Some New Observations on Wijsman <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.1996pt" id="M1" height="15.8918pt" version="1.1" viewBox="-0.0657574 -11.6922 22.1111 15.8918" width="22.1111pt"><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><g transform="matrix(.012,0,0,-0.012,15.393,4.134)"><path id="g50-51" d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z"/></g></svg>-</span>Lacunary Statistical Convergence of Double Set Sequences in Intuitionistic Fuzzy Metric Spaces

Kişi, Ömer

doi:https://doi.org/10.1155/2021/6897038

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

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

Ömer Kişi¹

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 [5–11].

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 [13–17].

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 [26–29].

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 [32–38]).

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 eachHence, when the above equality is examined, for every , we haveSimilarly, we obtainThat is, . As a result, we obtain .

Theorem 8. If and , then .

Proof. Take , , and . Then, for every , we acquireas . Then, for , there are such that for all , . Also, we can find such that and , . Let be an integer with and . Then,Since as , it follows that . Similarly, we can show that . Hence, .

Theorem 9. If , then .

Proof. Let and . Assume and in such way that . Then, for any , there are such thatfor all and . Also, there are such thatfor all and . Take and . Then, we take such thatTherefore, we getSince is arbitrary, we get for all , which yields .

Throughout the following definitions and theorems, we consider to be a separable IFMS and to be a strongly admissible ideal.

Definition 5. The sequence is strongly -lacunary Cauchy sequence (Wijsman sense) if for each , for each , and for all , there are such that

Theorem 10. Every Wijsman strongly -lacunary convergent sequence of closed sets is Wijsman strongly -lacunary Cauchy with regard to IFM .

Proof. Let . At that case, for each , for every , and for all ,Since is a strongly admissible ideal, the setis nonempty and belongs to . So, we select positive integers such that , and we getNow, presume thatConsider the inequalityNotice that if , therefore,From another point of view, since , we getWe reach thatHence, . This gives that for every and for each . Therefore, , so the sequence is Wijsman strongly -lacunary sequence.

Definition 6. The sequence is Wijsman strongly -lacunary convergent to iff there is a set such that for each ,In this case, we write .

Theorem 11. If the sequence is Wijsman strongly -lacunary convergent to , then is Wijsman strongly -lacunary convergent to .

Proof. Presume that . Then, there is a set such thatfor each ,for every and for all . Hereby, for each and , we getfor . Since is an admissible ideal, we obtainand so . Hence, .

Theorem 12. Let be a strongly admissible ideal involving feature . Then, implies .

Definition 7. The sequence is known as Wijsman strongly -lacunary Cauchy sequence if for each , for all , and for all , and there is a set such that and such thatfor every .

Theorem 13. Every Wijsman strongly -lacunary Cauchy sequence of closed sets is Wijsman strongly -lacunary Cauchy in IFMS with regard to .

Proof. If the hypothesis is provided, then for each , for each , and for all , there is a set such thatand such thatfor each . Let . It is clear that andAs be a strongly admissible ideal, thenTherefore, we obtain ; that is, is strongly -lacunary Cauchy sequence (Wijsman sense) with regard to .

Theorem 14. Let be an admissible ideal involving property . Then, the concept of Wijsman strongly -lacunary Cauchy sequence of sets coincides with Wijsman strongly -lacunary Cauchy sequence of sets.

Proof. If a set sequence is strongly -lacunary Cauchy sequence, then it is strongly -lacunary Cauchy sequence according to Theorem 13, where need not to have the feature .
Now, it is adequate to demonstrate that a sequence in is a strongly -lacunary Cauchy sequence under assumption that it is a strongly -lacunary Cauchy sequence. Let be a Wijsman strongly lacunary Cauchy sequence. In this case, for each , for all , there is a number such thatLetwhere and , . Clearly, for , . Since has the property , then by Theorem 3.3 in [9], there is so that and is finite for all . Now, we demonstrate that and for each and . To show these, let and such that . If , then is a finite set; therefore, there is so thatfor all . So, it follows thatTherefore, for each , , and , we getThis implies that is Wijsman strongly -lacunary Cauchy.

Definition 8. A sequence in IFMS is called to be Wijsman lacunary convergent to with regard to IFM if, for every and , there is such thatfor all and . We write .

Definition 9. Take as a separable IFMS and take . (a) is known as Wijsman lacunary -limit point of if there is set such that the setand . (b) is known as Wijsman lacunary -cluster point of if, for every and , we get

Here, denotes the set of all Wijsman lacunary -limit points and indicates the set of all Wijsman lacunary -cluster points in IFMS.

Theorem 15. For each sequence in IFMS, we have .

Proof. Let . So, there is a set such that , where and are as in Definition 9, satisfying . Hence, for every and , there are such thatfor all and . Therefore,Now, with being admissible, we must have and as such . Hence, .

3. Conclusion

In this study, we examined a version of ideal convergence, named Wijsman lacunary ideal convergence of double set sequences, in IFMS. We investigated new convergence concepts for double set sequences in IFMS and obtained some meaningful results. In addition, Wijsman lacunary -limit points and Wijsman lacunary -cluster points of double set sequences in IFMS were defined. Some of the results presented in this article are analogous to the research studies in the relevant topic, but in most situations, the proofs follow a different approach.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest.

References

H. Fast, “Sur la convergence statistique,” Colloquium Mathematicum, vol. 2, no. 3-4, pp. 241–244, 1951.
View at: Publisher Site | Google Scholar
M. Mursaleen and O. H. H. Edely, “Statistical convergence of double sequences,” Journal of Mathematical Analysis and Applications, vol. 288, no. 1, pp. 223–231, 2003.
View at: Publisher Site | Google Scholar
P. Kostyrko, T. Šalát, and W. Wilczyński, “I-convergence,” Real Analysis Exchange, vol. 26, no. 2, pp. 669–686, 2000.
View at: Publisher Site | Google Scholar
P. Das, P. Kostyrko, W. Wilczyński, and P. Malik, “I and I^∗-convergence of double sequences,” Mathematica Slovaca, vol. 58, no. 5, pp. 605–620, 2008.
View at: Publisher Site | Google Scholar
P. Kostyrko, M. Macaj, T. Šalát, and M. Sleziak, “-Convergence and extremal -limit points,” Mathematica Slovaca, vol. 55, pp. 443–464, 2005.
View at: Google Scholar
A. A. Nabiev, S. Pehlivan, and M. Gürdal, “On -Cauchy sequences,” Taiwanese Journal of Mathematics, vol. 11, no. 2, pp. 569–566, 2007.
View at: Publisher Site | Google Scholar
P. Das and S. K. Ghosal, “Some further results on I-Cauchy sequences and condition (AP),” Computers & Mathematics with Applications, vol. 59, no. 8, pp. 2597–2600, 2010.
View at: Publisher Site | Google Scholar
P. Das, E. Savas, and S. Kr. Ghosal, “On generalizations of certain summability methods using ideals,” Applied Mathematics Letters, vol. 24, no. 9, pp. 1509–1514, 2011.
View at: Publisher Site | Google Scholar
E. Dündar and B. Altay, “-convergence and -Cauchy double sequences,” Acta Mathematica Scientia, vol. 34, no. 2, pp. 343–353, 2014.
View at: Publisher Site | Google Scholar
C. Belen and M. Yildirim, “On generalized statistical convergence of double sequences via ideals,” Annali dell'Universita di Ferrara, vol. 58, no. 1, pp. 11–20, 2012.
View at: Publisher Site | Google Scholar
M. Gürdal and A. Şahiner, “Extremal -limit points of double sequences,” Applied Mathematics E-Notes, vol. 8, pp. 131–137, 2008.
View at: Google Scholar
J. Fridy and C. Orhan, “Lacunary statistical convergence,” Pacific Journal of Mathematics, vol. 160, no. 1, pp. 43–51, 1993.
View at: Publisher Site | Google Scholar
C. Çakan, B. Altay, and H. Çoşkun, “Double lacunary density and lacunary statistical convergence of double sequences,” Studia Scientiarum Mathematicarum Hungarica, vol. 47, no. 1, pp. 35–45, 2010.
View at: Publisher Site | Google Scholar
B. C. Tripathy, B. Hazarika, and B. Choudhary, “Lacunary I-convergent sequences,” Kyungpook Mathematical Journal, vol. 52, no. 4, pp. 473–482, 2012.
View at: Publisher Site | Google Scholar
S. Kumar, V. Kumar, and S. S. Bhatia, “On ideal version of lacunary statistical convergence of double sequences,” General Mathematics Notes, vol. 17, no. 1, pp. 32–44, 2013.
View at: Google Scholar
T. Yaying and B. Hazarika, “Lacunary arithmetic convergence,” Proceedings of the Jangjeon Mathematical Society, vol. 21, no. 3, pp. 507–513, 2018.
View at: Google Scholar
T. Yaying and B. Hazarika, “Lacunary arithmetic statistical convergence,” National Academy Science Letters, vol. 43, no. 6, pp. 547–551, 2020.
View at: Publisher Site | Google Scholar
L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 3, pp. 338–353, 1965.
View at: Publisher Site | Google Scholar
K. T. Atanassov, “Intuitionistic fuzzy sets,” Fuzzy Sets and Systems, vol. 20, no. 1, pp. 87–96, 1986.
View at: Publisher Site | Google Scholar
I. Kramosil and J. Michalek, “Fuzzy metric and statistical metric spaces,” Kybernetika, vol. 11, no. 5, pp. 336–344, 1975.
View at: Google Scholar
O. Kaleva and S. Seikkala, “On fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 12, no. 3, pp. 215–229, 1984.
View at: Publisher Site | Google Scholar
A. George and P. Veeramani, “On some results in fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 64, no. 3, pp. 395–399, 1994.
View at: Publisher Site | Google Scholar
A. George and P. Veeramani, “On some results of analysis for fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 90, no. 3, pp. 365–368, 1997.
View at: Publisher Site | Google Scholar
J. H. Park, “Intuitionistic fuzzy metric spaces,” Chaos, Solitons & Fractals, vol. 22, no. 5, pp. 1039–1046, 2004.
View at: Publisher Site | Google Scholar
F. Lael and K. Nourouzi, “Some results on the IF-normed spaces,” Chaos, Solitons & Fractals, vol. 37, no. 3, pp. 931–939, 2008.
View at: Publisher Site | Google Scholar
P. Debnath, “Lacunary ideal convergence in intuitionistic fuzzy normed linear spaces,” Computers & Mathematics with Applications, vol. 63, no. 3, pp. 708–715, 2012.
View at: Publisher Site | Google Scholar
E. Savaş and M. Gürdal, “A generalized statistical convergence in intuitionistic fuzzy normed spaces,” ScienceAsia, vol. 41, pp. 289–294, 2015.
View at: Google Scholar
E. Savaş and M. Gürdal, “Generalized statistically convergent sequences of functions in fuzzy 2-normed spaces,” Journal of Intelligent and Fuzzy Systems, vol. 27, no. 4, pp. 2067–2075, 2014.
View at: Google Scholar
E. Savaş and M. Gürdal, “Certain summability methods in intuitionistic fuzzy normed spaces,” Journal of Intelligent and Fuzzy Systems, vol. 27, no. 4, pp. 1621–1629, 2014.
View at: Google Scholar
F. Nuray and B. E. Rhoades, “Statistical convergence of sequences of sets,” Fasciculi Mathematici, vol. 49, pp. 87–99, 2012.
View at: Google Scholar
U. Ulusu and F. Nuray, “Lacunary statistical convergence of sequences of sets,” Progress in Applied Mathematics, vol. 4, pp. 99–109, 2012.
View at: Google Scholar
Ö. Kişi and F. Nuray, “New convergence definitions for sequences of sets,” Abstract and Applied Analysis, vol. 2013, Article ID 852796, 2013.
View at: Publisher Site | Google Scholar
F. Nuray, E. Dündar, and U. Ulusu, “Wijsman -convergence of double sequences of closed sets,” Pure and Applied Mathematics Letters, vol. 2, pp. 31–35, 2014.
View at: Google Scholar
E. Dündar, U. Ulusu, and N. Pancaroğlu, “Strongly -lacunary convergence and -lacunary Cauchy double sequences of sets,” Aligarh Bulletin of Mathematics, vol. 35, no. 1-2, pp. 1–17, 2016.
View at: Google Scholar
U. Ulusu, E. Dündar, and E. Gülle, “-Cesàro summability of double sequences of sets,” Palestine Journal of Mathematics, vol. 9, no. 1, pp. 561–568, 2020.
View at: Google Scholar
U. Ulusu and E. Dündar, “I-lacunary statistical convergence of sequences of sets,” Filomat, vol. 28, no. 8, pp. 1567–1574, 2014.
View at: Publisher Site | Google Scholar
E. Dündar, U. Ulusu, and B. Aydin, “-lacunary statistical convergence of sequences of sets,” Konuralp Journal of Mathematics, vol. 5, no. 1, pp. 1–10, 2017.
View at: Google Scholar
U. Ulusu and E. Gülle, “-statistically and -lacunary statistically convergent double set sequences of order ,” Bulletin of Mathematical Analysis and Applications, vol. 13, no. 1, pp. 1–15, 2021.
View at: Google Scholar
Ö. Kişi, “Lacunary statistical convergence for sequence of sets in intuitionistic fuzzy metric space,” Journal of Applied Mathematics and Informatics, In press.
View at: Google Scholar
A. Esi, V. A. Khan, M. Ahmad, and M. Alam, “Some results on Wijsman ideal convergence in intuitionistic fuzzy metric spaces,” Journal of Function Spaces, vol. 2020, Article ID 7892913, 2020.
View at: Publisher Site | Google Scholar

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Copyright © 2021 Ömer Kişi. 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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