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Volume 2021 |Article ID 6897038 | https://doi.org/10.1155/2021/6897038

Ömer Kişi, "Some New Observations on Wijsman -Lacunary Statistical Convergence of Double Set Sequences in Intuitionistic Fuzzy Metric Spaces", Journal of Mathematics, vol. 2021, Article ID 6897038, 17 pages, 2021. https://doi.org/10.1155/2021/6897038

Some New Observations on Wijsman -Lacunary Statistical Convergence of Double Set Sequences in Intuitionistic Fuzzy Metric Spaces

Academic Editor: Huseyin Isik
Received19 Jul 2021
Accepted08 Oct 2021
Published18 Oct 2021

Abstract

In this study, we investigate the notions of the Wijsman -statistical convergence, Wijsman -lacunary statistical convergence, Wijsman strongly -lacunary convergence, and Wijsman strongly -Cesàro convergence of double sequence of sets in the intuitionistic fuzzy metric spaces (briefly, IFMS). Also, we give the notions of Wijsman strongly -lacunary convergence, Wijsman strongly -lacunary Cauchy, and Wijsman strongly -lacunary Cauchy set sequence in IFMS and establish noteworthy results.

1. Introduction and Background

Statistical convergence was firstly examined by Henry Fast [1]. This notion was redefined for double sequences by Mursaleen and Edely [2]. As a consequence of the study of ideal convergence defined by Kostyrko et al. [3], there has been valuable research to discover summability works of the classical theories. Das et al. [4] rethought -convergence of double sequences and worked some features of this convergence. Ideal convergence became a noteworthy topic in summability theory after the studies of [511].

Fridy and Orhan [12] examined the notion of lacunary statistical convergence by using lacunary sequence. The publication of the paper affected deeply all the scientific fields. Çakan and Altay [13] demonstrated multidimensional similarities of the conclusions given by Fridy and Orhan [12]. Some works in lacunary statistical convergence can be found in [1317].

Theory of fuzzy sets (FSs) was firstly given by Zadeh [18]. This work affected deeply all the scientific fields. The theory of FSs has submitted to employ imprecise, vagueness, and inexact data [18]. FSs have been widely implemented in different disciplines and technologies. The theory of FSs cannot always cope with the lack of knowledge of membership degrees. That is why Atanassov [19] investigated the theory of IFS which is the extension of the theory of FSs. Kramosil and Michalek [20] investigated fuzzy metric space (FMS) utilizing the concepts fuzzy and probabilistic metric space. The FMS as a distance between two points to be a nonnegative fuzzy number was examined by Kaleva and Seikkala [21]. George and Veeramani [22] gave some qualifications of FMS. Some basic features of FMS were given, and significant theorems were proved in [23]. Moreover, FMS has used in practical research studies, for example, decision-making, fixed point theory, and medical imaging. Park [24] generalized FMSs and defined IF metric space (IFMS). Park utilized George and Veeramani’s [22] opinion of using t-norm and t-conorm to the FMS meantime describing IFMS and investigating its fundamental properties. The concept of IF-normed spaces (IFNS for shortly) was given by Lael and Nourouzi [25]. In order to have a different topology from the topology generated by the -norm , the condition was omitted from Park’s definition. Statistical convergence, ideal convergence, and different features of sequences in INFS were examined by several authors [2629].

Recently, convergence of sequences of sets was studied by several authors. Nuray and Rhoades [30] presented the idea of statistical convergence of set sequences and established some essential theorems. Ulusu and Nuray [31] examined the lacunary statistical convergence of sequence of sets. Convergence for sequences of sets became a notable topic in summability theory after the studies of (see [3238]).

Lacunary statistical convergence and lacunary strongly convergence for sequence of sets in IFMS were worked by Kisi [39]. Further, Wijsman -convergence and Wijsman -convergence for sequence of sets in IFMS were investigated by Esi et al. [40].

Throughout this work, we indicate to be the admissible ideal in , to be a double lacunary sequence, to be the IFMS, and to be nonempty closed subsets of .

2. Main Results

Definition 1. A sequence of nonempty closed subsets of is known as Wijsman -statistical convergent to or -convergent to with regard to IFM , if for every , , for each , and for all ,We demonstrate this symbolically by or . The set of all Wijsman -statistical convergent sequences in IFMS is indicated by .

Example 1. Let and double sequence be determined as follows:If ( is the class of with density of equal to 0), then the sequence is Wijsman -statistical convergent to with regard to IFM .

Definition 2. A sequence is Wijsman strong -Cesàro summable to or -summable to with regard to IFM , if for every , for each , and for all ,We write or .

Example 2. Let and double sequence be determined as follows:If ( is the class of finite subsets of ), then sequence is Wijsman strong -Cesàro summable to with regard to IFM .

Definition 3. The sequence is known as Wijsman -lacunary statistically convergent to or -convergent to with regard to IFM , if for every , , for each , and for all ,In that case, we write or .

Example 3. Let and double sequence be determined as follows:If we take , then the sequence is Wijsman -lacunary statistical convergent to with regard to IFM .

Definition 4. A double sequence is Wijsman strong -lacunary summable to or -summable to with regard to IFM , if for every , for all , and for each ,We shall write or .

Example 4. Let and double sequence be determined as follows:If we take , then the sequence is Wijsman strong -lacunary summable to with regard to IFM .

Theorem 1. Let be a double lacunary sequence. Then,

Proof. Let and . At that time, for every , we acquireand soThen, for any and for every ,This proof is concluded.

Theorem 2. Let be a double lacunary sequence. Then, is bounded () andThe set of all bounded double sequences of sets in IFMS is indicated by .

Proof. Presume that and . At this point, there is an such thatfor every and all . Given , we obtainAs a consequence, for each , we getThis proof is concluded.

Corollary 1. We have the following result:

Theorem 3. If and , then implies .

Proof. Presume that and . Then, there are such thatfor sufficiently large , which gives thatAssume that . For each , for all , and for each , we haveThus, for any ,Hence, by our supposition, the set on the right side belongs to , and clearly the set on the left side belongs to . As a result, we obtain .

Theorem 4. If and , then implies .

Proof. Presume that and . Then, there are such that and for all and . Assume that and letSince , it holds for each , , for every , and for all ,So, we can select positive integers such that for all . Now, takeand let and be integers providing and . Then, for every and each , we getSince as , it concludes that for each ,and as a result for any , the setIt gives that .

Theorem 5. Let be a double lacunary sequence. Ifthen if .

Proof. It obvious from Theorem 3 and Theorem 4.

Theorem 6. Let be a strongly admissible ideal providing feature , . If , then

Proof. Presume that and and . Letfor every . Since provides the feature , then there is (i.e., ) such that for every and for ,LetThen, . This gives that . Since and , we have to get .
Let . Then, for every and for , the th term of the statistical limit expressionis wherebecause . Since is a lacunary sequence, (34) is a regular weighted mean transform of ’s, and as a result, it is -convergent to 0 as , and also it has a subsequence which is convergent to 0 since provides the feature . However, since this a subsequence ofwe conclude thatwhich is not convergent to 1. This contradiction indicates that we cannot have .

Theorem 7. If and , then .

Proof. Let and . Then, there are such that and for all which gives thatPresume that . For each , we getSince , then for each