Some Results on Wijsman Ideal Convergence in Intuitionistic Fuzzy Metric Spaces

Abstract

In the present work, we study and extend the notion of Wijsman –convergence and Wijsman –convergence for the sequence of closed sets in a more general setting, i.e., in the intuitionistic fuzzy metric spaces (briefly, IFMS). Furthermore, we also examine the concept of Wijsman –Cauchy and –Cauchy sequence in the intuitionistic fuzzy metric space and observe some properties.

1. Introduction

In 1951, Fast [1] initiated the theory of statistical convergence. It is an extremely effective tool to study the convergence of numerical problems in sequence spaces by the idea of density. Statistical convergence of the sequence of sets was examined by Nuray and Rhoades [2]. Ulusu and Nuray [3] studied the Wijsman lacunary statistical convergence sequence of sets and connected with the Wijsman statistical convergence. Esi et al. [4] introduced the Wijsman -statistical convergence of interval numbers. Kostyrko et al. [5] generalized the statistical convergence and introduced the notion of ideal –convergence. Salát et al. [6, 7] investigated it from the sequence space viewpoint and associated with the summability theory. Further, it was analyzed by Khan et al. [8] with the help of a bounded operator. In 2008, Das et al. [9] analyzed and –convergence for double sequences. Kisi and Nuray [10] initiated new convergence definitions for the sequence of sets. Furthermore, Gümüş [11] studied the Wijsman ideal convergent sequence of sets using the Orlicz function.

In 1965, Zadeh [12] started the fuzzy sets theory. This theory has proved its usefulness and ability to solve many problems that classical logic was unable to handle. Karmosil et al. [13] introduced the fuzzy metric space, which has the most significant applications in quantum particle physics. Afterward, numerous researchers have studied the concept of fuzzy metric spaces in different ways. George et al. [14, 15] modified the notion of fuzzy metric space and determined a Hausdorff topology for fuzzy metric spaces. Atanassov [16] generalized the fuzzy sets and introduced the notion of intuitionistic fuzzy sets in 1986. Park [17] examined the notion of IFMS, and Saadati and Park [18] further analyzed the intuitionistic fuzzy topological spaces. Moreover, statistical convergence, ideal convergence, and different properties of sequences in intuitionistic fuzzy normed spaces were examined by Mursaleen et al. [19–21]. Also one can refer to Sengül and Et [22], Sengül et al. [23], Et and Yilmazer [24], Mohiuddine and Alamri [25], and Mohiuddine et al. [26, 27].

2. Preliminaries

We recall some concepts and results which are needed in sequel.

Definition 1 [5]. A family of subsets is known as an ideal in a non-empty set , if(1)(2)for any (3)for any and .

Remark 2 [5]. An ideal is known as non-trivial if . A nontrivial ideal is known as admissible if .

Definition 3 [5]. A nonempty subset is known as filter in if (1)for every ,(2)for every ,(3)for every with , one obtain

Proposition 4 [5]. For every ideal , there is a filter associated with defined as follows:

Definition 5 [5]. Let be a mutually disjoint sequence of sets of . Then, there is sequence of sets so that and each symmetric difference is finite. In this case, admissible ideal is known as property .

Lemma 6 [28]. Suppose be an admissible ideal alongside property . Let a countable collection of subsets of positive integer in such a way that . Then, there exists a set such that is finite for all and .

Definition 7 [29]. Let be a metric space and be a sequence of nonempty closed subsets of which is said to be Wijsman convergent to the closed of, if

In other words, .

In 2012, Nuray and Rhoades [2] initiated the theory of Wijsman statistical convergence for a sequence of sets. Furthermore, Kisi and Nuray [10] extended it into -convergence.

Definition 8 [10]. Suppose is a metric space. A nonempty closed subset of is known as Wijsman –convergent to a closed set , if for every , one has Hence, one writes .

Definition 9 [10]. Suppose is a metric space. A nonempty closed subsetofis known as Wijsman–Cauchy if for each, there exists a positive integerso that the set

Definition 10 [10]. Suppose is a separable metric space and is nonempty closed subsets of . A sequence is known as Wijsman –convergent to if and only if and in such a manner that One writes .

Definition 11 [10]. Suppose is a separable metric space and is an admissible ideal. A sequence of nonempty closed subsets of is known as the Wijsman –Cauchy sequence if there exists , where in such a way that subsequence is Wijsman Cauchy in , i.e.,

Remark 12 [10]. In general, the Wijsman topology is not first-countable, if sequence of nonempty sets is Wijsman convergent to set , then every subsequence of may not be convergent to . Every subsequence of the convergent sequence converges to the same limit provided that is a separable metric space.

Definition 13 [17]. Let be a nonempty set, and be fuzzy sets on , be a continuous -norm, and be a continuous -conorm. Then, the five-tuple is known as an intuitionistic fuzzy metric space (for short, IFMS) if it fulfills the subsequent conditions for all and for every : (a)(b)(c) if and only if ,(d),(e)(f) is continuous,(g),(h) if and only if ,(i)(j)(k) is continuous.In such situation, is called the intuitionistic fuzzy metric (briefly, IFM).

Example 14 [17]. Suppose is a metric space. Define and for all , and suppose and are fuzzy sets on defined as Then is an IFMS.

Definition 15 [18]. Let be an IFMS and be a nonempty subset of . For all and , we define and where and are the degree of nearness and nonnearness of to at , respectively.

Saadati and Park [18] studied the notion of convergence sequence with respect to IFMS which are defined as follows:

Definition 16 [18]. Let be an IFMS. A sequence is convergent to if for any and there exists in such a way that

Definition 17 [20]. An IFMS is known as separable if it contains a countable dense subset, i.e., there is a countable set along with subsequent property: for any and for all , there is at least one in order that

3. Wijsman and –convergence in IFMS

Throughout this section, we denote to be the admissible ideal in . We begin with the following definitions as follows.

Definition 18. Let be an IFMS. A sequence of sets is said be Wijsman convergent to if for every and there exists such that The set of all Wijsman limit point of the sequence is denoted by .

Definition 19. Let be an IFMS and be a proper ideal in . A sequence of nonempty closed subsets of is known as Wijsman –convergent to with respect to IFM , if for every , for each and for all such that We write .

Example 20. Suppose is an IFMS and is nonempty closed subsets of . Assume and are sequence defined by Since

Therefore, the sequence of sets is Wijsman statistical convergent to the set .

Now, define the set as

If we assume , then the Wijsman statistical convergence coincides with the Wijsman ideal convergence. Therefore,

Definition 21. Let be a separable IFMS and be an admissible ideal in . A sequence of nonempty closed subsets of is known as Wijsman –Cauchy with respect to IFM , if for each , for each and for all such that

Definition 22. Let be a separable IFMS and be any nonempty closed subset of . The sequence is known as Wijsman –Cauchy with respect to IFM , if there exists and with the result that the subsequence is Wijsman Cauchy in , i.e. and

Definition 23. Let be a separable IFMS and be an proper ideal in . Let be nonempty closed subsets of . The sequence is known as Wijsman –convergent to with respect to , if there exists , where such that for each , we have and In such case, we write .

In the following theorem, we prove that every Wijsman –convergent implies the Wijsman –Cauchy condition in IFMS:

Theorem 24. Let be a separable IFMS and let be an arbitrary admissible ideal. Then, every Wijsman –convergent sequence of closet sets is Wijsman –Cauchy with respect to IFM .

Proof. Suppose . Then, for every , for all and , the set belongs to . Since is an admissible ideal, then there exists with the result that . Now, suppose that Considering the inequality and Observe that if , therefore and From another point of view, since , we obtain We achieve that Hence, . This implies that for every and for all and . Therefore, , so the sequence is which is Wijsman –Cauchy.

Theorem 25. Let be a separable IFMS and let be an admissible ideal. Then, every Wijsman –Cauchy sequence of closed sets is Wijsman –Cauchy.

Proof. Suppose that sequence is Wijsman –Cauchy with respect to IFM . Then, for each and for each , there exists , where in such a way that and Suppose . Therefore, for each , one obtains and Now, suppose that . Clearly, and Hence, for all and for each , one can determine so that , that is, sequence is Wijsman –Cauchy.

Theorem 26. Let be an admissible ideal including property (AP) and be a separable IFMS. Then, the notion of Wijsman –Cauchy sequence of sets coincides with Wijsman –Cauchy with respect to and vice-versa.

Proof. The direct part is already proven in Theorem 25.
Now, suppose that sequence is Wijsman –Cauchy sequence with respect to IFM . Then by definition, if for every , for each and for all , there exists a such that Now, suppose that where , . Obviously, for , . Using Lemma 6, there exists so that and are finite for all .
Now, we prove that and To show the above equations, let , and such that . If , then is a finite set; therefore, there exists in order that and for all . Then, the above inequalities follow that for and Therefore, for each and , we achieve This proves that the sequence is a Wijsman –Cauchy.

Theorem 27. Let be a separable IFMS and let be an admissible ideal. Then implies that sequence is a Wijsman –Cauchy sequence with respect to IFM .

Proof. Suppose that . Then, there exists with so that and for any and .
Suppose and in such a way that . If , then is a finite set; therefore, there exists so that and Therefore, and Hence, sequence is Wijsman –Cauchy with respect to IFM .

4. Wijsman –cluster points and Wijsman –limit points in IFMS

Throughout this section, we denote to be the proper ideal in and define Wijsman –cluster and –limit points of the sequence of sets in intuitionistic fuzzy metric space and obtain some results.

Definition 28. Let be a separable IFMS. An element is known as the Wijsman -cluster point of if and only if for any and for all , one has

We denote as the collection of all Wijsman -cluster points.

Definition 29. Let be a separable IFMS. An element is known as Wijsman –limit point of sequence of nonempty closed subsets of provided that in such a way that , and for any and , we obtain

We denote as the collection of all Wijsman –limit points.

Theorem 30. Let be a separable IFMS. Then, for any sequence sets, , .

Proof. Suppose . Then, there exists such that and for all and , we have and According to Equations (54) and (55), there exists so that for each and for any and and Hence, Then, the right-hand side of (58) does not belong to , and then which means that .

Theorem 31. Let be a separable IFMS. Then, for any sequence , .

Proof. Let . Then, for each and for all and for each , one has Suppose for . is a descending sequence of subsets of . Hence, so that and which means that .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there is no conflict of interest.

Authors’ Contributions

All authors contributed equally and significantly in writing this article.