New results on lacunary ideal convergence in fuzzy cone normed spaces

Abstract

In this paper, some existing theories on convergence of sequences in fuzzy cone normed space (FCNS in short) are extended to lacunary ideal convergence in FCNS. An original concept, named lacunary convergence of sequence in FCNS, is investigated. Also, lacunary 𝐼-limit points and lacunary 𝐼-cluster points of sequences in FCNS are examined. Furthermore, la-cunary Cauchy and lacunary 𝐼-Cauchy sequences in FCNS are presented and relationships between them are studied.

Keywords

• Lacunary Ideal Convergence
• Lacunary I-Limit Points
• Lacunary I-Cluster Points
• Fuzzy Cone Normed Space

References

1. REFERENCES
2. [1] Zadeh LA. Fuzzy sets. Inform Control 1965;8:338–353. [CrossRef]
3. [2] Atanassov KT. Intuitionistic fuzzy sets. Fuzzy Sets Syst 1986;20:87–96. [CrossRef]
4. [3] Katsaras AK. Fuzzy topological vector spaces. Fuzzy Sets Syst 1984;12:143–154. [CrossRef]
5. [4] Felbin C. Finite dimensional fuzzy normed linear space. Fuzzy Sets Syst 1992;48:239–248. [CrossRef]
6. [5] Kramosil I, Michalek J. Fuzzy metric and statistical metric spaces. Kybernetika 1975;11:336–344.
7. [6] George A, Veeramani P. On some results in fuzzy metric spaces. Fuzzy Sets Syst 1994;64:395–399.[CrossRef]
8. [7] Park JH. Intuitionistic fuzzy metric spaces. Chaos Solit Fractals 2004;22:1039–1046. [CrossRef]

9. [8] Lael F, Nourouzi K. Some results on the IF-normed spaces. Chaos Solit Fractals 2008;37:931–939.[CrossRef]
10. [9] Ahmad S, Ullah A, Akgül A, Abdeljawad T. Numerical analysis of fractional human liver model in fuzzy environment. J Taibah Univ Sci 2021;15:840–851. [CrossRef]
11. [10] Ahmad S, Ullah A, Akgül A, Abdeljawad T. Computational analysis of fuzzy fractional order non-dimensional Fisher equation. Physica Scripta 2021;96:084004. [CrossRef]
12. [11] Ahmad S, Ullah A, Akgül A, Abdeljawad T. Semi-analytical solutions of the 3rd order fuzzy dispersive partial differential equations under fractional operators. Alex Eng J 2021;60:5861–5878. [CrossRef]
13. [12] Ullah Z, Ahmad S, Ullah A, Akgül A. On solution of fuzzy Volterra integro-differential equations. Arab J Basic Appl Sci 2021;28:330–339. [CrossRef]
14. [13] Long-Guang H, Xian Z. Cone metric spaces and fixed point theorems of contractive mappings. J Math Anal Appl 2007;332:1468–1476. [CrossRef]
15. [14] Bag T. Finite dimensional fuzzy cone normed linear spaces. Int J Math Sci Comput. 2013;3:9–14.
16. [15] Choudhury SB, Das P. A new contraction mapping principle in partially ordered fuzzy metric spaces. Ann Fuzzy Math Inform 2014;8:889–901.
17. [16] Mohinta S, Samanta TK. Coupled fixed point theorems in partially ordered non-Archimedean complete fuzzy metric spaces. Ann Fuzzy Math Inform 2016;11:829–840. [CrossRef]
18. [17] Somasundaram RM, Beaula T. Some aspects of 2-fuzzy 2-normed linear spaces. Bull Malays Math Soc 2009;32:211–221.
19. [18] Tamang P, Bag T. Some results on finite dimensional fuzzy cone normed linear space. Ann Fuzzy Math Inform 2017;13:123–134. [CrossRef]
20. [19] Güler AÇ. I-convergence in fuzzy cone normed spaces. Sahand Commun Math. 2021;18:45–57.
21. [20] Fast H. Sur la convergence statistique. Colloq Math 1951;2:241–244. [CrossRef]
22. [21] Karakuş S, Demirci K, Duman O. Statistical convergence on intuitionistic fuzzy normed spaces. Chaos Solit Fractals 2008;35:763–769. [CrossRef]
23. [22] Hazarika B, Alotaibi A, Mohiudine SA. Statistical convergence in measure for double sequences of fuzzy-valued functions. Soft Comput 2020;24:6613–6622. [CrossRef]
24. [23] Kadak U, Mohiuddine SA. Generalized statistically almost convergence based on the difference operator which includes the (p,q)-gamma function and related approximation theorems. Results Math 2018;73:1–31.[CrossRef]
25. [24] Mohiuddine SA, Alamri BAS. Generalization of equi-statistical convergence via weighted lacunary sequence with associated Korovkin and Voronovskaya type approximation theorems. Rev R Acad Cienc Exactas Fís Nat Ser A Math 2019;113:1955–1973. [CrossRef]
26. [25] Mohiuddine SA, Asiri A, Hazarika B. Weighted statistical convergence through difference operator of sequences of fuzzy numbers with application to fuzzy approximation theorems. Int J Gen Syst 2019;48:492–506. [CrossRef]
27. [26] Mohiuddine SA, Danish Lohani QM. On generalized statistical convergence in intuitionistic fuzzy normed space. Chaos Solit Fractals 2009;42:1731–1737. [CrossRef]
28. [27] Savaş E, Gürdal M. Certain summability methods in intuitionistic fuzzy normed spaces. J Intell Fuzzy Syst 2014;27:1621–1629. [CrossRef]
29. [28] Savaş E, Gürdal M. Generalized statistically convergent sequences of functions in fuzzy 2-normed spaces. J Intell Fuzzy Syst 2014;27:2067–2075. [CrossRef]
30. [29] Savaş E, Gürdal M. A generalized statistical convergence in intuitionistic fuzzy normed spaces. Science Asia 2015;41:289–294. [CrossRef]
31. [30] Fridy JA, Orhan C. Lacunary statistical convergence. Pacific J Math 1993;160:43–51. [CrossRef]
32. [31] Kostyrko P, Salát T, Wilczynsski W. I-convergence. Real Anal Exchange 2000;26:669–686. [CrossRef]
33. [32] Kostyrko P, Macaj M, Salát T, Sleziak M. I-convergence and extremal I-limit points. Math Slovaca 2005;55:443–464.
34. [33] Nabiev A, Pehlivan S, Gürdal M. On I-Cauchy sequence. Taiwanese J Math 2007;11:569–576. [CrossRef]
35. [34] Kumar K, Kumar V. On the I and I^*-convergence of sequences in fuzzy normed spaces. Adv Fuzzy Syst 2008;3:341–365.
36. [35] Kumar K, Kumar V. On the ideal convergence of sequences of fuzzy numbers. Inf Sci 2008;178:4670–4678. [CrossRef]
37. [36] Gürdal M. On ideal convergent sequences in 2-normed spaces. Thai J Math. 2006;4:85–91.
38. [37] Kumar K, Kumar V. On the ideal convergence of sequences in intuitionistic fuzzy normed spaces. Selçuk J Math 2009;10:27–41.
39. [38] Yamancı U, Gürdal M. On lacunary ideal convergence in random n-normed space. J Math 2013;868457:1–8. [CrossRef]
40. [39] Debnath P. Lacunary ideal convergence in intuitionistic fuzzy normed linear spaces. Comput Math Appl 2012;63:708–715. [CrossRef]
41. [40] Tripathy BC, Hazarika B, Choudhary B. Lacunary I-convergent sequences. Kyungpook Math J 2012;52:473–482. [CrossRef]
42. [41] Başar F. Summability theory and its applications. Istanbul: Bentham Science Publishers; 2020. p. 520.
43. [42] Mursaleen M, Başar F. Sequence spaces: topics in modern summability theory. Boca Raton, London, New York: CRC Press, Taylor & Francis Series: Mathematics and Its Applications; 2020. p. 312. [CrossRef]
44. [43] Kadak U, Başar F. Power series with real or fuzzy coefficients. Filomat 2012;25:519–528. [CrossRef]
45. [44] Talo Ö, Başar F. On the space bv_p (F) of sequences of p-bounded variation of fuzzy numbers. Acta Math Sin Eng Ser 2008;24:1205–1212. [CrossRef]
46. [45] Talo Ö, Başar F. Certain spaces of sequences of fuzzy numbers defined by a modulus function. Demonstratio Math 2010;43:139–149. [CrossRef]
47. [46] Talo Ö, Başar F. Quasilinearity of the classical sets of sequences of fuzzy numbers and some related results. Taiwanese J Math 2010;14:1799–1819. [CrossRef]
48. [47] Menger K. Statistical metrics. Proc Nat Acad Sci 1942;28:535–537. [CrossRef]