Lacunary Invariant Statistical Convergence in Fuzzy Normed Spaces

Year 2024, Volume: 7 Issue: 2, 76 - 82, 23.05.2024

Şeyma Yalvaç

https://doi.org/10.32323/ujma.1424201

Cited By: 1

Abstract

In the study done here regarding the theory of summability, we introduce some new concepts in fuzzy normed spaces. First, at the beginning of the original part of our study, we define the lacunary invariant statistical convergence. Then, we examine some characteristic features like uniqueness, linearity of this new notion and give its important relation with pre-given concepts.

Keywords

Fuzzy normed spaces, Invariant convergence, Lacunary convergence, Statistical convergence

References

• [1] I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66 (1959), 361-375.
• [2] H. Fast, Sur la convergence statistique, Colloq. Math., 2(3-4) (1951), 241-244.
• [3] T. Šalát, On statistically convergent sequences of real numbers, Math. Slovaca, 30(2) (1980), 139-150.
• [4] J. A. Fridy, On statistical convergence, Analysis, 5(4) (1985), 301-314.
• [5] J. S. Connor, The statistical and strong p-Cesàro convergence of sequences, Analysis, 8(1-2) (1988), 47-64.
• [6] A. R. Freedman, J. J. Sember, M. Raphael, Some Cesàro-type summability spaces, Proc. Lond. Math. Soc., 3(3) (1978), 508-520.
• [7] G. Das, B. K. Patel, Lacunary distribution of sequences, Indian J. Pure Appl. Math., 20(1) (1989), 64-74.
• [8] J. A. Fridy, C. Orhan, Lacunary statistical convergence, Pacific. J. Math., 160 (1993), 43-51.
• [9] J. A. Fridy, C. Orhan Lacunary statistical summability, J. Math. Anal. Appl., 173(2) (1993), 497-504.
• [10] U. Ulusu, F. Nuray, Lacunary statistical convergence of sequences of sets, Progr. Appl. Math., 4(2) (2012), 99-109.
• [11] U. Ulusu, F. Nuray, Statistical lacunary summability of sequences of sets, AKU J. Sci. Eng., 13 (2013), 9-14.
• [12] U. Ulusu, F. Nuray, On strongly lacunary summability of sequences of sets, J. Appl. Math. Bioinform., 3(3) (2013), 75-88.
• [13] S. Banach, Théorie des Opérations Linéaires, Warszawa, 1932.
• [14] G. Lorentz, A contribution to the theory of divergent sequences, Acta Math., 80 (1948), 167-190.
• [15] D. Dean, R. A. Raimi, Permutations with comparable sets of invariant means, Duke Math. J., 27 (1960), 467-479.
• [16] R. A. Raimi, Invariant means and invariant matrix methods of summability, Duke Math. J., 30 (1963), 81-94.
• [17] M. Mursaleen, O. H. H. Edely, On the invariant mean and statistical convergence, Appl. Math. Lett., 22(11) (2009), 1700-1704.
• [18] M. Mursaleen, On some new invariant matrix methods of summability, Q. J. Math., 34(1) (1983), 77-86.
• [19] E. Savaş, Some sequence spaces involving invariant means, Indian J. Math., 31 (1989), 1-8.
• [20] E. Savaş, Strong s-convergent sequences, Bull. Calcutta Math., 81 (1989), 295-300.
• [21] P. Schaefer, Infinite matrices and invariant means, Proc. Mer. Math. Soc., 36 (1972), 104-110.
• [22] E. Savaş, On lacunary strong s􀀀convergence, Indian J. Pure Appl. Math., 21 (1990), 359-365.
• [23] E. Savaş, F. Nuray, On s-statistically convergence and lacunary s-statistically convergence, Math. Slovaca, 43(3) (1993), 309-315.
• [24] L. A. Zadeh, Fuzzy sets, Inform. and Control, 8 (1965), 338-353.
• [25] D. Dubois, H. Prade, Operations on fuzzy numbers, Internat. J. Systems Sci., 9(6) (1978), 613-626.
• [26] D. Dubois, H. Prade, Fuzzy real algebra: some results, Fuzzy Sets and Systems, 2(4) (1979), 327-348.
• [27] C. L. Chang, Fuzzy topolojical spaces, J. Math. Anal. Appl., 24(1) (1968), 182-190.
• [28] C. K. Wong, Covering properties of fuzzy topological spaces. J. Math. Anal. Appl., 43(3) (1973), 697-704.
• [29] C. K. Wong, Fuzzy topology: product and quotient theorems. J. Math. Anal. Appl., 45(2) (1974), 512-521.
• [30] O. Kaleva, S. Seikkala, On fuzzy metric spaces, Fuzzy Sets and Sytems, 12(3) (1984), 215-229.
• [31] I. Kramosil, J. Michálek, Fuzzy metrics and statstical metric spaces, Kybernetika, 11(5) (1975), 336-344.
• [32] C. Felbin, Finite dimensional fuzzy normed linear space, Fuzzy Sets and Systems, 48(2) (1992), 293-248.
• [33] A. K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets and Systems, 12(2) (1984), 143-154.
• [34] C. Şençimen, S. Pehlivan, Statistical convergence in fuzzy normed linear spaces, Fuzzy Sets and Systems, 159 (2008), 361-370.
• [35] M. R. Türkmen, M. Çınar, Lacunary statistical convergence in fuzzy normed linear spaces, Appl. Comput. Math., 6(5) (2017), 233-237.
• [36] M. R. Türkmen, M. Çınar, l-statistical convergence in fuzzy normed linear spaces, J. Intell. Fuzzy Syst., 34(6) (2018), 4023-4030.
• [37] M. R. Türkmen, E. Dündar, On lacunary statistical convergence of double sequences and some properties in fuzzy normed spaces, J. Intell. Fuzzy Syst., 36(2) (2019), 1683-1690.
• [38] Ş. Yalvaç, E. Dündar, Invariant convergence in fuzzy normed spaces, Honam Math. J., 43(3) (2021), 433-440.
• [39] Ş Yalvaç, E. Dündar, Lacunary strongly invariant convergence in fuzzy normed spaces, Math. Sci. Appl. E-Notes, 11(2) (2023), 89-96