Lacunary Strongly Invariant Convergence in Fuzzy Normed Spaces

Year 2023, Volume: 11 Issue: 2, 89 - 96, 30.06.2023

Şeyma Yalvaç Erdinç Dündar

Abstract

In this study, firstly, we defined the notions of lacunary invariant convergence and lacunary invariant Cauchy sequence in fuzzy normed spaces. Then, we introduced lacunary strongly invariant convergence in fuzzy normed spaces and we investigated some properties of these new concepts

Keywords

Fuzzy normed space, Invariant convergence, Lacunary convergence

References

• [1] Freedman, A.R., Sembrer, J.J., Raphael, M.: Some Cesaro-type summability spaces. Proc. London Math. Soc. 37 3, 508–520 (1978).
• [2] Bag, T., Samanta, S.K.: Finite dimensional fuzzy normed linear spaces. J. Fuzzy Math. 11 (3) 687–705 (2003).
• [3] Banach, S.: Théorie des Operations Lineaires. Warszawa (1932).
• [4] Chang, C.L.: Fuzzy topolojical spaces. J. Math. Anal. Appl. 24, 182–190 (1968).
• [5] Cheng, S.C., Mordeson, J.N.: Fuzzy linear operator and fuzzy normed linear spaces. Bull. Calcutta Math. Soc. 86, 429–436 (1994).
• [6] Das, N.R., Das, P.: Fuzzy topology generated by fuzzy norm. Fuzzy Sets and Systems. 107, 349–354 (1999).
• [7] Das, G., Patel, B.K.: Lacunary distribution of sequences. Indian J. Pure Appl. Math. 20 (1), 64–74 (1989).
• [8] Dean, D., Raimi, R.A.: Permutations with comparable sets of invariant means. Duke Math. 27, 467–480 (1960).
• [9] Diamond, P., Kloeden, P.: Metric spaces of fuzzy sets: theory and aplications. World Scientific, Singapore(1994).
• [10] Fang, J.-X.: A note on the completions of fuzzy metric spaces and fuzzy normed space. Fuzzy Sets and Systems. 131, 399–407 (2002).
• [11] Fang, J.-X., Huang, H.: On the level convergence of a sequence of fuzzy numbers. Fuzzy Sets and Systems. 147, 417–435 (2004).
• [12] Felbin, C.: Finite dimensional fuzzy normed linear space. Fuzzy Sets and Systems. 48, 293–248 (1992).
• [13] George, A., Veeramani, P.: On some results in fuzzy metric spaces. Fuzzy Sets and Sytems. 64, 395–399 (1994).
• [14] Goetschel, R., Voxman, W.: Elementary fuzzy calculus. Fuzzy Sets and Systems. 18, 31–43 (1986).
• [15] Itoh, M., Cho, M.: Fuzzy bounded operators. Fuzzy Sets and Systems. 93, 353–362 (1998).
• [16] Kaleva, O., Seikkala, S.: On fuzzy metric spaces. Fuzzy Sets and Sytems. 12, 215–229 (1984).
• [17] Katsaras, A.K.: Fuzzy topological vector spaces II. Fuzzy Sets and Systems. 12, 143–154 (1984).
• [18] Kramosil, I., Michalek, J.: Fuzzy metrics and statistical metric spaces. Kybernetika. 11, 336–344 (1975).
• [19] Lorentz, G.: A contribution to the theory of divergent sequences. Acta Math. 80, 167–190 (1948).
• [20] Michalek, J.: Fuzzy topologies. Kybernetika. 11, 345–354 (1975).
• [21] Mizumoto, M., Tanaka, K.: Some properties of fuzzy numbers. Advances in Fuzzy Set Theory and applications. North-Holland, Amsterdam. 153–164 (1979).
• [22] Mursaleen, M.: On some new invariant matrix methods of summability. Quart. J. Math. Oxford. 34, 77–86 (1983).
• [23] Mursaleen, M.: Matrix transformations between some new sequence spaces. Houston J. Math. 9, 505–509 (1983).
• [24] Mursaleen, M., Edely, O. H. H.: On the invariant mean and statistical convergence. Appl. Math. Lett. 22, 1700–1704 (2009).
• [25] Raimi, R.A.: Invariant means and invariant matrix methods of summability. Duke Math. J. 30, 81–94 (1963).
• [26] Savaş, E.: Strong -convergent sequences. Bull. Calcutta Math. 81, 295–300 (1989).
• [27] Savaş, E.: On lacunary strong -convergence. Indian J. Pure Appl. Math. 21, 359–365 (1990).
• [28] Schaefer, P.: Infinite matrices and invariant means. Proc. Amer. Math. Soc. 36, 104–110 (1972).
• [29] Şençimen, C., Pehlivan, S.: Statistical convergence in fuzzy normed linear spaces. Fuzzy Sets and Systems 159, 361–370 (2008).
• [30] Türkmen, M. R., Cinar, M.: Lacunary statistical convergence in fuzzy normed linear spaces. App. and Comp. Math. 6 (5), 233–237 (2017).
• [31] Xiao, J., Zhu, X.: On linearly topological structure and property of fuzzy normed linear space. Fuzzy Sets and Systems 125, 153–161 (2002).
• [32] Yalvaç, Ş., Dündar, E.: Invariant convergence in fuzzy normed spaces. Honam Math. J. 43(3), 433–440 (2021).
• [33] Zadeh, L. A.: Fuzzy sets. Inform. and Control. 8, 338–353 (1965).