Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 257181, 9 pages

http://dx.doi.org/10.1155/2013/257181

Research Article

Wijsman Orlicz Asymptotically Ideal -Statistical Equivalent Sequences

Bipan Hazarika

Department of Mathematics, Rajiv Gandhi University, Rono Hills, Doimukh, Arunachal Pradesh 791112, India

Received 27 May 2013; Accepted 17 September 2013

Academic Editor: Mitsuru Sugimoto

Copyright © 2013 Bipan Hazarika. 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.

An ideal is a family of subsets of positive integers which is closed under taking finite unions and subsets of its elements. In this paper, we introduce a new definition of asymptotically ideal -statistical equivalent sequence in Wijsman sense and present some definitions which are the natural combination of the definition of asymptotic equivalence, statistical equivalent, -statistical equivalent sequences in Wijsman sense. Finally, we introduce the notion of Cesaro Orlicz asymptotically -equivalent sequences in Wijsman sense and establish their relationship with other classes.

1. Preliminaries and Notations

Let us start with a basic definition from the literature. Let , the set of all natural numbers, and . Then, the natural density of is defined by if the limit exists, where the vertical bars indicate the number of elements in the enclosed set.

Fast [1] presented the following definition of statistical convergence for the sequences of real numbers. The sequence is said to be statistically convergent to if for every , the set has natural density zero, that is, for each ,

In this case, we write - or and denotes the set of all statistically convergent sequences. Note that every convergent sequence is statistically convergent but not conversely. In 1985, Fridy [2] presented the notion of statistically Cauchy sequence and determined that it is equivalent to statistical convergence. Some basic properties related to the concept of statistical convergence were studied in [3–5] where many important references can be found.

Kostyrko et al. [6] introduced the notion of -convergence with the help of an admissible ideal which denotes the ideal of subsets of , which is a generalization of statistical convergence. Quite recently, Das et al. [7] unified these two approaches to introduce new concepts such as-statistical convergence and -lacunary statistical convergence and investigated some of their consequences. For more applications of ideals we refer to [8–21] where many important references can be found.

A family of sets (power sets of ) is said to be an ideal if and only if

(a), (b)for each , we have ,(c)for each and each , we have .

A nonempty family of sets is said to be a filter on if and only if

(a), (b)for each , we have ,(c)for each and each , we have .

An ideal is called a nontrivial ideal if and . Clearly is a non-trivial ideal if and only if is a filter on .

A non-trivial ideal is called

(a)admissible if and only if ,(b)maximal if there cannot exist any non-trivial ideal containing as a subset.

Recall that a sequence of points in is said to be -convergent to a real number if for every , (see [6]). In this case, we write .

Throughout the paper, we denote that is an admissible ideal of subsets of , unless otherwise stated.

Pobyvanets [22] introduced the concept of asymptotically regular matrices, which preserve the asymptotic equivalence of two nonnegative numbers sequences. Marouf [23] presented definitions for asymptotically equivalent and asymptotic regular matrices. Li [24] introduced the concept asymptotic equivalent sequences using summability. Patterson [25] extend these concepts by presenting an asymptotically statistical equivalent analog of these definitions and natural regularity conditions for nonnegative summability matrices. Patterson and Savaş [26] introduced the concept of an asymptotically lacunary statistical equivalent sequences of real numbers. Braha [27] presented the notion of -lacunary statistical equivalent real sequences. Savaş [28] introduced the concept of asymptotically lacunary statistical equivalent sequences via ideals. Kumar and Sharma [29] introduced the generalized equivalent sequences of real numbers using ideals and studied some basic properties of this notion.

The concept of the convergence of sequences of points has been extended by several authors to the convergence of sequences of sets. One of these extensions considered in this paper is the concept of Wijsman convergence. The concept of Wijsman statistical convergence which is the implementation of the concept of statistical convergence to sequences of sets is presented by Nuray and Rhoades [30]. Hazarika and Esi [31], introduced the concept of statistical almost -convergence of sequences of sets. For more works on convergence of sequences of sets, we refer to [32–38].

Now we recall the definitions that are being used throughout the paper.

Definition 1 (see [23]). Two nonnegative sequences and are said to be asymptotically equivalent if

denoted by .

Definition 2 (see [25]). Two nonnegative sequences and are said to be asymptotically statistical equivalent of multiple provided that, for every ,

denoted by and simply asymptotically statistical equivalent if .

Definition 3 (see [29]). Two nonnegative sequences and are said to be strongly asymptotically -equivalent of multiple provided that, for each ,

denoted by and simply strongly asymptotically -equivalent if .

Definition 4 (see [29]). Two non-negative sequences and are said to be -asymptotically statistical equivalent of multiple provided that, for every and for every ,

denoted by and simply -asymptotically statistical equivalent if .

Definition 5 (see [7]). A sequence of real numbers is said to be -statistically convergent to a real number for each and ,

In this case, we write .

Let be a metric space. For any point and any non-empty subset , the distance from to is defined by

Definition 6 (see [33]). Let be a metric space. For any non-empty closed subsets , we say that the sequence is Wijsman convergent to if for each . In this case, we write .

The concepts of Wijsman statistical convergence and boundedness for the sequence were given by Nuray and Rhoades [30] as follows.

Definition 7 (see [30]). Let be a metric space. For any non-empty closed subsets , we say that the sequence is Wijsman statistical convergent to if the sequence is statistically convergent to , that is, for and for each

In this case, we write or .

The sequence is bounded if for each . The set of all bounded sequences of sets is denoted by .

Ulusu and Nuray in [39] defined asymptotically equivalent and asymptotically statistical equivalent sequences of sets as follows.

Definition 8. Let be a metric space. For any non-empty closed subsets such that and for each . We say that the sequences and are asymptotically equivalent (Wijsman sense) if, for each ,

denoted by .

Definition 9. Let be a metric space. For any non-empty closed subsets such that and for each . We say that the sequences and are asymptotically statistical equivalent (Wijsman sense) of multiple if, for every and for each ,

denoted by , and simply asymptotically statistical equivalent (Wijsman sense) if .

2. Wijsman Orlicz Asymptotically Ideal -Statistical Equivalent Sequences

In this section, we define the notion of Cesaro Orlicz asymptotically -statistical equivalent and Orlicz asymptotically ideal -statistical equivalent sequences in Wijsman sense and establish some interesting relationships between these notions.

Let denote the space whose elements are finite sets of distinct positive integers. Given any element of , we denote by the sequence such that for and otherwise. Further,

that is, is the set of those whose support has cardinality at most , and we get

Let be a metric space. For any non-empty closed subsets . We define

Now, we give the following definitions.

Definition 10. Let be a metric space. For any non-empty closed subsets , we say that the sequence is -summable to (Wijsman sense) if . In this case, we write .

Definition 11. Let be a metric space. For any non-empty closed subsets , we say that the sequence is strongly -summable to (Wijsman sense) if

In this case, we write or and denote the set of all Wijsman strongly -summable sequences.

Definition 12. Let . The number

is said to be the -density of . It is clear that .

Definition 13. Let be a metric space. For any non-empty closed subsets , we say that the sequence is -statistical convergent to (Wijsman sense) if, for each ,

In this case we write or and denote the set of all Wijsman -statistically convergent sequences.

Definition 14. Let be a metric space. For any non-empty closed subsets such that and for each . We say that the sequences and are

(i)strongly asymptotically -equivalent (Wijsman sense) of multiple provided that

denoted by , and simply Wijsman strongly asymptotically -equivalent if ;(ii)asymptotically -statistical equivalent (Wijsman sense) of multiple provided that, for every ,

denoted by , and simply Wijsman asymptotically -statistical equivalent if ;(iii)-asymptotically -statistical equivalent (Wijsman sense) of multiple provided that, for every and for every ,

denoted by , and simply Wijsman -asymptotically -statistical equivalent if .

Recall that an Orlicz function is a function which is continuous, nondecreasing, and convex with , for and as . An Orlicz function is said to satisfy the -condition for all values of , if there exists a constant such that , . Note that if , then for all (see [40]).

Now, we give the following definitions.

Definition 15. Let be a metric space. For any non-empty closed subsets such that and for each . We say that the sequences and are as follows.

(i)Cesaro Orlicz asymptotically equivalent (Wijsman sense) of multiple provided that

denoted by , and simply Wijsman Cesaro Orlicz asymptotically equivalent if .(ii)Cesaro Orlicz -asymptotically equivalent of multiple provided that, for every

denoted by and simply Wijsman Cesaro Orlicz -asymptotically equivalent if .(iii)Orlicz asymptotically -equivalent (Wijsman sense) of multiple provided that

denoted by , and simply Wijsman Orlicz asymptotically -equivalent if .(iv)Orlicz -asymptotically -equivalent (or -equivalent) (Wijsman sense) of multiple provided that, for every ,

denoted by , and simply Wijsman Orlicz -asymptotically -equivalent if .(v)Orlicz asymptotically -statistical equivalent (Wijsman sense) of multiple provided that, for every ,

denoted by , and simply Wijsman Orlicz asymptotically -statistical equivalent if .(vi)Orlicz -asymptotically -statistical equivalent (Wijsman sense) of multiple provided that, for every and for every

denoted by , and simply Wijsman Orlicz -asymptotically -statistical equivalent if .

Theorem 16. Let be a metric space. For any non-empty closed subsets such that and for each and be an Orlicz function. Then we have

(a);(b)if satisfies the -condition and such that , then ;(c)if satisfies the -condition, then , where .

Proof . (a) Suppose that . Let be given. Then we can write

Consequently, for any , we have

Hence, .

(b) Suppose that is bounded and . Since is bounded then there exists a real number such that . Moreover, for any we can write

Now, for any , we get

Hence, .

(c) The proof of this part follows from parts (a) and (b).

Theorem 17. Let be a metric space. For any non-empty closed subsets such that and for each and be a nondecreasing sequence of positive real numbers such that as and for every . Then, .

Proof. By the definition of the sequences , it follows that . Then there exists such that

Suppose that ; then, for every and sufficiently large , we have

Consequently, for any , we have

This completes the proof of the theorem.

Theorem 18. Let be a metric space. For any non-empty closed subsets such that and for each , and let be an Orlicz function which satisfies the -conditions. Then .

Proof. By the definition of the sequences , it follows that . Then there exists such that

Suppose that , then for every and sufficiently large we have

Since satisfies the -condition, it follows that

for some constant in both cases where and .

In the first case it follows from the definition of Orlicz function, and for the second case we have

such that . Using the -condition of Orlicz functions we get the following estimation

where and are constants. The proof of the theorem follows from the relations (34) and (37).

Theorem 19. Let be a metric space. For any non-empty closed subsets such that and for each . Let be an Orlicz function and such that . Then .

Proof. If , then there exists such that for all . Let be an integer such that . Then for every , we have

Consequently for any , we have

This established the result.

Theorem 20. Let be a metric space. For any non-empty closed subsets such that and for each . Let be an Orlicz function. Then we have

(a);(b) for every , then, .

Proof. (a) From the definition of the sequence it follows that . Then there exists such that

Then we get the following relation:

Since and are continuous, then for any , from the last relation we get

Hence, .

(b) For this part we assume that the sequence satisfies the condition that for any set . Suppose that then there exists such that for all . Suppose that . Then for every we put

From our assumption it is clear that . Further, observe that

Let be any integer with for some . Then we have

where are sets of integer which have more than elements for . Choosing and in view of the fact that , where . It follows from our assumption that the set . This completes the proof of the theorem.

Theorem 21. Let be a metric space. For any non-empty closed subsets such that and for each . Let be an Orlicz function. Then we have

(a);(b)if satisfies the -condition and such that then ;(c)if satisfies the -condition, then .

Proof. The proof of this theorem follows from the same techniques used in the proofs of Theorems 16 and 20. It is omitted here.

Acknowledgments

The author expresses his heartfelt gratitude to the anonymous reviewer for such excellent comments and suggestions which have enormously enhanced the quality and presentation of this paper.

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