Lacunary <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.3241005pt" id="M1" height="12.0163pt" version="1.1" viewBox="-0.0657574 -11.6922 15.5818 12.0163" width="15.5818pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g198-10" d="M63 95C70 84 85 83 93 83C121 83 136 106 136 131C136 160 113 179 85 179C52 179 26 150 26 107C26 41 92 -15 199 -15C302 -15 399 18 484 106C535 159 576 245 617 333C684 348 739 389 774 457L758 467C722 411 681 381 633 368C679 468 726 563 779 603C807 603 888 604 888 655C888 667 875 675 859 675C848 675 818 671 783 648C758 651 718 656 665 656C605 656 537 649 482 616C434 585 396 536 396 471C396 392 448 345 520 330C511 311 502 292 493 271C400 61 306 16 181 16C110 16 61 68 61 95H63ZM729 601C648 547 595 474 536 362C465 370 437 423 437 462C437 518 477 560 522 580S625 604 676 604C696 604 714 604 729 603V601Z"/></g></svg>-</span>Invariant Convergence of Sequence of Sets in Intuitionistic Fuzzy Metric Spaces

Huban, Mualla Birgül

doi:https://doi.org/10.1155/2021/7302292

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Volume 2021 | Article ID 7302292 | https://doi.org/10.1155/2021/7302292

Lacunary -Invariant Convergence of Sequence of Sets in Intuitionistic Fuzzy Metric Spaces

Mualla Birgül Huban¹

Academic Editor: Huseyin Isik

Received28 Jul 2021

Revised12 Aug 2021

Accepted16 Oct 2021

Published02 Nov 2021

Abstract

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 [1] investigated the concept of statistical convergence. The publication of the study is affected deeply all the scientific fields. Nuray and Ruckle [2] 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. [3] 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 [9] worked the notion of lacunary statistical convergence by using lacunary sequence.

Various authors involving Raimi [10], Schaefer [11], and Mursaleen [12] worked invariant convergent sequences. Nuray et al. [13] investigated -convergence with the help of -uniform density. Mursaleen [14] put forward the idea of strongly -convergence. Savaş and Nuray [15] presented the opinion of -statistical convergence and lacunary -statistical convergence and proved some correlation theorems. Nuray and Ulusu [16] defined lacunary -invariant convergence and lacunary -invariant Cauchy sequence of real numbers.

After the original study of Zadeh [17], 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 [18]. 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 [23] worked fuzzy metric space (FMS) using the concepts fuzzy and probabilistic metric space. Park [24] rethought FMSs and investigated intuitionistic fuzzy metric space (IFMS). Park utilized George and Veeramani’s [25] opinion of using -norm and -conorm to the FMS meantime describing IFMS and investigating its fundamental properties. In [26], 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 [32] presented a new convergence concept for sequences of sets called Wijsman statistical convergence. Ulusu and Nuray [33] examined the lacunary statistical convergence of sequence of sets. Kişi and Nuray [34] 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 [41]. Furthermore, Wijsman -convergence and Wijsman -convergence for sequence of sets in IFMS were investigated by Esi et al. [42].

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 [4]). 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, .(ii)Presume that , then . Let . By supposition, we get . Since is bounded, there is , such that for each and for all , . Then, we acquire For each . Hence, we obtain Uniformly in . As a result, we get .(iii)It is clear from the consequence of (i) and (ii)

Definition 7. A sequence is named to be lacunary invariant statistical convergent or -convergent (Wijsman sense) to Y, if for each , for each , and for all ,uniformly in .

Theorem 8. A sequence is -convergent to Y iff it is -convergent to Y.

3. Conclusion

Fuzzy set theory is based on the assumption that reasoning is not crystal clear. This theory has a significant role in the areas of technology and science. Intuitionistic fuzzy set has many application areas; for example, sale analysis, new product marketing, and financial services. This study aims to find out the use of the notion of lacunary -invariant convergence of sequence of sets for demonstrating some results in the area of intuitionistic fuzzy metric space. With the help of its applications, we give the notions of Wijsman lacunary -invariant convergent, Wijsman lacunary -invariant convergent, and Wijsman -strongly lacunary invariant convergent sequences in IFMS and acquired meaningful results for these notions. Also, we have examined the notions of Wijsman lacunary -invariant Cauchy and Wijsman -invariant Cauchy sequence in IFMS. The elements of IFMS have been studied. The results acquired here are more common than corresponding results for metric spaces. It is expected that new results will help to understand deeply the concept of this new type of convergence on IFMS. It could also be possible to work with the opinion of “Lacunary -invariant convergence of sequence of sets in probabilistic metric space” utilizing intuitionistic probability theory in the prospective studies.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2021 Mualla Birgül Huban. 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.

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