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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
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 . This notion was redefined for double sequences by Mursaleen and Edely . As a consequence of the study of ideal convergence defined by Kostyrko et al. , there has been valuable research to discover summability works of the classical theories. Das et al.  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 [5–11].
Fridy and Orhan  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  demonstrated multidimensional similarities of the conclusions given by Fridy and Orhan . Some works in lacunary statistical convergence can be found in [13–17].
Theory of fuzzy sets (FSs) was firstly given by Zadeh . This work affected deeply all the scientific fields. The theory of FSs has submitted to employ imprecise, vagueness, and inexact data . 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  investigated the theory of IFS which is the extension of the theory of FSs. Kramosil and Michalek  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 . George and Veeramani  gave some qualifications of FMS. Some basic features of FMS were given, and significant theorems were proved in . Moreover, FMS has used in practical research studies, for example, decision-making, fixed point theory, and medical imaging. Park  generalized FMSs and defined IF metric space (IFMS). Park utilized George and Veeramani’s  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 . 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 [26–29].
Recently, convergence of sequences of sets was studied by several authors. Nuray and Rhoades  presented the idea of statistical convergence of set sequences and established some essential theorems. Ulusu and Nuray  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 [32–38]).
Lacunary statistical convergence and lacunary strongly convergence for sequence of sets in IFMS were worked by Kisi . Further, Wijsman -convergence and Wijsman -convergence for sequence of sets in IFMS were investigated by Esi et al. .
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 .
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