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Research Article | Open Access
Mualla Birgül Huban, "Lacunary -Invariant Convergence of Sequence of Sets in Intuitionistic Fuzzy Metric Spaces", Journal of Mathematics, vol. 2021, Article ID 7302292, 10 pages, 2021. https://doi.org/10.1155/2021/7302292
Lacunary -Invariant Convergence of Sequence of Sets in Intuitionistic Fuzzy Metric Spaces
The concepts of invariant convergence, invariant statistical convergence, lacunary invariant convergence, and lacunary invariant statistical convergence for set sequences were introduced by Pancaroğlu and Nuray (2013). We know that ideal convergence is more general than statistical convergence for sequences. This has motivated us to study the lacunary -invariant convergence of sequence of sets in intuitionistic fuzzy metric spaces (briefly, IFMS). In this study, we examine the notions of lacunary -invariant convergence (Wijsman sense), lacunary -invariant convergence (Wijsman sense), and -strongly lacunary invariant convergence (Wijsman sense) of sequences of sets in IFMS. Also, we give the relationships among Wijsman lacunary invariant convergence, , , and in IFMS. Furthermore, we define the concepts of -Cauchy sequence and -Cauchy sequence of sets in IFMS. Furthermore, we obtain some features of the new type of convergences in IFMS.
1. Introduction and Background
Fast  investigated the concept of statistical convergence. The publication of the study is affected deeply all the scientific fields. Nuray and Ruckle  redefined this concept which is known as generalized statistical convergence. A lot of development has been made in area about statistical convergence. Kostyrko et al.  defined ideal convergence, as a generalization of statistical convergence and worked some features of this convergence. Ideal convergence became a remarkable topic in summability theory after the studies of [4–8]. Fridy and Orhan  worked the notion of lacunary statistical convergence by using lacunary sequence.
Various authors involving Raimi , Schaefer , and Mursaleen  worked invariant convergent sequences. Nuray et al.  investigated -convergence with the help of -uniform density. Mursaleen  put forward the idea of strongly -convergence. Savaş and Nuray  presented the opinion of -statistical convergence and lacunary -statistical convergence and proved some correlation theorems. Nuray and Ulusu  defined lacunary -invariant convergence and lacunary -invariant Cauchy sequence of real numbers.
After the original study of Zadeh , a huge number of research works have appeared on fuzzy theory and its applications as well as fuzzy analogues of the classical theories. Fuzzy sets (FSs) have been extensively applied in different disciplines and technologies. The theory of intuitionistic fuzzy sets (IFS) was presented by Atanassov . The fuzzy sets and intuitionistic fuzzy sets have been widely used to solve many complex problems connected to different areas, especially in decision-making [19–22]. Kramosil and Michalek  worked fuzzy metric space (FMS) using the concepts fuzzy and probabilistic metric space. Park  rethought FMSs and investigated intuitionistic fuzzy metric space (IFMS). Park utilized George and Veeramani’s  opinion of using -norm and -conorm to the FMS meantime describing IFMS and investigating its fundamental properties. In , motivated by Park’s definition of an IF-metric, Lael and Nourouzi first defined an IF-normed space and then investigated, among other results, the fundamental theorems: open mapping, closed graph, and uniform boundedness in IF-normed spaces. 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 [27–31]. For the extraction of information by reflecting and modeling the hesitancy present in real-life situation, intuitionistic fuzzy set theory has been playing a significant role. The implementation of IF sets in place of fuzzy sets means the introduction of another degree of freedom into set description. IF fixed point theory has become a subject of great interest for expert in fixed point theory because this branch of mathematics has covered new possibilities for summability theory.
Convergence of sequences of sets has been examined by several authors. Nuray and Rhoades  presented a new convergence concept for sequences of sets called Wijsman statistical convergence. Ulusu and Nuray  examined the lacunary statistical convergence of sequence of sets. Kişi and Nuray  investigated ideal convergence for sequences of sets (Wijsman sense) and established some essential theorems. Convergence for sequences of sets became a notable topic in summability theory after the studies of [35–40].
Lacunary statistical convergence and lacunary strongly convergence for sequence of sets in IFMS were examined by Kişi . Furthermore, Wijsman -convergence and Wijsman -convergence for sequence of sets in IFMS were investigated by Esi et al. .
The purpose of this study is to present some recent development in IFMS. The aim of the study is to examine some features of this new kind of convergence in IFMS. Also, it is demonstrated that the new kind of convergence in IFMS is generally different from the known convergence in classical metric space. However; it is indicated that if certain conditions are met, every classical metric space can be a IFMS at the same time.
Throughout this work, we indicate to be the admissible ideal in , be a 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 called to be lacunary invariant convergent (Wijsman sense) to with regards to IFM , if for every , for each and for all , such thatuniformly in .
Definition 2. A sequence of nonempty closed subsets of is known as lacunary -invariant convergent or -convergent (Wijsman sense) to Y with regards to IFM , if for every , for each and for all , the setthat is, . We demonstrate this symbolically by .
Theorem 1. Let be a bounded sequence. If is -convergent to , then is lacunary invariant convergent (Wijsman sense) to with regards to IFM .
Proof. Let be arbitrary and . For each and for all , we estimateThen, for each and for all , we getwhereFor every and for every , it is obvious that . Since is bounded sequence, there is a , such thatand so, we haveHence, we obtainSimilarly, we haveHence, is lacunary invariant convergent to Y (Wijsman sense) with regards to IFM .
Definition 3. Let be a separable IFMS, and be a proper ideal in . The sequence is known as lacunary -invariant convergent or -convergent (Wijsman sense) to with regards to IFM , if there is a setsuch that for each and for all ,In that case, we write .
Theorem 2. If a sequence is -convergent to , then is -convergent to with regards to IFM .
Proof. Presume that . Then, (i.e., (say)), such thatBut then, for each and , there is , such thatfor all . Sinceis included in and the ideal is admissible, we getHence,for all and . Therefore, we conclude that .
Theorem 3. Let the ideal fulfill the property . If is a sequence in , such that , then .
Proof. Assume that provides the feature and . Then, for every , for each and for all ,We define the set for and asClearly, is countable and belongs to and for . By the feature , there is a sequence of , such that the symmetric differences are finite sets for and . Now, to conclude the proof, it is enough to show that for and for each , we getfor . Let . Select , such that . For each , we acquireSince are finite sets, there is a , such thatIf and , thenHence, for each and , we haveSince is arbitrary, we obtain .
Definition 4. A sequence is known as lacunary -invariant Cauchy sequence or -Cauchy sequence (Wijsman sense) with regards to IFM if for each , for each and for all , there is , such thatthat is, .
Definition 5. A sequence is known as lacunary -invariant Cauchy sequence or -Cauchy sequence (Wijsman sense) with regards to IFM provided that there is a setsuch thatfor each and for all .
We give following theorems which indicate relationships among -convergence, -Cauchy sequence, and -Cauchy sequence with regards to IFM .
Theorem 4. If a sequence is -convergent, then is -Cauchy sequence with regards to IFM .
Proof. Presume that . Then, for every , for each and for all , the setbelongs to . Since is an admissible ideal, then there is with the result that . Now, assume thatThinking the inequalityObserve that if , therefore,From another standpoint, since , we obtainWe reach thatHence, . This gives that for every and . Therefore, , so is -Cauchy sequence with regards to IFM .
Theorem 5. Let be a separable IFMS and be an admissible ideal. If a sequence is -Cauchy sequence, then is -Cauchy sequence with regards to IFM .
Proof. Assume that sequence is -Cauchy with regards to IFM . Then, for each and for each , there is , where , such thatPresume that . Therefore, for each , one getsfor all . Now, assume that . Obviously, andAs a consequence, for all and for each , one can identify , such that , that is, sequence is -Cauchy with regards to IFM .
Lemma 1 (see ). Assume be an admissible ideal with the feature . Let there be a countable collection of subsets of in such a way that . As a result, there is a set , such that is finite for all and .
Theorem 6. Let provides the feature . Then, the notions -Cauchy sequence and -Cauchy sequence with regards to IFM coincide.
Proof. The direct part has been proved in Theorem 5.
Now, assume that the sequence is -Cauchy sequence with regards to IFM . Then, for each , for each , and for all , there is a , such thatNow, presume thatwhere , . Clearly, for , . Using Lemma 1, there is , so that and are finite for all .
Now, we denote thatTo demonstrate the above equations, let and , such that . If , then is a finite set; so, there is in order thatfor all . From the above inequalities, we obtainfor all .
Therefore, for every , , and , we acquireThis give that the sequence is -Cauchy.
Definition 6. The sequence is named to be -strongly lacunary invariant convergent (Wijsman sense) to , provided that for each and for all ,uniformly in , where . We indicate this symbolically by .
Theorem 7. Let be an admissible ideal and .(i)If , then (ii)If is bounded and , then (iii)If , then iff
Proof. (i)If , then for every , for each and for all , we obtain And so, For every . This gives that , and hence,