ISRN Mathematical Analysis

Volume 2013 (2013), Article ID 602963, 6 pages

http://dx.doi.org/10.1155/2013/602963

Research Article

On -Asymptotically Statistical Equivalence of Sequences of Sets

Ömer Kışı¹ and Fatıh Nuray²

¹Mathematics Education Department, Faculty of Education, Cumhuriyet University, Sıvas, Turkey

²Department of Mathematics, Faculty of Science and Literature, Afyon Kocatepe University, 03200 Afyonkarahısar, Turkey

Received 10 June 2013; Accepted 13 August 2013

Academic Editors: R. Avery and G. Schimperna

Copyright © 2013 Ömer Kışı and Fatıh Nuray. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper presents the notion of -asymptotically statistical equivalence, which is a natural combination of asymptotic -equivalence, and -statistical equivalence for sequences of sets. We find its relations to -asymptotically statistical convergence, strong -asymptotically equivalence, and strong Cesaro -asymptotically equivalence for sequences of sets.

1. Introduction and Background

Let be a nondecreasing sequence of positive numbers tending to , such that , . The generalized de la Vallee-Poussin mean is defined by

where .

A sequence is said to be -summable to a number if

If , then -summability reduces to -summability.

We write

for the sets of sequences , which are strongly Cesaro summable and strongly -summable to , that is, and , respectively. Let denote the set of all nondecreasing sequences of positive numbers tending to , such that and .

Statistical convergence of sequences of points was introduced by Fast (see [1]), and under different names, it has been discussed in number theory, trigonometric series, and summability. 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. Mursalen defined -statistical convergence by using the sequence. He denoted this new method by , and found its relation to statistical convergence, -summability, and -summability (see [2]). Savaş introduced and studied the concepts of strongly -summability and -statistical convergence for fuzzy numbers (see [3]). He also presented asymptotically -statistical equivalent sequences of fuzzy numbers (see [4]). Kostyrko et al. (see [5, 6]) introduced the concept of -convergence of sequences in a metric space and studied some properties of this convergence. In addition to these definitions, natural inclusion theorems are also presented. The concept of convergence of sequences of points has been extended by several authors to convergence of sequences of sets. One of these extensions that we will consider in this paper is Wijsman convergence. The concept of Wijsman statistical convergence is an implementation of the concept of statistical convergence presented by Nuray and Rhoades (see [7]).

Definition 1. The sequence is said to be statistically convergent to the number if for every ,

In this case one writes (see [8]).

Definition 2 2. A family of sets is called an ideal if and only if

 (i), (ii) for each one has , (iii) for each and each one has (see [5]).

An ideal is called nontrivial if , and nontrivial ideal is called admissible if for each .

Definition 3. A family of sets is a filter in if and only if

 (i), (ii) for each one has , (iii) for each and each one has (see [5]).

Proposition 4. is a nontrivial ideal in if and only if

is a filter in (see [5]).

Definition 5. Let be a nontrivial ideal of subsets of , and let be a metric space. A sequence of elements of is said to be -convergent to . Therefore if and only if for each the set

belongs to . The number is called the limit of the sequence (see [5]).

Definition 6. Let be a metric space. For any non-empty closed subsets , one says that the sequence is Wijsman convergent to :

for each . In this case one writes (see [9, 10]).

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

As the sequence is Wijsman convergent to the -axis .

Definition 7. Let be a metric space. For any non-empty closed subsets , one says that the sequence is Wijsman statistically convergent to if for and for each ,

In this case one writes or (see [7]).

where denotes the set of Wijsman statistical convergence sequences.

Also, the concept of bounded sequence for sequences of sets was given by Nuray and Rhoades (see [7]). Let be a metric space. For any non-empty closed subsets of , we say that the sequence is bounded if for each .

Definition 8. Let be a metric space. For any non-empty closed subsets , we say that the sequence is Wijsman Cesaro summable to if is Cesaro summable to ; that is, for each ,

and one says that the sequence is Wijsman strongly Cesaro summable to if is strongly summable to ; that is, for each ,

(see [7]).

Definition 9. Let be a metric space. For any non-empty closed subsets such that and for each , one says that the sequences and are asymptotically equivalent (Wijsman sense) if for each ,

(denoted by ) (see [11]).

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

Since

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

Definition 10. Let be a metric space. For non-empty closed subsets such that and for each , one says that the sequences and are asymptotically statistically equivalent (Wijsman sense) of multiple provided that for every and for each ,

(denoted by ) and simply asymptotically statistically equivalent (Wijsman sense) if (see [11]).

2. Main Results

Definition 11 11 (see [12]). Let be a metric space and let be a proper ideal in . For any non-empty closed subsets , we say that the sequence is Wijsman -convergent to , if for each and for each , the set

belongs to . In this case one writes or , and the set of  Wijsman -convergent sequences of sets will be denoted by

As an example, consider the following sequence. Let , and let be the following sequence:

and . The sequence is not Wijsman convergent to the set . But, if we take , then is Wijsman -convergent to set , where is the ideal of sets that have zero density.

Definition 12. Let be a metric space. For any non-empty closed subsets , we say that the sequence is said to be Wijsman -statistically convergent or -convergent to if for every and for each ,

In this case one writes , or and,

If , then Wijsman -statistical convergence is the same as Wijsman statistical convergence for the sequences of sets.

Definition 13. Let be a metric space. For any non-empty closed subsets , we say that the sequence is said to be Wijsman strongly summable to if for every and for each ,

In this case one writes .

If , then -summability reduces to -summability for sequences of sets.

Theorem 14. Let , be a metric space. For any non-empty closed subsets , then

 (i) and the inclusion is proper for sequences of sets, (ii) if is bounded and , then , (iii),

where denotes the set of bounded sequences of sets.

Proof. (i) Let and . One has

Therefore .

The following example shows that for sequences of sets:

Then and for every ,

that is, . On the other hand,

that is, .

(ii) Suppose that is bounded and . Then there is a such that for all . Given , one has

which implies that .

(iii) This immediately follows from (i) and (ii).

Definition 15. Let be a metric space, and let be an admissible ideal. For non-empty closed subsets such that and for each , one says that the sequences and are said to be asymptotically Wijsman -equivalent of multiple if for every and for each ,

This  will  be  denoted  by  .

Definition 16. Let be a metric space, and let be an admissible ideal. For non-empty closed subsets such that and for each , one says that the sequences and are said to be strong Cesaro -asymptotically equivalent (Wijsman sense) of multiple if every and for each ,

This will be denoted by .

Definition 17. Let be a metric space. For non-empty closed subsets such that and for each , one says that the sequences and are Wijsman -asymptotically statistically equivalent of multiple if for every and for each ,

This will be denoted by .

Example 18. Let be a proper ideal in , and let   be a metric space, then are non-empty closed subsets. Let , ,  be the following sequences:

If we take we have

Thus, the sequences and are asymptotically -equivalent (Wijsman sense); that is, , where is the ideal of sets that have zero density.

Example 19. Let be a proper ideal in and let,  be a metric space, then are non-empty closed subsets. Let , ,  be the following sequences:

If we take we have

Thus, the sequences and are asymptotically -equivalent (Wijsman sense); that is, , where is the ideal of sets which have zero density.

Definition 20. Let be a metric space. For non-empty closed subsets such that and for each, one says that the sequences and are strongly -asymptotically equivalent (Wijsman sense) of multiple if for every and for each ,

This will be denoted by .

Definition 21. Let be a metric space. For non-empty closed subsets such that and for each , one says that the sequences and are -asymptotically -statistically equivalent (Wijsman sense) of multiple , provided that for every and for each ,

This will be denoted by .

Theorem 22. Let and let be an admissible ideal in . If , then .

Proof. Assume that and . Then,

and so,

Then for any ,

Since right hand belongs to , then left hand also belongs to , and this completes the proof.

Theorem 23. Let and let be an admissible ideal in . If and are bounded and , then .

Proof. Let be bounded sequences and let . Then there is an such that

for all . For each ,

Then,

Therefore, .

The following example shows that if and are not bounded, then Theorem 23 cannot be true.

Example 24. Take and define to be,

where denotes the greatest integer function and for all . Note that is not bounded. Then , but is not true.

Theorem 25. Let and let be an admissible ideal in . If , then .

Proof. Assume that and . Then,

and so,

Hence .

Theorem 26. If , then implies .

Proof. Assume that lim inf and there exists a such that for sufficiently large . For given one has,

Therefore,

then for any we get

and this completes the proof.

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