Double Wijsman asymptotically statistical equivalence of order α

Abstract

In this paper, we study the concepts of Wijsman asymptotically statistical equivalence of order α, Wijsman asymptotically strongly p-Cesàro equivalence of order α and Hausdorff asymptotically statistical equivalence of order α for double set sequences. Also, we research some properties of these concepts and give the inclusion relationships between them.

Keywords

Asymptotic equivalence statistical convergence Cesàro summability Wijsman convergence Hausdorff convergence double set sequences

1 Introduction

The concept of statistical convergence was introduced by Fast [12]. Then, this concept has been discussed over the years by many researchers (see, [5 , 43]). More recently, a new type of convergence called statistical convergence of order α which extended from statistical convergence was studied by Çolak [7] and Çolak and Bektaş [8]. Similar concepts can be found in [11].

Pringsheim [37] introduced the concept of convergence for double sequences. Recently, this concept was extended to the concept of statistical convergence for double sequences by Mursaleen and Edely [20]. Within a decade, a new type of convergence for double sequences called statistical convergence of order α which extended from statistical convergence was studied by Çolak and Altın [9]. More investigations on double sequences can be found in [4 , 38].

In [18], Marouf presented definitions for asymptotically equivalent and asymptotic regular matrices. Then, Patterson [34] extended these concepts by presenting an asymptotically statistical equivalent analog of these definitions. Also, the concepts of asymptotically equivalence for double sequences were studied by Patterson and Savaş [35]. More developments on the concept of asymptotically equivalence can be found in [17, 36].

The convergence idea for set sequences arose from the convergence concepts for real sequences. The concepts of Wijsman convergence and Hausdorff convergence which are convergence types for set sequences are handled in this study (see, [1–3 , 55]). Nuray and Rhoades [22] extended the concepts of Wijsman convergence and Hausdorff convergence to the concept of statistical convergence for set sequences. Recently, some convergence concepts of order α for set sequences were studied by both Savaş [39] and Şengül and Et [42], independently. More investigations on set sequences can be found in [14, 15 , 50].

The concepts of Wijsman convergence and Wijsman strongly Cesàro summability for double set sequences were defined by Nuray et al. [24]. Also Nuray et al. [27] introduced Wijsman statistical convergence and Hausdorff statistical convergence for double set sequences. Lately, some convergence concepts of order α for double set sequences were introduced by Gülle and Ulusu [16] and Ulusu and Gülle [54]. More developments on double set sequences can be found in [10 , 53].

Ulusu and Nuray [45] firstly introduced the concepts of Wijsman asymptotically equivalence for set sequences. After a while, the concepts of Wijsman asymptotically equivalence, Wijsman asymptotically strongly Cesàro equivalence and Wijsman asymptotically statistical equivalence for double set sequences were studied by Nuray et al. [26]. More investigations on the concept of asymptotically equivalence for set sequences can be found in [29 , 52].

2 Definitions and Notations

The following basic definitions and notations will be needed in this study. So, we start giving these concepts, primarily (see, [1–3 , 55]).

A double sequence (x_ij) is said to be convergent to L in Pringsheim’s sense if for every ɛ > 0, there exists $N_{ɛ} \in ℕ$ such that |x_ij - L| < ɛ, whenever i, j > N_ɛ.

A double sequence (x_ij) is said to be statistically convergent to the number L if for every ɛ > 0,

$lim_{m, n \to \infty} \frac{1}{mn} | {(i, j) : i \leq n, j \leq m, | x_{ij} - L | \geq ɛ} | = 0 .$ A double sequence (x_ij) is said to be strongly Cesàro summable to L if $lim_{m, n \to \infty} \frac{1}{mn} \sum_{i, j = 1, 1}^{m, n} | x_{ij} - L | = 0 .$

Let X be any non-empty set. The function $f : ℕ \to P (X)$ is defined by f (i) = U_i ∈ P (X) for each $i \in ℕ$ , where P (X) is power set of X. The sequence {U_i} = (U₁, U₂, . . .), which is the range’s elements of f, is said to be set sequences.

Let (X, d) be a metric space. For any point x ∈ X and any non-empty subset U of X, the distance from x to U is defined by $ρ (x, U) = inf_{u \in U} d (x, u) .$

A double sequence {U_ij} is said to be Wijsman convergent to U if for each x ∈ X, $lim_{i, j \to \infty} ρ (x, U_{ij}) = ρ (x, U) .$

A double sequence {U_ij} is said to be Hausdorff convergent to U if for each x ∈ X, $lim_{i, j \to \infty} sup_{x \in X} | ρ (x, U_{ij}) - ρ (x, U) | = 0 .$

Two nonnegative double sequences (x_ij) and (y_ij) are said to be asymptotically equivalent if $lim_{i, j \to \infty} \frac{x_{ij}}{y_{ij}} = 1 .$

Throughout the study, we take (X, d) be a metric space and U_ij, V_ij be any non-empty closed subsets of X. We define ρ (x ; U_ij, V_ij) as follows: $ρ (x; U_{ij}, V_{ij}) = {\begin{matrix} \frac{ρ (x, U_{ij})}{ρ (x, V_{ij})} & , & x \notin U_{ij} \cup V_{ij} \\ L & , & x \in U_{ij} \cup V_{ij} . \end{matrix}$

Double sequences {U_ij} and {V_ij} are said to be Wijsman asymptotically equivalent of multiple L if for each x ∈ X, $lim_{i, j \to \infty} ρ (x; U_{ij}, V_{ij}) = L .$

Double sequences {U_ij} and {V_ij} are said to be Wijsman asymptotically statistical equivalent of multiple L if for every ɛ > 0 and each x ∈ X, $\begin{matrix} lim_{m, n \to \infty} \frac{1}{mn} | {(i, j) : \\ i \leq n, j \leq m, | ρ (x; U_{ij}, V_{ij}) - L | \geq ɛ} | = 0 . \end{matrix}$

The class of all Wijsman asymptotically statistical equivalent of multiple L double sequences denotes by simply W₂ (S^L).

Double sequences {U_ij} and {V_ij} are said to be Wijsman asymptotically strongly Cesàro equivalent of multiple L if for each x ∈ X, $lim_{m, n \to \infty} \frac{1}{mn} \sum_{i, j = 1, 1}^{m, n} | ρ (x; U_{ij}, V_{ij}) - L | = 0 .$

3 Main results

In this section, we study the concepts of Wijsman asymptotically statistical equivalence of order α, Wijsman asymptotically strongly p-Cesàro equivalence of order α and Hausdorff asymptotically statistical equivalence of order α for double set sequences. Also, we research some properties of these concepts and give the inclusion relationships between them.

3.1 Definition

Let 0 < α ≤ 1. Double sequences {U_ij} and {V_ij} are Wijsman asymptotically statistical equivalent of multiple L of order α if for every ɛ > 0 and each x ∈ X, $\begin{matrix} lim_{m, n \to \infty} \frac{1}{(mn)^{α}} | {(i, j) : \\ i \leq m, j \leq n, | ρ (x; U_{ij}, V_{ij}) - L | \geq ɛ} | = 0 . \end{matrix}$ In this case we write $U_{ij} \overset{W_{2} (S_{L}^{α})}{\sim} V_{ij}$ , and simply Wijsman asymptotically statistical equivalent if L = 1.

The class of all Wijsman asymptotically statistical equivalent of multiple L of order α double sequences will be denoted by simply $W_{2} (S_{L}^{α})$ .

3.2 Example

Let $X = ℝ^{2}$ and double sequences {U_ij} and {V_ij} be defined as following: $U_{ij} : = {\begin{matrix} {(x, y) \in ℝ^{2} : x^{2} + y^{2} - ijx = 0} \\ if ij is a square integer, \\ {(- 1, 1)} \\ otherwise . \end{matrix}$ and $V_{ij} : = {\begin{matrix} {(x, y) \in ℝ^{2} : x^{2} + y^{2} + ijx = 0} \\ if ij is a square integer, \\ {(- 1, 1)} \\ otherwise . \end{matrix}$

Then, the double sequences {U_ij} and {V_ij} are Wijsman asymptotically statistical equivalent of order α (0 < α ≤ 1).

3.3 Remark

For α = 1, the concept of Wijsman asymptotically statistical equivalence of multiple L of order α coincides with the concept of Wijsman asymptotically statistical equivalence of multiple L for double set sequences, [26].

3.4 Theorem

If 0 < α ≤ β ≤ 1, then $W_{2} (S_{L}^{α}) \subseteq W_{2} (S_{L}^{β})$ .

Proof. Let 0 < α ≤ β ≤ 1 be given and suppose that $U_{ij} \overset{W_{2} (S_{L}^{α})}{\sim} V_{ij}$ . For every ɛ > 0 and each x ∈ X, we have

$\begin{matrix} \frac{1}{(mn)^{β}} | {(i, j) : \\ i \leq m, j \leq n, | ρ (x; U_{ij}, V_{ij}) - L | \geq ɛ} | \\ \leq \frac{1}{(mn)^{α}} | {(i, j) : \\ i \leq m, j \leq n, | ρ (x; U_{ij}, V_{ij}) - L | \geq ɛ} | . \end{matrix}$ Hence, by our assumption, we get $U_{ij} \overset{W_{2} (S_{L}^{β})}{\sim} V_{ij}$ . Consequently, $W_{2} (S_{L}^{α}) \subseteq W_{2} (S_{L}^{β})$ .

If we take β = 1 in Theorem 3.4, then we obtain the following corollary.

3.5 Corollary

Let α ∈ (0, 1]. If the double sequences {U_ij} and {V_ij} are Wijsman asymptotically statistical equivalent of multiple L of order α, then the double sequences are Wijsman asymptotically statistical equivalent of multiple L, i.e., $W_{2} (S_{L}^{α}) \subseteq W_{2} (S^{L})$ .

3.6 Definition

Let 0 < α ≤ 1 and 0< p < ∞. Double sequences {U_ij} and {V_ij} are Wijsman asymptotically strongly p-Cesàro equivalent of multiple L of order α if for each x ∈ X, $lim_{m, n \to \infty} \frac{1}{(mn)^{α}} \sum_{i, j = 1, 1}^{m, n} | ρ (x; U_{ij}, V_{ij}) - L |^{p} = 0 .$ In this case we write $U_{ij} \overset{W_{2} [C_{L}^{α}]^{p}}{\sim} V_{ij}$ , and simply Wijsman asymptotically strongly p-Cesàro equivalent if L = 1.

If p = 1, the double sequences {U_ij} and {V_ij} are Wijsman asymptotically strongly Cesàro equivalent of multiple L of order α, and we write $U_{ij} \overset{W_{2} [C_{L}^{α}]}{\sim} V_{ij}$ .

The class of all Wijsman asymptotically strongly p-Cesàro equivalent of multiple L of order α double sequences will be denoted by simply $W_{2} [C_{L}^{α}]^{p}$ .

3.7 Example

Let $X = ℝ^{2}$ and double sequences {U_ij} and {V_ij} be defined as following: $U_{ij} : = {\begin{matrix} {(x, y) \in ℝ^{2} : (x - 1)^{2} + y^{2} = \frac{1}{ij}} \\ if ij is a square integer, \\ {(0, 1)} \\ otherwise . \end{matrix}$ and $V_{ij} : = {\begin{matrix} {(x, y) \in ℝ^{2} : (x + 1)^{2} + y^{2} = \frac{1}{ij}} \\ if ij is a square integer, \\ {(0, 1)} \\ otherwise . \end{matrix}$

Then, the double sequences {U_ij} and {V_ij} are Wijsman asymptotically strongly Cesàro equivalent of order α (0 < α ≤ 1).

3.8 Remark

For α = 1, the concepts of Wijsman asymptotically strongly Cesàro equivalence of multiple L of order α and Wijsman asymptotically strongly p-Cesàro equivalence of multiple L of order α coincide with the concepts of Wijsman asymptotically strongly Cesàro equivalence of multiple L and Wijsman asymptotically strongly p-Cesàro equivalence of multiple L for double set sequences which the first one of these can be found in [26] and second one has not studied until now.

3.9 Theorem

If 0 < α ≤ β ≤ 1, then $W_{2} [C_{L}^{α}]^{p} \subseteq W_{2} [C_{L}^{β}]^{p}$ .

Proof. Let 0 < α ≤ β ≤ 1 be given and suppose that $U_{ij} \overset{W_{2} [C_{L}^{α}]^{p}}{\sim} V_{ij}$ . For each x ∈ X, we have $\begin{matrix} \frac{1}{(mn)^{β}} \sum_{i, j = 1, 1}^{m, n} | ρ (x; U_{ij}, V_{ij}) - L |^{p} \\ \leq \frac{1}{(mn)^{α}} \sum_{i, j = 1, 1}^{m, n} | ρ (x; U_{ij}, V_{ij}) - L |^{p} . \end{matrix}$ Hence, by our assumption, we get $U_{ij} \overset{W_{2} [C_{L}^{β}]^{p}}{\sim} V_{ij}$ . Consequently, $W_{2} [C_{L}^{α}]^{p} \subseteq W_{2} [C_{L}^{β}]^{p}$ .

If we take β = 1 in Theorem 3.9, then we obtain the following corollary.

3.10 Corollary

Let α ∈ (0, 1]. If double sequences {U_ij} and {V_ij} are Wijsman asymptotically strongly p-Cesàro equivalent of multiple L of order α, then the double sequences are Wijsman asymptotically strongly p-Cesàro equivalent of multiple L, which has not studied until now.

Now, we shall state a theorem that gives a relation between $W_{2} [C_{L}^{α}]^{q}$ and $W_{2} [C_{L}^{α}]^{p}$ , where 0 < α ≤ 1 and 0< p < q < ∞.

3.11 Theorem

Let 0 < α ≤ 1 and 0< p < q < ∞. Then, $W_{2} [C_{L}^{α}]^{q} \subset W_{2} [C_{L}^{α}]^{p}$ .

Proof. Let 0 < α ≤ 1 and 0< p < q < ∞. Suppose that $U_{ij} \overset{W_{2} [C_{L}^{α}]^{q}}{\sim} V_{ij}$ . For each x ∈ X, by Hölder inequality, we have $\begin{matrix} \frac{1}{(mn)^{α}} \sum_{i, j = 1, 1}^{m, n} | ρ (x; U_{ij}, V_{ij}) - L |^{p} \\ < \frac{1}{(mn)^{α}} \sum_{i, j = 1, 1}^{m, n} | ρ (x; U_{ij}, V_{ij}) - L |^{q} . \end{matrix}$ Hence, by our assumption, we get $U_{ij} \overset{W_{2} [C_{L}^{α}]^{p}}{\sim} V_{ij}$ . Consequently, $W_{2} [C_{L}^{α}]^{q} \subset W_{2} [C_{L}^{α}]^{p}$ .

3.12 Theorem

Let 0 < α ≤ β ≤ 1 and 0< p < ∞. If double sequences {U_ij} and {V_ij} are Wijsman asymptotically strongly p-Cesàro equivalent of multiple L of order α, then the double sequences are Wijsman asymptotically statistical equivalent of multiple L of order β.

Proof. Let 0 < α ≤ β ≤ 1 and 0< p < ∞. Assume that the double sequences {U_ij} and {V_ij} are Wijsman asymptotically strongly p-Cesàro equivalent of multiple L of order α. For every ɛ > 0 and each x ∈ X, we have

$\begin{matrix} \sum_{i, j = 1, 1}^{m, n} | ρ (x; U_{ij}, V_{ij}) - L |^{p} \\ = \underset{| ρ (x; U_{ij}, V_{ij}) - L | \geq ɛ}{\sum_{i, j = 1, 1}^{m, n}} | ρ (x; U_{ij}, V_{ij}) - L |^{p} \\ + \underset{| ρ (x; U_{ij}, V_{ij}) - L | < ɛ}{\sum_{i, j = 1, 1}^{m, n}} | ρ (x; U_{ij}, V_{ij}) - L |^{p} \\ \geq \underset{| ρ (x; U_{ij}, V_{ij}) - L | \geq ɛ}{\sum_{i, j = 1, 1}^{m, n}} | ρ (x; U_{ij}, V_{ij}) - L |^{p} \\ \geq ɛ^{p} | {(i, j) : \\ i \leq m, j \leq n, | ρ (x; U_{ij}, V_{ij}) - L | \geq ɛ} | \end{matrix}$ and so

$\begin{matrix} \frac{1}{(mn)^{α}} \sum_{i, j = 1, 1}^{m, n} | ρ (x; U_{ij}, V_{ij}) - L |^{p} \\ \geq \frac{ɛ^{p}}{(mn)^{α}} | {(i, j) : \\ i \leq m, j \leq n, | ρ (x; U_{ij}, V_{ij}) - L | \geq ɛ} | \\ \geq \frac{ɛ^{p}}{(mn)^{β}} | {(i, j) : \\ i \leq m, j \leq n, | ρ (x; U_{ij}, V_{ij}) - L | \geq ɛ} | . \end{matrix}$ Hence, by our assumption, we get that the double sequences {U_ij} and {V_ij} are Wijsman asymptotically statistical equivalent of multiple L of order β.

If we take β = α in Theorem 3.12, then we obtain the following corollary.

3.13 Corollary

Let α ∈ (0, 1] and 0< p < ∞. If the double sequences {U_ij} and {V_ij} are Wijsman asymptotically strongly p-Cesàro equivalent of multiple L of order α, then the double sequences are Wijsman asymptotically statistical equivalent of multiple L of order α.

3.14 Definition

Let 0 < α ≤ 1. Double sequences {U_ij} and {V_ij} are Hausdorff asymptotically statistical equivalent of multiple L of order α if for every ɛ > 0 and each x ∈ X, $\begin{matrix} lim_{m, n \to \infty} \frac{1}{(mn)^{α}} | {(i, j) : \\ i \leq m, j \leq n, sup_{x \in X} | ρ (x; U_{ij}, V_{ij}) - L | \geq ɛ} | = 0 . \end{matrix}$ In this case we write $U_{ij} \overset{H_{2} (S_{L}^{α})}{\sim} V_{ij}$ .

The class of all Hausdorff asymptotically statistical equivalent of multiple L of order α double sequences will be denoted by simply $H_{2} (S_{L}^{α})$ .

3.15 Remark

For α = 1, the concept of Hausdorff asymptotically statistical equivalence of multiple L of order α coincides with the concept of Hausdorff asymptotically statistical equivalence of multiple L which has not studied until now.

3.16 Theorem

If 0 < α ≤ β ≤ 1, then $H_{2} (S_{L}^{α}) \subseteq H_{2} (S_{L}^{β})$ .

Proof. Let 0 < α ≤ β ≤ 1 be given and suppose that $U_{ij} \overset{H_{2} (S_{L}^{α})}{\sim} V_{ij}$ . For every ɛ > 0 and each x ∈ X, we have

$\begin{matrix} \frac{1}{(mn)^{β}} | {(i, j) : \\ i \leq m, j \leq n, sup_{x \in X} | ρ (x; U_{ij}, V_{ij}) - L | \geq ɛ} | \\ \leq \frac{1}{(mn)^{α}} | {(i, j) : \\ i \leq m, j \leq n, sup_{x \in X} | ρ (x; U_{ij}, V_{ij}) - L | \geq ɛ} | . \end{matrix}$ Hence, by our assumption, we get $U_{ij} \overset{H_{2} (S_{L}^{β})}{\sim} V_{ij}$ . Consequently, $H_{2} (S_{L}^{α}) \subseteq H_{2} (S_{L}^{β})$ .

If we take β = 1 in Theorem 3.16, then we obtain the following corollary.

3.17 Corollary

Let α ∈ (0, 1]. If double sequences {U_ij} and {V_ij} are Hausdorff asymptotically statistical equivalent of multiple L of order α, then the double sequences are Hausdorff asymptotically statistical equivalent of multiple L.

3.18 Theorem

Let 0 < α ≤ β ≤ 1. If double sequences {U_ij} and {V_ij} are Hausdorff asymptotically statistical equivalent of multiple L of order α, then the double sequences are Wijsman asymptotically statistical equivalent of multiple L of order β.

Proof. Let 0 < α ≤ β ≤ 1 and assume that $U_{ij} \overset{H_{2} (S_{L}^{α})}{\sim} V_{ij}$ . For every ɛ > 0 and each x ∈ X, we have

$\begin{matrix} \frac{1}{(mn)^{β}} | {(i, j) : \\ i \leq m, j \leq n, | ρ (x; U_{ij}, V_{ij}) - L | \geq ɛ} | \\ \leq \frac{1}{(mn)^{β}} | {(i, j) : \\ i \leq m, j \leq n, sup_{x \in X} | ρ (x; U_{ij}, V_{ij}) - L | \geq ɛ} | \\ \leq \frac{1}{(mn)^{α}} | {(i, j) : \\ i \leq m, j \leq n, sup_{x \in X} | ρ (x; U_{ij}, V_{ij}) - L | \geq ɛ} | . \end{matrix}$ Hence, by our assumption, we achieve the desired result.

If we take β = α in Theorem 3.18, then we obtain the following corollary.

3.19 Corollary

Let α ∈ (0, 1]. If double sequences {U_ij} and {V_ij} are Hausdorff asymptotically statistical equivalent of multiple L of order α, then the double sequences are Wijsman asymptotically statistical equivalent of multiple L of order α.

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