On the Logarithmic Summability of Sequences in Intuitionistic Fuzzy Normed Spaces

Abstract

We introduce logarithmic summability in intuitionistic fuzzy normed spaces($IFNS$) and give some Tauberian conditions for which logarithmic summability yields convergence in $IFNS$. Besides, we define the concept of slow oscillation with respect to logarithmic summability in $IFNS$, investigate its relation with the concept of q-boundedness and give Tauberian theorems by means of q-boundedness and slow oscillation with respect to logarithmic summability. A comparison theorem between Ces\`{a}ro summability method and logarithmic summability method in $IFNS$ is also proved in the paper.

Keywords

• Intuitionistic fuzzy normed space
• Logarithmic summability
• Slow oscillation
• Tauberian theorem

References

1. [1] L. A. Zadeh, Fuzzy sets, Inf. Control, 8 (1965), 338–353.
2. [2] K. Atanassov, Intuitionistic fuzzy sets, In VII ITKR’s Session, Sofia, June 1983 (Deposed in Central Sci.-Techn. Library of Bulg. Acad. of Sci., 1697/84) (in Bulgarian). Reprinted: International Journal of Bioautomation 2016; 20(S1): S1-S6 (in English).
3. [3] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst., 20 (1986), 87–96.
4. [4] J. H. Park, Intuitionistic fuzzy metric spaces, Chaos Solitons Fractals, 22 (2004), 1039–1046.
5. [5] R. Saadati, J. H. Park, On the intuitionistic fuzzy topological spaces, Chaos Solitons Fractals, 27 (2006), 331–344.
6. [6] F. Lael, K. Nourouzi, Some results on the IF􀀀normed spaces, Chaos Solitons Fractals, 37 (2008), 931–939.
7. [7] S. Karakus, K. Demirci, O. Duman, Statistical convergence on intuitionistic fuzzy normed spaces, Chaos Solitons Fractals, 35 (2008), 763–769.
8. [8] M. Mursaleen, S. A. Mohiuddine, On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space, J. Comput. Appl. Math., 233 (2009), 142–149.

9. [9] M. Mursaleen, S. A. Mohiuddine, Statistical convergence of double sequences in intuitionistic fuzzy normed spaces, Chaos Solitons Fractals, 41 (2009), 2414–2421.
10. [10] S. A. Mohiuddine, Q. M. Danish Lohani, On generalized statistical convergence in intuitionistic fuzzy normed space, Chaos Solitons Fractals, 42 (2009), 1731–1737.
11. [11] M. Mursaleen, S. A. Mohiuddine, H. H. E. Osama, On the ideal convergence of double sequences in intuitionistic fuzzy normed spaces, Comput. Math. Appl., 59 (2010), 603–611.
12. [12] Ö. Talo, E. Yavuz, Cesa`ro summability of sequences in intuitionistic fuzzy normed spaces and related Tauberian theorems, Soft Comput., (2020), doi: 10.1007/s00500-020-05301-z.
13. [13] H. Efe, C. Alaca, Compact and bounded sets in intuitionistic fuzzy metric spaces, Demonstr. Math., 40(2) (2007), 449–456.
14. [14] E. Yavuz, H. Çoşkun, On the logarithmic summability method for sequences of fuzzy numbers, Soft Comput., 21 (2017), 5779–5785.
15. [15] E. Yavuz, Tauberian theorems for statistical summability methods of sequences of fuzzy numbers, Soft Comput., 23 (2019), 5659–5665.
16. [16] S. A. Sezer, Logarithmic means of sequences of fuzzy numbers and a Tauberian theorem, Soft Comput., 24 (2020), 367–374.
17. [17] S. A. Sezer, Statistical harmonic summability of sequences of fuzzy numbers, Soft Comput., (2020), doi: 10.1007/s00500-020-05151-9.
18. [18] E. Dündar, Ö. Talo, F. Başar, Regularly (I2;I)􀀀convergence and regularly (I2;I)􀀀Cauchy double sequences of fuzzy numbers, International Journal of Analysis, (2013), Article ID 749684, 7 pages.
19. [19] E. Dündar, Ö. Talo, I2-convergence of double sequences of fuzzy numbers, Iran. J. Fuzzy Syst., 10(3) (2013), 37–50.
20. [20] M. R. Türkmen, E. Dündar, U. Ulusu, Fuzzy n-normlu uzaylarda c¸ift dizilerin Lacunary ideal yakınsaklı˘gı, International Congresson Science and Education (ICSE 2018), Afyonkarahisar, Turkey, 2018.
21. [21] U. Ulusu, E. Dündar, Asymptotically I-Ces`aro equivalence of sequences of sets, Univers. J. Math. Appl., 1(2) (2018), 101–105.
22. [22] M. R. T¨urkmen, 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.
23. [23] E. Dündar, M. R. Türkmen, On I2-convergence and I-2 -convergence of double sequences in fuzzy normed spaces, Konuralp J. Math., 7(2) (2019), 405–409.
24. [24] E. Dündar, M. R. Türkmen, On I2-Cauchy double sequences in fuzzy normed spaces, Commun. Adv. Math. Sci., 2(2) (2019), 154–160.
25. [25] E. Dündar, M. R. Türkmen, N. P. Akın, Regularly ideal convergence of double sequences in fuzzy normed spaces, Bull. Math. Anal. Appl., 12(2) (2020), 12–26.
26. [26] Ü. Totur, İ. Çanak, Tauberian theorems for (¯N; p;q) summable double sequences of fuzzy numbers, Soft Comput., 24 (2020), 2301–2310.
27. [27] F. Moricz, Necessary and sufficient Tauberian conditions for the logarithmic summability of functions and sequences, Studia Math., 219 (2013), 109–121.
28. [28] F. Moricz, On the harmonic averages of numerical sequences, Arch. Math. (Basel), 86 (2006), 375–384.