Deferred Cesàro summability and statistical convergence for double sequences of sets

Abstract

The main purpose of this paper is introduced the concept of deferred Cesàro mean in the Wijsman sense for double sequences of sets and then presented the concepts of strongly deferred Cesàro summability and deferred statistical convergence in the Wijsman sense for double sequences of sets. Also, investigate the relationships between these concepts and then to prove some theorems associated with the concepts of deferred statistical convergence in the Wijsman sense for double sequences of sets is purposed.

Keywords

Deferred Cesàro summability deferred statistical convergence double sequences of sets convergence in the Wijsman sense

1 Introduction and backgrounds

Long after the concept of convergence for double sequences was introduced by Pringsheim [19], Mursaleen and Edely [11] presented the concept of statistical convergence for double sequences. After that, using the concepts of double lacunary sequence, Patterson and Savaş [17] studied on the concept of lacunary statistical convergence for double sequences. See [5 , 22] for more details.

Over the years, on the various convergence concepts for sequences of sets have been studied by many authors. One of them, discussed in this study, is the concept of convergence in the Wijsman sense [3 , 23]. The concepts of convergence and Cesàro summability in the Wijsman sense for double sequences of sets were introduced by Nuray et al. [13] and after that, for double sequences of sets, the concepts of statistical convergence and lacunary statistical convergence in the Wijsman sense were given by Nuray et al. [14, 15].

Long after the concept of deferred Cesàro mean for real (or complex) sequences was introduced by Agnew [1], Küçükaslan and Yılmaztürk [10] presented the concept of deferred statistical convergence for single sequences. After that, for double sequences, the concepts of deferred Cesàro summabiliy and deferred statistical convergence were introduced by Daǧadur and Sezgek [4]. Furthermore, for sequences of sets, the concepts of strongly deferred Cesàro summability and deferred statistical convergence in the Wijsman sense were given by Altınok et al. [2]. See [7–9 , 20] for more details.

Let’s start by giving the basic definitions necessary for a better understanding of our paper.

For a metric space (Y,ρ),d (y,U) denote the distance fromy toU where $d (y, U) : = d_{y} (U) = inf_{u \in U} ρ (y, u)$ for anyy ∈ Y and any non-emptyU ⊆ Y.

For a non-empty setY, let a function $f : ℕ \to 2^{Y}$ (the power set ofY) is defined byf (i) = U_i ∈ 2^Y for each $i \in ℕ$ . Then, the sequence {U_i} = {U₁,U₂, …}, which is the codomain elements off, is called sequences of sets.

Throughout the study, (Y,ρ) will be considered as a metric space andU,U_ij will be considered as any non-empty closed subsets ofY.

A double sequence of sets {U_ij} is said to be convergent in the Wijsman sense to the setU if for eachy ∈ Y, $lim_{i, j \to \infty} d_{y} (U_{ij}) = d_{y} (U)$ and it is denoted by $U_{ij} \overset{W_{2}}{⟶} U$ .

A double sequence of sets {U_ij} is said to be Cesàro summable in the Wijsman sense to the setU if for eachy ∈ Y, $lim_{m, n \to \infty} \frac{1}{mn} \sum_{i, j = 1, 1}^{m, n} d_{y} (U_{ij}) = d_{y} (U) .$

A double sequence of sets {U_ij} is said to be statistically convergent in the Wijsman sense to the setU if for everyɛ > 0 and eachy ∈ Y,

$\begin{matrix} lim_{m, n \to \infty} | {(i, j) : \\ i \leq m, j \leq n : | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} | = 0 . \end{matrix}$

The class of all statistical convergence double sequences of sets in the Wijsman sense is denoted by {W₂S}.

A sequence of sets {U_i} is said to be strongly deferred Cesàro summable in the Wijsman sense to the setU if for eachy ∈ Y, $lim_{m \to \infty} \frac{1}{q (m) - p (m)} \sum_{i = p (m) + 1}^{q (m)} | d_{y} (U_{i}) - d_{y} (U) | = 0,$ where {p (m)} and {q (m)} are sequences of non-negative integers satisfying $p (m) < q (m) and lim_{m \to \infty} q (m) = \infty$ and it is denoted by $U_{i} \overset{[WD]}{⟶} U$ .

A sequence of sets {U_i} is said to be deferred statistically convergent in the Wijsman sense to the setU if for everyɛ > 0 and eachy ∈ Y,

$\begin{matrix} lim_{m \to \infty} \frac{1}{q (m) - p (m)} | {p (m) < i \leq q (m) : \\ | d_{y} (U_{i}) - d_{y} (U) | \geq ɛ} | = 0 \end{matrix}$ and it is denoted by $U_{i} \overset{WDS}{⟶} U$ .

A double sequenceθ₂ = {(k_m,l_n)} is said to be double lacunary sequence if there exist two increasing sequences (k_m) and (l_n) of integers such that

$\begin{matrix} k_{0} = 0, h_{m} = k_{m} - k_{m - 1} \to \infty and \\ l_{0} = 0, {\bar{h}}_{n} = l_{n} - l_{n - 1} \to \infty as m, n \to \infty . \end{matrix}$

2 New concepts

In this section, we first introduced the concept of deferred Cesàro mean in the Wijsman sense for double sequences of sets and then presented the concepts of strongly deferred Cesàro summability and deferred statistical convergence in the Wijsman sense for double sequences of sets.

2.1 Definition

The deferred Cesàro meanD_φ,ψ in the Wijsman sense of a double sequence of sets $U = {U_{ij}}$ is defined by

$\begin{matrix} (D_{φ, ψ} U)_{mn} & = \frac{1}{φ (m) ψ (n)} \sum_{i = p (m) + 1}^{q (m)} \sum_{j = r (n) + 1}^{s (n)} d_{y} (U_{ij}) \\ : = \frac{1}{φ (m) ψ (n)} \underset{j = r (n) + 1}{\sum_{i = p (m) + 1}^{q (m), s (n)}} d_{y} (U_{ij}) \end{matrix}$ for eachy ∈ Y where {p (m)}, {q (m)}, {r (n)} and {s (n)} are sequences of non-negative integers satisfying following conditions:

$p (m) < q (m), lim_{m \to \infty} q (m) = \infty; [- 0.4 cm] [- 0.4 cm] r (n) < s (n), lim_{n \to \infty} s (n) = \infty$ and $q (m) - p (m) = φ (m); s (n) - r (n) = ψ (n) .$ (1)

Throughout the paper, unless otherwise specified, {p (m)}, {q (m)}, {r (n)} and {s (n)} are considered as sequences of non-negative integers satisfying (1) and (2).

2.2 Definition

The double sequence of sets {U_ij} is said to be deferred Cesàro summable in the Wijsman sense to the setU if for eachy ∈ Y, $lim_{m, n \to \infty} \frac{1}{φ (m) ψ (n)} \underset{j = r (n) + 1}{\sum_{i = p (m) + 1}^{q (m), s (n)}} d_{y} (U_{ij}) = d_{y} (U) .$ This case is denoted in $U_{ij} \overset{W_{2} D}{⟶} U$ format.

2.3 Definition

The double sequence of sets {U_ij} is said to be strongly deferred Cesàro summable in the Wijsman sense to the setU if for eachy ∈ Y, $lim_{m, n \to \infty} \frac{1}{φ (m) ψ (n)} \underset{j = r (n) + 1}{\sum_{i = p (m) + 1}^{q (m), s (n)}} | d_{y} (U_{ij}) - d_{y} (U) | = 0 .$ This case is denoted in $U_{ij} \overset{[W_{2} D]}{⟶} U$ format.

The set of all strongly deferred Cesàro summable double sequences of sets in the Wijsman sense is denoted by {[W₂D]}.

2.4 Remark

Forp (m) =0,q (m) = m andr (n) = nolinebreak0,s (n) = n, the concept of strongly deferred Cesàro summability in the Wijsman sense coincides with the concept of Wijsman strongly Cesàro summability for double sequences of sets in [15].

Forp (m) = k_m-1,q (m) = k_m andr (n) = l_n-1,s (n) = l_n where {(k_m,l_n)} is a double lacunary sequence, the concept of strongly deferred Cesàro summability in the Wijsman sense coincides with the concept of Wijsman strongly lacunary summability for double sequences of sets in [14].

2.5 Definition

The double sequence of sets {U_ij} is said to be deferred statistical convergent in the Wijsman sense to the setU if for everyɛ > 0 and eachy ∈ Y,

$\begin{matrix} lim_{m, n \to \infty} \frac{1}{φ (m) ψ (n)} | {(i, j) : p (m) < i \leq q (m), \\ r (n) < j \leq s (n), | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} | = 0 . \end{matrix}$ This case is denoted in $U_{ij} \overset{W_{2} DS}{⟶} U$ format.

The set of all deferred statistical convergent double sequences of sets in the Wijsman sense is denoted by {W₂DS}.

2.6 Remark

Forp (m) =0,q (m) = m andr (n) = nolinebreak0,s (n) = n, the concept of deferred statistical convergence in the Wijsman sense coincides with the concept of Wijsman statistical convergence for double sequences of sets in [15].

Forp (m) = k_m-1,q (m) = k_m andr (n) = l_n-1,s (n) = l_n where {(k_m,l_n)} is a double lacunary sequence, the concept of deferred statistical convergence in the Wijsman sense coincides with the concept of Wijsman lacunary statistical convergence for double sequences of sets in [14].

3 Main results

In this section, we first investigated the relationships between the concepts given in section 2 and then we proved some theorems associated with the concepts of deferred statistical convergence in the Wijsman sense for double sequences of sets.

3.1 Theorem

If a double sequence of sets {U_ij} is strongly deferred Cesàro summable in the Wijsman sense to a setU, then this sequence is deferred statistical convergent in the Wijsman sense to the setU.

Proof. Let $U_{ij} \overset{[W_{2} D]}{⟶} U$ . For everyɛ > 0 and eachy ∈ Y, we have

$\begin{matrix} \underset{j = r (n) + 1}{\sum_{i = p (m) + 1}^{q (m), s (n)}} | d_{y} (U_{ij}) - d_{y} (U) | \\ \geq \underset{| d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ}{j = r (n) + 1 \sum_{i = p (m) + 1}^{q (m), s (n)}} | d_{y} (U_{ij}) - d_{y} (U) | \\ \geq ɛ | {(i, j) : p (m) < i \leq q (m), \\ r (n) < j \leq s (n), | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} | \end{matrix}$ and so

$\begin{matrix} \frac{1}{ɛ} \frac{1}{φ (m) ψ (n)} \underset{i = r (n) + 1}{\sum_{i = p (m) + 1}^{q (m), s (n)}} | d_{y} (U_{ij}) - d_{y} (U) | \\ \geq \frac{1}{φ (m) ψ (n)} | {(i, j) : p (m) < i \leq q (m), \\ r (n) < j \leq s (n), | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} | . \end{matrix}$ Form,n→ ∞, by our assumption, the statement on the left side of the above inequality convergent to 0. Thus, we get $U_{ij} \overset{W_{2} DS}{⟶} U$ .

3.2 Corollary

If $U_{ij} \overset{W_{2}}{⟶} U$ , then $U_{ij} \overset{W_{2} DS}{⟶} U$ .

The converse of Theorem 3.1 is not true in general. We can consider the following example to explain this situation.

3.3 Example

Let $X = ℝ^{2}$ and a double sequences of sets {U_ij} be defined as following: $\begin{matrix} U_{ij} : = {\begin{matrix} {ij} & ; & \begin{matrix} p (m) < i \leq p (m) + [| \sqrt{φ (m)} |] \\ r (n) < j \leq s (n) + [| \sqrt{ψ (n)} |], \\ (i, j) \in ℕ \times ℕ, \end{matrix} [0.5 cm] {0} & ; & otherwise . \end{matrix} \end{matrix}$ This sequence is not bounded. Also, this sequences is deferred statistical convergent in the Wijsman sense to the setU = {0}, but is not strongly deferred Cesàro summable in the Wijsman sense.

A double sequence of sets {U_ij} is said to be bounded if sup d_y (U_ij)< ∞ for eachy ∈ Y. Also, $L_{\infty}^{2}$ denotes the set of all bounded double sequences of sets.

3.4 Theorem

If a double sequence of sets {U_ij} is bounded and deferred statistical convergent in the Wijsman sense to a setU, then this sequence is strongly deferred Cesàro summable in the Wijsman sense to the setU.

Proof. Let {U_ij} is bounded and $U_{ij} \overset{W_{2} DS}{⟶} U$ . Since ${U_{ij}} \in L_{\infty}^{2}$ , there is an $M > 0$ such that $| d_{y} (U_{ij}) - d_{y} (U) | \leq M$ for alli,j and eachy ∈ Y. Thus, for everyɛ > 0 and eachy ∈ Y, we have

$\begin{matrix} \frac{1}{φ (m) ψ (n)} \underset{j = r (n) + 1}{\sum_{i = p (m) + 1}^{q (m), s (n)}} | d_{y} (U_{ij}) - d_{y} (U) | \\ = \frac{1}{φ (m) ψ (n)} \underset{| d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ}{j = r (n) + 1 \sum_{i = p (m) + 1}^{q (m), s (n)}} | d_{y} (U_{ij}) - d_{y} (U) | \\ + \frac{1}{φ (m) ψ (n)} \underset{| d_{y} (U_{ij}) - d_{y} (U) | < ɛ}{j = r (n) + 1 \sum_{i = p (m) + 1}^{q (m), s (n)}} | d_{y} (U_{ij}) - d_{y} (U) | \\ \leq \frac{M}{φ (m) ψ (n)} | {(i, j) : p (m) < i \leq q (m), \\ r (n) < j \leq s (n), | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} | + ɛ . \end{matrix}$ Form,n→ ∞, by our assumption, the statement on the right side of the above inequality convergent to 0. Thus, we get $U_{ij} \overset{[W_{2} D]}{⟶} U$ .

3.5 Corollary

${[W_{2} D]} ⋂ L_{\infty}^{2} = {W_{2} DS} ⋂ L_{\infty}^{2}$ .

3.6 Theorem

Let {T_ij}, {U_ij} and {V_ij} be three double sequences of sets such thatT_ij ⊂ U_ij ⊂ V_ij for all $(i, j) \in ℕ \times ℕ$ . Then, $T_{ij} \overset{W_{2} DS}{⟶} U and V_{ij} \overset{W_{2} DS}{⟶} U \Rightarrow U_{ij} \overset{W_{2} DS}{⟶} U .$

Proof. Suppose that $T_{ij} \overset{W_{2} DS}{⟶} U$ , $V_{ij} \overset{W_{2} DS}{⟶} U$ andT_ij ⊂ U_ij ⊂ V_ij. For all $(i, j) \in ℕ \times ℕ$ and eachy ∈ Y, $T_{ij} \subset U_{ij} \subset V_{ij} \Rightarrow d_{y} (V_{ij}) \leq d_{y} (U_{ij}) \leq d_{y} (T_{ij})$ is hold. Hence, for everyɛ > 0 we have

$\begin{matrix} {(i, j) : p (m) < i \leq q (m), r (n) < j \leq s (n), \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} \\ = {(i, j) : p (m) < i \leq q (m), r (n) < j \leq s (n), \\ d_{y} (U_{ij}) \geq d_{y} (U) + ɛ} \\ \cup {(i, j) : p (m) < i \leq q (m), r (n) < j \leq s (n), \\ d_{y} (U_{ij}) \leq d_{y} (U) - ɛ} \\ \subset {(i, j) : p (m) < i \leq q (m), r (n) < j \leq s (n), \\ d_{y} (T_{ij}) \geq d_{y} (U) + ɛ} \\ \cup {(i, j) : p (m) < i \leq q (m), r (n) < j \leq s (n), \\ d_{y} (V_{ij}) \leq d_{y} (U) - ɛ} \end{matrix}$ and so

$\begin{matrix} \frac{1}{φ (m) ψ (n)} | {(i, j) : p (m) < i \leq q (m), \\ r (n) < j \leq s (n), | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} | \\ < \frac{1}{φ (m) ψ (n)} | {(i, j) : p (m) < i \leq q (m), \\ r (n) < j \leq s (n), | d_{y} (T_{ij}) - d_{y} (U) | \geq ɛ} | \\ + \frac{1}{φ (m) ψ (n)} | {(i, j) : p (m) < i \leq q (m), \\ r (n) < j \leq s (n), | d_{y} (V_{ij}) - d_{y} (U) | \geq ɛ} | . \end{matrix}$ Form,n→ ∞, by our assumption, the statements on the right side of the above inequality convergent to 0. Thus, we get $U_{ij} \overset{W_{2} DS}{⟶} U$ .

3.7 Theorem

Let ${\frac{p (m)}{φ (m)}}$ and ${\frac{r (n)}{ψ (n)}}$ be bounded. Then, $U_{ij} \overset{W_{2} S}{⟶} U \Rightarrow U_{ij} \overset{W_{2} DS}{⟶} U .$

Proof. Suppose thatU_ijoversetW₂S ⟶ U. Then, for everyɛ > 0 and eachy ∈ Y we have

$\begin{matrix} lim_{m, n \to \infty} \frac{1}{mn} | {(i, j) : i \leq m, j \leq n, \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} | = 0 . \end{matrix}$ Thus, it is well-known fact that

$\begin{matrix} lim_{m, n \to \infty} \frac{1}{q (m) s (n)} | {(i, j) : i \leq q (m), j \leq s (n), \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} | = 0 . \end{matrix}$ Also, since

$\begin{matrix} {(i, j) : p (m) < i \leq q (m), r (n) < j \leq s (n), \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} \\ \subseteq {(i, j) : i \leq q (m), j \leq s (n), \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ}, \end{matrix}$ we have

$\begin{matrix} \frac{1}{φ (m) ψ (n)} | {(i, j) : p (m) < i \leq q (m), \\ r (n) < j \leq s (n), | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} | \\ \leq \frac{q (m) s (n)}{φ (m) ψ (n)} \frac{1}{q (m) s (n)} | {(i, j) : i \leq q (m), \\ j \leq s (n), | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} | \\ = (1 + \frac{2 p (m)}{φ (m)}) (1 + \frac{2 r (n)}{ψ (n)}) \frac{1}{q (m) s (n)} \cdot \\ | {(i, j) : i \leq q (m), j \leq s (n), \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} | . \end{matrix}$ If ${\frac{p (m)}{φ (m)}}$ and ${\frac{r (n)}{ψ (m)}}$ are bounded in above inequality, by the assumption, we getU_ijoversetW₂DS ⟶ U form,n→ ∞.

3.8 Theorem

Letq (m) = m andr (n) = n for all $m, n \in ℕ$ . Then, $U_{ij} \overset{W_{2} DS}{⟶} U \Rightarrow U_{ij} \overset{W_{2} S}{⟶} U .$

Proof. Letq (m) = m ands (n) = n for all $m, n \in ℕ$ . Also, we suppose that $U_{ij} \overset{W_{2} DS}{⟶} U$ . Using the technique given by Agnew [1], we can define the following sequences:

$\begin{matrix} p (m) = m^{(1)} > p (m^{(1)}) = m^{(2)} > p (m^{(2)}) = m^{(3)} > \dots \\ r (n) = n^{(1)} > r (n^{(1)}) = n^{(2)} > r (n^{(2)}) = n^{(3)} > \dots \end{matrix}$ for all $m, n \in ℕ$ . Then, for everyɛ > 0 and eachy ∈ Y, we can write

$\begin{matrix} {(i, j) : i \leq m, j \leq n, | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} \\ = {(i, j) : i \leq m^{(1)}, j \leq n^{(1)}, \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} \\ \cup {(i, j) : m^{(1)} < i \leq m, j \leq n^{(1)}, \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} \\ \cup {(i, j) : i \leq m^{(1)}, n^{(1)} < j \leq n, \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} \\ \cup {(i, j) : m^{(1)} < i \leq m, n^{(1)} < j \leq n, \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} . \end{matrix}$ Here, we can rewrite some of the above sets, respectively, as follows:

$\begin{matrix} {(i, j) : i \leq m^{(1)}, j \leq n^{(1)}, | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} \\ = {(i, j) : i \leq m^{(2)}, j \leq n^{(2)}, \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} \\ \cup {(i, j) : m^{(2)} < i \leq m^{(1)}, j \leq n^{(2)}, \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} \\ \cup {(i, j) : i \leq m^{(2)}, n^{(2)} < j \leq n^{(1)}, \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} \\ \cup {(i, j) : m^{(2)} < i \leq m^{(1)}, n^{(2)} < j \leq n^{(1)}, \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ}, \end{matrix}$ $\begin{matrix} {(i, j) : m^{(1)} < i \leq m, j \leq n^{(1)}, \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} \\ = {(i, j) : m^{(1)} < i \leq m, j \leq n^{(2)}, \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} \\ \cup {(i, j) : m^{(1)} < i \leq m, n^{(2)} < j \leq n^{(1)}, \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} \end{matrix}$ and

$\begin{matrix} {(i, j) : i \leq m^{(1)}, n^{(1)} < j \leq n, \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} \\ = {(i, j) : i \leq m^{(2)}, n^{(1)} < j \leq n, \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} \\ \cup {(i, j) : m^{(1)} < i \leq m^{(2)}, n^{(1)} < j \leq n^{(2)}, \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} . \end{matrix}$ Similarly, we can write

$\begin{matrix} {(i, j) : i \leq m^{(2)}, j \leq n^{(2)}, \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} \\ = {(i, j) : i \leq m^{(3)}, j \leq n^{(3)}, \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} \\ \cup {(i, j) : m^{(3)} < i \leq m^{(2)}, j \leq n^{(3)}, \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} \\ \cup {(i, j) : i \leq m^{(3)}, n^{(3)} < j \leq n^{(2)}, \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} \\ \cup {(i, j) : m^{(3)} < i \leq m^{(2)}, n^{(3)} < j \leq n^{(2)}, \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} . \end{matrix}$ If this process is continued similarly, then we obtain

$\begin{matrix} {(i, j) : i \leq m^{(t_{1} - 1)}, j \leq n^{(t_{2} - 1)}, \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} \\ = {(i, j) : i \leq m^{(t_{1})}, j \leq n^{(t_{2})}, \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} \\ \cup {(i, j) : m^{(t_{1})} < i \leq m^{(t_{1} - 1)}, j \leq n^{(t_{2})}, \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} \\ \cup {(i, j) : i \leq m^{(t_{1})}, n^{(t_{2})} < j \leq n^{(t_{2} - 1)}, \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} \\ \cup {(i, j) : m^{(t_{1})} < i \leq m^{(t_{1} - 1)}, \\ n^{(t_{2})} < j \leq n^{(t_{2} - 1)}, | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} \end{matrix}$ wheret₁,t₂ are fixed positive integers such thatm^(t₁) ≥ 1,m^(t₁+1) = 0 andn^(t₂) ≥ 1,n^(t₂+1) = 0. From this whole process, for all $m, n \in ℕ$ and eachy ∈ Y we have

$\begin{matrix} \frac{1}{mn} | {(i, j) : i \leq m, j \leq n, | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} | \\ = \sum_{(k, l) = (0, 0)}^{(t_{1} + 1, t_{2} + 1)} \frac{(m^{(k)} - m^{(k + 1)}) (n^{(l)} - n^{(l + 1)})}{mn} H_{mn} \end{matrix}$ where

$\begin{matrix} H_{mn} : = \frac{1}{(m^{(k)} - m^{(k + 1)}) (n^{(l)} - n^{(l + 1)})} \cdot \\ | {(i, j) : m^{(k + 1)} < i \leq m^{(k)}, \\ n^{(l + 1)} < j \leq n^{(l)}, | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} | . \end{matrix}$ Considering the following matrix $T_{mn}^{kl}$ ,

$\begin{matrix} T_{mn}^{kl} : = \\ {\begin{matrix} \frac{(m^{(k)} - m^{(k + 1)}) (n^{(l)} - n^{(l + 1)})}{mn} & , & \begin{matrix} k = 0, 1, 2, \dots, t_{1} \\ l = 0, 1, 2, \dots, t_{2} \end{matrix}, [0.5 cm] 0 & , & ifnot, \end{matrix} \end{matrix}$ wherem⁽⁰⁾ = m andn⁽⁰⁾ = n, it is clear that the statistical convergence in the Wijsman sense of the double sequence of sets {U_ij} is equivalent to the convergence of transform under the matrix $T_{mn}^{kl}$ of the sequenceH_mn. Since the matrix $T_{mn}^{kl}$ is regular, by the assumption, we get $U_{ij} \overset{W_{2} S}{⟶} U$ form,n→ ∞.

3.9 Corollary

Let ${\frac{p (m)}{m - p (m)}}$ and ${\frac{r (n)}{n - r (n)}}$ be bounded. Then, $U_{ij} \overset{W_{2} S}{⟶} U \Leftrightarrow U_{ij} \overset{W_{2} DS}{⟶} U .$

The following theorems will be considered under the restrictions:

$\begin{matrix} p (m) \leq p^{'} (m) < q^{'} (m) \leq q (m) and \\ r (n) \leq r^{'} (n) < s^{'} (n) \leq s (n) \end{matrix}$ for all $m, n \in ℕ$ where all of these are sequences of non-negative integers.

3.10 Theorem

If ${\frac{φ (m) ψ (n)}{φ^{'} (m) ψ^{'} (n)}}$ is bounded, then ${W_{2} DS}_{[φ, ψ]} \subset {W_{2} DS}_{[φ^{'}, ψ^{'}]} .$

Proof. LetU_ij ⟶ U ({W₂DS} _[φ,ψ]). For everyɛ > 0 and eachy ∈ Y, since

$\begin{matrix} {(i, j) : p^{'} (m) < i \leq q^{'} (m), r^{'} (n) < j \leq s^{'} (n), \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} \\ \subset {(i, j) : p (m) < i \leq q (m), r (n) < j \leq s (n), \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ}, \end{matrix}$ we have,

$\begin{matrix} \frac{1}{φ^{'} (m) ψ^{'} (n)} | {(i, j) : p^{'} (m) < i \leq q^{'} (m), \\ r^{'} (n) < j \leq s^{'} (n), | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} | \\ \leq \frac{φ (m) ψ (n)}{φ^{'} (m) ψ^{'} (n)} (\frac{1}{φ (m) ψ (n)} | {(i, j) : \\ p (m) < i \leq q (m), r (n) < j \leq s (n), \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} |) . \end{matrix}$ If ${\frac{φ (m) ψ (n)}{φ^{'} (m) ψ^{'} (n)}}$ is bounded in above inequality, by the assumption, we getU_ij ⟶ U ({W₂DS} _{[φ′,ψ′]}) forn,m→ ∞.

3.11 Theorem

If the sets {i : p (m) < i ≤ p′ (m)}, {i : q′ (m) < i ≤ q (m)}, {j : r (n) < j ≤ r′ (n)} and {j : s′ (n) < j ≤ s (n)} are finite for all $m, n \in ℕ$ , then ${W_{2} DS}_{[φ^{'}, ψ^{'}]} \subset {W_{2} DS}_{[φ, ψ]} .$

Proof LetU_ij ⟶ U ({W₂DS} _{[φ′,ψ′]}). For everyɛ > 0 and eachy ∈ Y, since

$\begin{matrix} {(i, j) : p (m) < i \leq q (m), r (n) < j \leq s (n), \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} \\ = {(i, j) : p (m) < i \leq p^{'} (m), r (n) < j \leq r^{'} (n), \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} \\ \cup {(i, j) : p (m) < i \leq p^{'} (m), r^{'} (n) < j \leq s^{'} (n), \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} \\ \cup {(i, j) : p (m) < i \leq p^{'} (m), s^{'} (n) < j \leq s (n), \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} \\ \cup {(i, j) : p^{'} (m) < i \leq q^{'} (m), r (n) < j \leq r^{'} (n), \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} \\ \cup {(i, j) : p^{'} (m) < i \leq q^{'} (m), r^{'} (n) < j \leq s^{'} (n), \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} \\ \cup {(i, j) : p^{'} (m) < i \leq q^{'} (m), s^{'} (n) < j \leq s (n), \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} \\ \cup {(i, j) : q^{'} (m) < i \leq q (m), r (n) < j \leq r^{'} (n), \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} \\ \cup {(i, j) : q^{'} (m) < i \leq q (m), r^{'} (n) < j \leq s^{'} (n), \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} \\ \cup {(i, j) : q^{'} (m) < i \leq q (m), s^{'} (n) < j \leq s (n), \\ | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ}, \end{matrix}$

we have

$\begin{matrix} \frac{1}{φ (m) ψ (n)} | {(i, j) : p (m) < i \leq q (m), \\ r (n) < j \leq s (n), | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} | \\ \leq \frac{1}{φ^{'} (m) ψ^{'} (n)} | {(i, j) : p (m) < i \leq p^{'} (m), \\ r (n) < j \leq r^{'} (n), | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} | \\ + \frac{1}{φ^{'} (m) ψ^{'} (n)} | {(i, j) : p (m) < i \leq p^{'} (m), \\ r^{'} (n) < j \leq s^{'} (n), | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} | \\ + \frac{1}{φ^{'} (m) ψ^{'} (n)} | {(i, j) : p (m) < i \leq p^{'} (m), & s^{'} (n) < j \leq s (n), | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} | \\ + \frac{1}{φ^{'} (m) ψ^{'} (n)} | {(i, j) : p^{'} (m) < i \leq q^{'} (m), \\ r (n) < j \leq r^{'} (n), | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} | \\ + \frac{1}{φ^{'} (m) ψ^{'} (n)} | {(i, j) : p^{'} (m) < i \leq q^{'} (m), \\ r^{'} (n) < j \leq s^{'} (n), | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} | \\ + \frac{1}{φ^{'} (m) ψ^{'} (n)} | {(i, j) : p^{'} (m) < i \leq q^{'} (m), \\ s^{'} (n) < j \leq s (n), | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} | \\ + \frac{1}{φ^{'} (m) ψ^{'} (n)} | {(i, j) : q^{'} (m) < i \leq q (m), & r (n) < j \leq r^{'} (n), | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} | \\ + \frac{1}{φ^{'} (m) ψ^{'} (n)} | {(i, j) : q^{'} (m) < i \leq q (m), & r^{'} (n) < j \leq s^{'} (n), | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} | \\ + \frac{1}{φ^{'} (m) ψ^{'} (n)} | {(i, j) : q^{'} (m) < i \leq q (m), & s^{'} (n) < j \leq s (n), | d_{y} (U_{ij}) - d_{y} (U) | \geq ɛ} | . \end{matrix}$ If the sets {i : p (m) < i ≤ p′ (m)}, {i : q′ (m) < i ≤ q (m)}, {j : r (n) < j ≤ r′ (n)} and {j : s′ (n) < j ≤ s (n)} are finite for all $n, m \in ℕ$ in above inequality, by the assumption, we getU_ij ⟶ U ({W₂DS} _[φ,ψ]) form,n→ ∞.

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