Abstract

This paper presents three definitions which are natural combination of the definitions of asymptotic equivalence, statistical convergence, lacunary statistical convergence, and Wijsman convergence. In addition, we also present asymptotically equivalent (Wijsman sense) analogs of theorems in Patterson and Savaş (2006).

1. Introduction

In 1993, Marouf presented definitions for asymptotically equivalent and asymptotic regular matrices. In 2003, Patterson extended these concepts by presenting an asymptotically statistical equivalent analog of these definitions and natural regularity conditions for nonnegative summability matrices. In 2006, Patterson and Savaş extended the definitions presented in [1] to lacunary sequences. In addition to these definitions, natural inclusion theorems were presented. The concept of Wijsman statistical convergence is implementation of the concept of statistical convergence to sequences of sets presented by Nuray and Rhoades in 2012. Similar to this concept, the concept of Wijsman lacunary statistical convergence was presented by Ulusu and Nuray in 2012. This paper extends the definitions presented in [2] to Wijsman statistical convergent sequences and Wijsman lacunary statistical convergent sequences. In addition to these definitions, natural inclusion theorems will also be presented.

2. Definitions and Notations

Definition 1 (see Marouf [3]). Two nonnegative sequences and are said to be asymptotically equivalent if (denoted by ).

Definition 2 (see Fridy [4]). The sequence is said to be statistically convergent to the number if for every , In this case we write .

The next definition is natural combination of Definitions 1 and 2.

Definition 3 (see Patterson [1]). Two nonnegative sequences and are said to be asymptotically statistical equivalent of multiple provided that for every (denoted by ) and simply asymptotically statistically equivalent if .

By a lacunary sequence we mean an increasing integer sequence such that and as . Throughout this paper the intervals determined by will be denoted by , and ratio will be abbreviated by

Definition 4 (see Patterson and Savaş [2]). Let be a lacunary sequence; the two nonnegative sequences and are said to be asymptotically lacunary statistical equivalent of multiple provided that for every (denoted by ) and simply asymptotically lacunary statistically equivalent if .

Definition 5 (see Patterson & Savaş [2]). Let be a lacunary sequence; two nonnegative number sequences and are strongly asymptotically lacunary equivalent of multiple provided that (denoted by ) and strongly simply asymptotically lacunary equivalent if .

Let be a metric space. For any point and any nonempty subset of , we define the distance from to by

Definition 6 (see Baronti & Papini [5]). Let be a metric space. For any nonempty closed subset , we say that the sequence is Wijsman convergent to if for each . In this case we write .

Definition 7 (see Nuray & Rhoades [6]). Let a metric space. For any nonempty closed subset , we say that the sequence is Wijsman statistically convergent to if is statistically convergent to ; that is, for and for each , In this case we write or .

Also the concept of bounded sequence for sequences of sets was given by Nuray and Rhoades.

Definition 8 (see Nuray & Rhoades [6]). Let be a metric space. For any nonempty closed subset of , we say that the sequence is bounded if for each . In this case we write .

Definition 9 (see Ulusu & Nuray [7]). Let be a metric space and let be a lacunary sequence. For any non-empty closed subset , we say that the sequence is Wijsman lacunarily statistically convergent to if is lacunarily statistically convergent to ; that is, for and for each , In this case we write or .

Following these results we introduce three new notions that are asymptotically statistical equivalent (Wijsman sense) of multiple , asymptotically lacunary statistical equivalent (Wijsman sense) of multiple , and strongly asymptotically lacunary equivalent (Wijsman sense) of multiple .

Definition 10. Let be a metric space. For any non-empty closed subset , such that and for each . We say that the sequences and are asymptotically equivalent (Wijsman sense) if for each , (denoted by ).

As an example, consider the following sequences of circles in the -plane: Since the sequences and are asymptotically equivalent (Wijsman sense); that is, .

Definition 11. Let be a metric space. For any non-empty closed subset , such that and for each . We say that the sequences and are asymptotically statistically equivalent (Wijsman sense) of multiple if for every and for each , (denoted by ) and simply asymptotically statistical equivalent (Wijsman sense) if .

As an example, consider the following sequences of circles in the -plane:

Since the sequences and are asymptotically statistically equivalent (Wijsman sense); that is, .

Definition 12. Let be a metric space and let be a lacunary sequence. For any non-empty closed subset , such that and for each . We say that the sequences and are asymptotically lacunary equivalent (Wijsman sense) of multiple if for each , (denoted by ) and simply asymptotically lacunarily equivalent (Wijsman sense) if .

Definition 13. Let be a metric space and let be a lacunary sequence. For any non-empty closed subset , such that and for each . We say that the sequences and are strongly asymptotically lacunary equivalent (Wijsman sense) of multiple if for each , (denoted by ) and simply strongly asymptotically lacunarily equivalent (Wijsman sense) if .

As an example, consider the following sequences:

Since the sequences and are strongly asymptotically lacunarily equivalent (Wijsman sense); that is, .

Definition 14. Let be a metric space and let be a lacunary sequence. For any non-empty closed subset such that and for each . We say that the sequences and are asymptotically lacunarily statistical equivalent (Wijsman sense) of multiple if for every and each , (denoted by ) and simply asymptotically lacunarily statistically equivalent (Wijsman sense) if .

As an example, consider the following sequences: Since the sequences and are asymptotically lacunarily statistically equivalent (Wijsman sense); that is, .

3. Main Results

Theorem 15. Let be a metric space, let be a lacunary sequence, and let , be non-empty closed subsets of :(i)    is a proper subset of ,(ii) and ,(iii), where denotes the set of bounded sequences of sets.

Proof. (i)-(a). Let and . Then we can write which yields the result.
(ii)-(b). Suppose that . Let and be the following sequences:
Note that is not bounded. We have, for every and for each , That is, . On the other hand, Hence .
(ii) Suppose that and . Then we can assume that for each and all .
Given , we get Therefore .(iii) This is an immediate consequences of and.

Theorem 16. Let be a metric space and let , be non-empty closed subsets of . If is a lacunary sequence with , then

Proof. Suppose first that , then there exists a such that for sufficiently large , which implies that If , then for every , for each , and for sufficiently large , we have This completes the proof.

Theorem 17. Let be a metric space and let , be non-empty closed subsets of . If is a lacunary sequence with , then

Proof. Let . Then there is an such that for all . Let and . There exists such that for every We can also find such that for all . Now let be any integer satisfying , where .
Then we can write This completes the proof.

Combining Theorems 16 and 17 we have the following.