty} lim_{n \to \infty} d (x, A_{mn}) \end{matrix}$ exist and are equal to d(x, A). In this case, we write $\begin{matrix} R (I_{W_{2}}, I_{W}) - lim_{m, n \to \infty} d (x, A_{mn}) = d (x, A) or \\ A_{mn} \overset{R (I_{W_{2}}, I_{W})}{⟶} A . \end{matrix}$

2.6 Theorem

If a double sequence {A_mn} is $R (I_{W_{2}}^{*}, I_{W}^{*})$ -convergent, then {A_mn} is $R (I_{W_{2}}, I_{W})$ -convergent.

Proof. Assume that {A_mn} is $R (I_{W_{2}}^{*}, I_{W}^{*})$ -convergent. Then, it is $I_{W_{2}}^{*}$ -convergent and so, by Lemma 1.1, it is $I_{W_{2}}$ -convergent. Also, there exist the sets $M_{1}, M_{2} \in F (I)$ such that for every ɛ > 0 and each x ∈ X, $\begin{matrix} (\forall ɛ > 0) (\exists m_{0} \in ℕ) (\forall m \geq m_{0}) (m \in M_{1}) \\ | d (x, A_{mn}) - d (x, K_{n}) | < ɛ, \end{matrix}$ for some K_n ∈ X and each $n \in ℕ$ , $\begin{matrix} (\forall ɛ > 0) (\exists n_{0} \in ℕ) (\forall n \geq n_{0}) (n \in M_{2}) \\ | d (x, A_{mn}) - d (x, L_{m}) | < ɛ, \end{matrix}$ for some L_m ∈ X and each $m \in ℕ .$ Hence, for every ɛ > 0 and each x ∈ X we have $\begin{matrix} A (ɛ) & = & {m \in ℕ : | d (x, A_{mn}) - d (x, K_{n}) | \geq ɛ} \\ \subset & H_{1} \cup {1, 2, \dots, (m_{0} - 1)}, (n \in ℕ), \end{matrix}$ $\begin{matrix} B (ɛ) & = & {n \in ℕ : | d (x, A_{mn}) - d (x, L_{m}) | \geq ɛ} \\ \subset & H_{2} \cup {1, 2, \dots, (n_{0} - 1)}, (m \in ℕ), \end{matrix}$ for $H_{1}, H_{2} \in I$ . Since $I$ is an admissible ideal, we have $\begin{matrix} H_{1} \cup {1, 2, \dots, (m_{0} - 1)} \in I, \end{matrix}$ and $\begin{matrix} H_{2} \cup {1, 2, \dots, (n_{0} - 1)} \in I \end{matrix}$ and therefore $A (ɛ) \in I$ and $B (ɛ) \in I$ . This shows that {A_mn} is $R (I_{W_{2}}, I_{W})$ -convergent.

2.7 Theorem

Let $I \subset 2^{N}$ be an admissible ideal with property (AP) and $I_{2} \subset 2^{N \times N}$ be a strongly admissible ideal with property (AP2). If a double sequence {A_mn} is $R (I_{W_{2}}, I_{W})$ -convergent, then {A_mn} is $R (I_{W_{2}}^{*}, I_{W}^{*})$ -convergent.

Proof. Let {A_mn} be $R (I_{W_{2}}, I_{W})$ -convergent. Then, {A_mn} is $I_{W_{2}}$ -convergent and so, by Lemma 1.2, {A_mn} is $I_{W_{2}}^{*}$ -convergent. Also, for every ɛ > 0 and each x ∈ X we have $\begin{matrix} A (ɛ) = {m \in ℕ : | d (x, A_{mn}) - d (x, K_{n}) | \geq ɛ} \in I, \end{matrix}$ for some K_n ∈ X and each $n \in ℕ$ , and $\begin{matrix} B (ɛ) = {n \in ℕ : | d (x, A_{mn}) - d (x, L_{m}) | \geq ɛ} \in I, \end{matrix}$ for some L_m ∈ X and each $m \in ℕ .$ Now, for each x ∈ X put

$\begin{matrix} A_{1} = {m \in ℕ : | d (x, A_{mn}) - d (x, K_{n}) | \geq 1} and \end{matrix}$ $\begin{matrix} A_{k} = {m \in ℕ : \frac{1}{k} \leq | d (x, A_{mn}) - d (x, K_{n}) | < \frac{1}{k - 1}} \end{matrix}$

for k ≥ 2, for some K_n ∈ X and for each $n \in ℕ$ . It is clear that A_i∩ A_j = ∅ for i ≠ j and $A_{i} \in I$ for each $i \in ℕ$ . By the property (AP), there is a countable family of sets {B₁, B₂, …} in $I$ such that A_j △ B_j is a finite set for each $j \in ℕ$ and $B = ⋃_{j = 1}^{\infty} B_{j} \in I$ .

We prove that for some K_n and each $n \in ℕ$ , $\begin{matrix} \underset{m \in M}{lim_{m \to \infty}} d (x, A_{mn}) = d (x, K_{n}), \end{matrix}$ for $M = ℕ ∖ B \in F (I)$ and each x ∈ X. Let δ > 0 be given. Choose $k \in ℕ$ such that 1/k < δ. Then, for each x ∈ X we have $\begin{matrix} {m \in ℕ : | d (x, A_{mn}) - d (x, K_{n}) | \geq δ} \subset ⋃_{j = 1}^{k} A_{j}, \end{matrix}$ for some K_n ∈ X and each $n \in ℕ .$ Since A_j △ B_j is a finite set for j ∈ {1, 2, …, k}, there exists $m_{0} \in ℕ$ such that $\begin{matrix} (⋃_{j = 1}^{k} B_{j}) \cap {m : m \geq m_{0}} \\ = (⋃_{j = 1}^{k} A_{j}) \cap {m : m \geq m_{0}} . \end{matrix}$ If m ≥ m₀ and m ∉ B then $\begin{matrix} m \notin ⋃_{j = 1}^{k} B_{j} and so m \notin ⋃_{j = 1}^{k} A_{j} . \end{matrix}$ Thus, for every δ > 0 and each x ∈ X we have $\begin{matrix} | d (x, A_{mn}) - d (x, K_{n}) | < \frac{1}{k} < δ, \end{matrix}$ for some K_n ∈ X and each $n \in ℕ .$ This implies that $lim_{m \to \infty} d (x, A_{mn}) = d (x, K_{n}),$ for m ∈ M . Hence, for each x ∈ X we have $\begin{matrix} I_{W}^{*} - lim_{m \to \infty} d (x, A_{mn}) = d (x, K_{n}), \end{matrix}$ for some K_n ∈ X and each $n \in ℕ$ . Similarly, for the set $\begin{matrix} B (ɛ) = {n \in ℕ : | d (x, A_{mn}) - d (x, L_{m}) | \geq ɛ} \in I, \end{matrix}$ we have $\begin{matrix} I_{W}^{*} - lim_{n \to \infty} d (x, A_{mn}) = d (x, L_{m}), \end{matrix}$ for some L_m ∈ X and each $m \in ℕ,$ for each x ∈ X Hence, {A_mn} is $R (I_{W_{2}}^{*}, I_{W}^{*})$ -convergent.

Now, we give the definitions of $R (I_{W_{2}}, I_{W})$ -Cauchy sequence and $R (I_{W_{2}}^{*}, I_{W}^{*})$ -Cauchy sequence.

2.8 Definition

A double sequence {A_mn} is said to be regularly $(I_{W_{2}}, I_{W})$ -Cauchy ( $R (I_{W_{2}}, I_{W})$ -Cauchy) if it is $I_{2}$ -Cauchy in Pringsheim’s sense and for every ɛ > 0 and each x ∈ X, there exist $k_{n} = k_{n} (ɛ, x) \in ℕ, l_{m} = l_{m} (ɛ, x) \in ℕ$ such that the following statements hold: $\begin{matrix} {m \in ℕ : | d (x, A_{mn}) - d (x, A_{k_{n} n}) | \geq ɛ} \in I, (n \in ℕ) \end{matrix}$ and $\begin{matrix} {n \in ℕ : | d (x, A_{mn}) - d (x, A_{{ml}_{m}}) | \geq ɛ} \in I, (m \in ℕ) . \end{matrix}$

2.9 Theorem

If a double sequence {A_mn} is $R (I_{W_{2}}, I_{W})$ -convergent, then {A_mn} is $R (I_{W_{2}}, I_{W})$ -Cauchy double sequence.

Proof. Let {A_mn} be $R (I_{W_{2}}, I_{W})$ -convergent. Then, {A_mn} is $I_{W_{2}}$ -convergent and by Lemma 1.4, it is $I_{W_{2}}$ -Cauchy double sequence. Also, for every ɛ > 0 and each x ∈ X the following sets belong to the $I$ $\begin{matrix} A_{1} (\frac{ɛ}{2}) = {m \in ℕ : | d (x, A_{mn}) - d (x, K_{n}) | \geq \frac{ɛ}{2}}, \end{matrix}$ for some K_n ∈ X and each $n \in ℕ,$ and $\begin{matrix} A_{2} (\frac{ɛ}{2}) = {n \in ℕ : | d (x, A_{mn} - d (x, L_{m}) | \geq \frac{ɛ}{2}} \end{matrix}$ for some L_m ∈ X and each $m \in ℕ$ , belong to the $I$ .

Since $I$ is an admissible ideal, for each x ∈ X the sets $\begin{matrix} A_{1}^{c} (\frac{ɛ}{2}) = {m \in ℕ : | d (x, A_{mn}) - d (x, K_{n}) | < \frac{ɛ}{2}}, \end{matrix}$ for some K_n ∈ X and each $n \in ℕ,$ $\begin{matrix} A_{2}^{c} (\frac{ɛ}{2}) = {n \in ℕ : | d (x, A_{mn}) - d (x, L_{m}) | < \frac{ɛ}{2}}, \end{matrix}$ for some L_m ∈ X and each $m \in ℕ$ , are nonempty and belong to $F (I)$ . For $k_{n} \in A_{1}^{c} (\frac{ɛ}{2})$ ( $n \in ℕ$ and k_n > 0), we have $\begin{matrix} | d (x, A_{k_{n} n}) - d (x, K_{n}) | < \frac{ɛ}{2}, \end{matrix}$ for some K_n ∈ X and each $n \in ℕ .$ Now, for every ɛ > 0 and each x ∈ X we define the set $\begin{matrix} B_{1} (ɛ) = {m \in ℕ : | d (x, A_{mn}) - d (x, A_{k_{n} n}) | \geq ɛ}, \end{matrix}$ ( $n \in ℕ$ ), where $k_{n} = k_{n} (ɛ) \in ℕ$ . Let m ∈ B₁ (ɛ). Then, for $k_{n} \in A_{1}^{c} (\frac{ɛ}{2})$ ( $n \in ℕ$ and k_n > 0) we have $\begin{matrix} ɛ & \leq & | d (x, A_{mn}) - d (x, A_{k_{n} n}) | \\ \leq & | d (x, A_{mn}) - d (x, K_{n}) | \\ + | d (x, A_{k_{n} n}) - d (x, K_{n}) | \\ < & | d (x, A_{mn}) - d (x, K_{n}) | + \frac{ɛ}{2}, \end{matrix}$ for some K_n ∈ X and each x ∈ X. This shows that $\begin{matrix} \frac{ɛ}{2} < | d (x, A_{mn}) - d (x, K_{n}) | \end{matrix}$ and so $m \in A_{1} (\frac{ɛ}{2}) .$ Hence, we have $B_{1} (ɛ) \subset A_{1} (\frac{ɛ}{2})$ .

Similarly, for every ɛ > 0 and each x ∈ X, and for $l_{m} \in A_{2}^{c} (\frac{ɛ}{2})$ ( $m \in ℕ$ and l_m > 0) we have $\begin{matrix} | d (x, A_{{ml}_{m}}) - d (x, L_{m}) | < \frac{ɛ}{2}, \end{matrix}$ for some L_m ∈ X and each $m \in ℕ .$ Therefore, for each x ∈ X, it can be seen that $\begin{matrix} B_{2} (ɛ) & = & {m \in ℕ : | d (x, A_{{ml}_{m}}) - d (x, L_{m}) | \geq ɛ} \\ \subset & A_{2} (\frac{ɛ}{2}) . \end{matrix}$ Hence, we have $B_{1} (ɛ) \in I$ and $B_{2} (ɛ) \in I$ , and so {A_mn} is $R (I_{W_{2}}, I_{W})$ -Cauchy double sequence.

2.10 Definition

A double sequence {A_mn} is said to be regularly $(I_{W_{2}}^{*}, I_{W}^{*})$ -Cauchy ( $R (I_{W_{2}}^{*}, I_{W}^{*})$ -Cauchy) if there exist the sets $M \in F (I_{2})$ , $M_{1} \in F (I)$ and $M_{2} \in F (I)$ (i.e., $ℕ \times ℕ \ M \in I_{2}$ , $ℕ \ M_{1} \in I$ and $ℕ \ M_{2} \in I$ ), and for every ɛ > 0 and each x ∈ X there exist N = N (ɛ, x), s = s (ɛ, x), t = t (ɛ, x), (s, t) ∈ M, k_n = k_n (ɛ), l_m = l_m (ɛ) $\in ℕ$ such that whenever m, n, s, t, k_n, l_m ≥ N, we have $\begin{matrix} | d (x, A_{mn}) - d (x, A_{st}) | < ɛ, for (m, n), (s, t) \in M, \end{matrix}$ $\begin{matrix} | d (x, A_{mn}) - d (x, A_{k_{n} n}) | < ɛ, \end{matrix}$ for each m ∈ M₁ and each $n \in ℕ$ , and $\begin{matrix} | d (x, A_{mn}) - d (x, A_{{ml}_{m}} | < ɛ, \end{matrix}$ for each n ∈ M₂ and each $m \in ℕ .$

2.11 Theorem

If a double sequence {A_mn} is $R (I_{W_{2}}^{*}, I_{W}^{*})$ -Cauchy then it is $R (I_{W_{2}}, I_{W})$ -Cauchy.

Proof. Since a double sequence {A_mn} is $R (I_{W_{2}}^{*}, I_{W}^{*})$ -Cauchy, it is $I_{W_{2}}^{*}$ -Cauchy. We know that $I_{W_{2}}^{*}$ -Cauchy implies $I_{W_{2}}$ -Cauchy by Lemma 1.3. Also, since the double sequence {A_mn} is $R (I_{W_{2}}^{*}, I_{W}^{*})$ -Cauchy, there exist the sets $M_{1}, M_{2} \in F (I)$ , and for every ɛ > 0 and each x ∈ X there exist $k_{n} = k_{n} (ɛ, x) \in ℕ$ and $l_{m} = l_{m} (ɛ, x) \in ℕ$ such that $\begin{matrix} | d (x, A_{mn}) - d (x, A_{k_{n} n}) | & < & ɛ, \end{matrix}$ for each m ∈ M₁ and each $n \in ℕ,$ $\begin{matrix} | d (x, A_{mn}) - d (x, A_{{ml}_{m}}) | & < & ɛ, \end{matrix}$ for each n ∈ M₂, each $m \in ℕ,$ $N = N (ɛ) \in ℕ$ and m, n, k_n, l_m ≥ N. Therefore, for $H_{1} = ℕ ∖ M_{1} \in I$ and $H_{2} = ℕ ∖ M_{2} \in I$ we have $\begin{matrix} A_{1} (ɛ) & = & {m \in ℕ : | d (x, A_{mn}) - d (x, A_{k_{n} n}) | \geq ɛ} \\ \subset & H_{1} \cup {1, 2, \dots, (N - 1)}, (n \in ℕ) \end{matrix}$ for m ∈ M₁ and $\begin{matrix} A_{2} (ɛ) & = & {n \in ℕ : | d (x, A_{mn}) - d (x, A_{{ml}_{m}}) | \geq ɛ} \\ \subset & H_{2} \cup {1, 2, \dots, (N - 1)}, (m \in ℕ) \end{matrix}$ for n ∈ M₂. Since $I$ is an admissible ideal, $H_{1} \cup {1, 2, \dots, (N - 1)} \in I$ and H₂ ∪ {1, 2, …, $N - 1} \in I .$ Hence, we have $A_{1} (ɛ) \in I$ and $A_{2} (ɛ) \in I$ , and so {A_mn} is $R (I_{W_{2}}, I_{W})$ -Cauchy double sequence.

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