Statistical Convergence of Double Sequences in Intuitionistic Fuzzy Metric Spaces

Abstract

Statistical convergence has been a prominent research area in mathematics since this concept was independently introduced by Fast and Steinhaus in 1951. Afterward, the statistical convergence of double sequences in metric spaces and fuzzy metric spaces has been widely studied. The main goal of the present study is to introduce the concepts of statistical convergence and statistical Cauchy for double sequences in intuitionistic fuzzy metric spaces. Moreover, this study characterizes the statistical convergence of a double sequence by an ordinary convergent of a subsequence of the double sequence. Besides, the current study theoretically contributes to the mentioned concepts and investigates some of their basic properties. Finally, the paper handles whether the aspects should be further investigated.

Keywords

• statistical convergence
• statistical Cauchy sequences
• double sequences
• intuitionistic fuzzy metric spaces

References

1. H. Fast, \emph{Sur la Convergence Statistique}, Colloquium Mathematicae 2 (1951) 241{--}244.
2. H. Steinhaus, \emph{Sur la Convergence Ordinaire et la Convergence Asymptotique}, Colloquium Mathematicae 2 (1951) 73{--}74.
3. T. Salat, \emph{On Statistically Convergent Sequences of Real Numbers}, Mathematica Slovaca 30 (1980) 139{--}150.
4. A. R. Freedman, J. J. Sember, \emph{Densities and Summability}, Pacific Journal of Mathematics 95 (1981) 293{--}305.
5. J. A. Fridy, \emph{On Statistical Convergence}, Analysis 5 (1985) 301--313.
6. J. S. Connor, \emph{The Statistical and Strong p-Cesaro Convergence of Sequences}, Analysis 8 (1988) 47{--}63.
7. E. Kolk, \emph{Matrix Summability of Statistically Convergent Sequences}, Analysis 13 (1993) 77{--}83.
8. J. A. Fridy, C. Orhan, \emph{Lacunary Statistical Convergence}, Pacific Journal of Mathematics 160 (1993) 43{--}51.

9. S. Bulut, A. Or, \emph{$\mathcal{I}$-Statistical Rough Convergence of Order $\alpha$}, Journal of New Theory (38) (2022) 34{--}41.
10. A. Pringsheim, \emph{Zur Ttheorie der Zweifach Unendlichen Zahlenfolgen}, Mathematische Annalen 53 (1900) 289{--}321.
11. M. Mursaleen, O. H. H. Edely, \emph{Statistical Convergence of Double Sequences}, Journal of Mathematical Analysis and Applications 288 (2003) 223{--}231.
12. L. A. Zadeh, \emph{Fuzzy Sets}, Information and Control 8 (1965) 338{--}353.
13. J. Kramosil, J. Michalek, \emph{Fuzzy Metric and Statistical Metric Spaces}, Kybernetika 11 (1975) 336{--}334.
14. O. Kaleva, S. Seikkala, \emph{On Fuzzy Metric Spaces}, Fuzzy Sets and Systems 12 (1984) 215{--}229.
15. A. George, P. Veeramani, \emph{On Some Results in Fuzzy Metric Spaces}, Fuzzy Sets and Systems 64 (1994) 395{--}399.
16. D. Mihet, \emph{On Fuzzy Contractive Mappings in Fuzzy Metric Spaces}, Fuzzy Sets and Systems 158 (2007) 915{--}921.
17. V. Gregori, J. J. Mi$\check{\text{n}}$ana, S. Morillas, \emph{A Note on Convergence in Fuzzy Metric Spaces}, Iranian Journal of Fuzzy System 11 (4) (2014) 75{--}85.
18. S. Morillas, A. Sapena, \emph{On Standard Cauchy Sequences in Fuzzy Metric Spaces}, in: Proceedings of the Conference in Applied Topology WiAT'13, Bilbao, 2013, pp. 101{--}108.
19. V. Gregori, J. J. Mi$\check{\text{n}}$ana, \emph{Strong Convergence in Fuzzy Metric Spaces}, Filomat 31 (6) (2017) 1619{--}1625.
20. C. Li, Y. Zhang, J. Zhang, \emph{On Statistical Convergence in Fuzzy Metric Spaces}, Journal of Intelligent and Fuzzy Systems 39 (3) (2020) 3987{--}3993.
21. J. H. Park, \emph{Intuitionistic Fuzzy Metric Spaces}, Chaos Solitions and Fractals 22 (2004) 1039{--}1046.
22. K. T. Atanasov, \emph{Intuitionistic Fuzzy Sets}, Fuzzy Sets and Systems 20 (1) (1986) 87{--}96.
23. B. Schweizer, A. Sklar, \emph{Statistical Metric Spaces}, Pacific Journal of Mathematics 10 (1) (1960) 314--334.
24. B. P. Varol, \emph{Statistical Convergent Sequences in Intuitionistic Fuzzy Metric Spaces}, Axioms 11 (2022) 159.
25. R. Sava\c{s}, \emph{On Double Statistical Convergence in Fuzzy Metric Spaces}, in: 8th International Conference on Recent Advances in Pure and Applied Mathematics ICRAPAM, Muğla, 2021, pp. 234{--}243.