On $\mathcal{I}_{\theta }$-convergence in Neutrosophic Normed Spaces

Abstract

The purpose of this article is to investigate lacunary ideal convergence of sequences in neutrosophic normed space (NNS). Also, an original notion, named lacunary convergence of sequence in NNS, is defined. Also, lacunary $% \mathcal{I}$-limit points and lacunary $\mathcal{I}$-cluster points of sequences in NNS have been examined. Furthermore, lacunary Cauchy and lacunary $\mathcal{I}$-Cauchy sequences in NNS are introduced and some properties of these notions are studied.

Keywords

• Lacunary ideal convergence
• Lacunary convergence
• Lacunary $% \mathcal{I}$-limit points
• Lacunary $\mathcal{I}$-cluster points
• Neutrosophic normed space

References

1. [1] L.A. Zadeh, Fuzzy sets, Inform. Control, 8 (1965), 338-353.
2. [2] K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Syst., 20 (1986), 87-96.
3. [3] K.T. Atanassov, G. Pasi, R. Yager, Intuitionistic fuzzy interpretations of multi-person multicriteria decision making, Proceedings of First International IEEE Symposium Intelligent Systems, 1 (2002), 115-119.
4. [4] I. Kramosil, J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetika, 11 (1975), 336-344.
5. [5] O. Kaleva, S. Seikkala, On fuzzy metric spaces, Fuzzy Sets Syst., 12 (1984), 215-229.
6. [6] A. George, P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets Syst., 64 (1994), 395-399.
7. [7] A. George, P. Veeramani, On some results of analysis for fuzzy metric spaces, Fuzzy Sets and Syst., 90 (1997), 365-368.
8. [8] J. H. Park, Intuitionistic fuzzy metric spaces, Chaos Solitons Fractals, 22 (2004), 1039-1046.

9. [9] R. Saadati, J. H. Park, On the intuitionistic fuzzy topological spaces, Chaos Solitons Fractals, 27 (2006), 331-344.
10. [10] H. Fast, Sur la convergence statistique, Colloq. Math., 2 (1951), 241-244.
11. [11] S. Karaku¸s, K. Demirci, O. Duman, Statistical convergence on intuitionistic fuzzy normed spaces, Chaos Solitons Fractals, 35 (2008), 763-769.
12. [12] M. Kiri¸sci, Fibonacci statistical convergence on intuitionistic fuzzy normed spaces, J. Intell. Fuzzy Systems, 36 (2019), 5597-5604.
13. [13] S.A. Mohiuddine, Q. M. Danish Lohani, On generalized statistical convergence in intuitionistic fuzzy normed space, Chaos Solitons Fractals, 42 (2009), 1731-1737.
14. [14] E. Savaş, M. Gürdal, Certain summability methods in intuitionistic fuzzy normed spaces, J. Intell. Fuzzy Systems, 27(4) (2014), 1621-1629.
15. [15] E. Savaş, M. Gürdal, Generalized statistically convergent sequences of functions in fuzzy 2-normed spaces, J. Intell. Fuzzy Systems, 27(4) (2014), 2067-2075.
16. [16] E. Savaş, M. Gürdal, A generalized statistical convergence in intuitionistic fuzzy normed spaces, Science Asia, 41 (2015), 289-294.
17. [17] E. Yavuz, On the logarithmic summability of sequences in intuitionistic fuzzy normed spaces, Fundam. J. Math. Appl., 3(2) (2020), 101-108.
18. [18] J.A. Fridy, C. Orhan, Lacunary statistical convergence, Pac. J. Math., 160(1) (1993), 43-51.
19. [19] J.A. Fridy, C. Orhan, Lacunary statistical summability, J. Math. Anal. Appl., 173(2) (1993), 497-504.
20. [20] F. Nuray, Lacunary statistical convergence of sequences of fuzzy numbers, Fuzzy Sets Syst., 99(3) (1998), 353-355.
21. [21] P. Debnath, Lacunary ideal convergence in intuitionistic fuzzy normed linear spaces, Comput. Math. Appl., 63 (2012), 708-715.
22. [22] U. Yamancı, M. Gürdal, On lacunary ideal convergence in random n-normed space, J. Math., (2013), Article ID 868457, 8 pages.
23. [23] M. Mursaleen, S.A. Mohiuddine, On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space, J. Comput. Appl. Math., 233(2) (2009), 142-149.
24. [24] F. Smarandache, Neutrosophic set, a generalisation of the intuitionistic fuzzy sets, Int. J. Pure Appl. Math., 24 (2005), 287-297.
25. [25] F. Smarandache, Neutrosophy. Neutrosophic Probability, Set, and Logic, ProQuest Information & Learning, Ann Arbor, Michigan, USA (1998).
26. [26] T. Bera, N.K. Mahapatra, On neutrosophic soft linear spaces, Fuzzy Inform. Engineering, 9 (3) (2017), 299-324.
27. [27] T. Bera, N.K. Mahapatra, Neutrosophic soft normed linear spaces, Neutrosophic Sets and Systems, 23 (2018), 52-71.
28. [28] T. Bera, N.K. Mahapatra, On neutrosophic soft metric space, Int. J. Adv. Math., 2018(1) (2018), 180-200.
29. [29] T. Bera, N.K. Mahapatra, Compactness and Continuity on Neutrosophic Soft Metric Space, Int. J. Adv. Math., 2018(4) (2018), 1-24.
30. [30] T. Bera, N. K. Mahapatra, Continuity and Convergence on neutrosophic soft normed linear spaces, Int. J. Fuzzy Comput. Modelling, 3(2) (2020), 156-186.
31. [31] T.K. Samanta, Iqbal H. Jebril, Finite dimensional intuitionistic fuzzy normed linear space, Int. J. Open Problems Compt. Math., 2(4) (2009), 574-591.
32. [32] T. Bag, S.K. Samanta, Finite dimensional fuzzy normed linear spaces, Ann. Fuzzy Math. Inform., 6(2) (2013), 271-283.
33. [33] M. Kirisci, N. Şimşek, Neutrosophic metric spaces, Math. Sci, 14 (2020), 241-248.
34. [34] M. Kirisci, N. Şimşek, Neutrosophic normed spaces and statistical convergence, The Journal of Analysis, 28 (2020), 1059-1073.
35. [35] F. Smarandache, A unifying field in logics: Neutrosophic logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability and Statistics, Phoenix: Xiquan (2003).
36. [36] F. Smarandache, Introduction to neutrosophic measure, neutrosophic integral, and neutrosophic probability, Sitech-Education, Columbus, Craiova, (2013), 1-143.
37. [37] N. Şimşek, M. Kirişci, Fixed point theorems in neutrosophic metric spaces, Sigma J. Eng. Nat. Sci., 10(2) (2019), 221-230.
38. [38] M. Kirisci, N. Şimşek, M. Akyiğit, Fixed point results for a new metric space, Math. Meth. Appl. Sci., 2020 1-7. doi: 10.1002/mma.6189.
39. [39] Ö. Ki¸si, Lacunary statistical convergence of sequences in neutrosophic normed spaces, 4th International Conference on Mathematics: An Istanbul Meeting for World Mathematicians, Istanbul, 2020, 345-354.
40. [40] P. Kostyrko, T. Salát and W. Wilczynsski, I-convergence, Real Anal. Exchange, 26(2) (2000), 669-686.
41. [41] P. Kostyrko, M. Macaj, T. Salát, M. Sleziak, I-convergence and extremal I-limit points, Math. Slovaca, 55 (2005), 443-464.
42. [42] A. A. Nabiev, S. Pehlivan, M. Gürdal, On I-Cauchy sequences, Taiwanese J. Math., 11(2) (2007), 569-566.
43. [43] M. Gürdal, On ideal convergent sequences in 2-normed spaces, Thai J. Math., 4(1) (2012), 85-91.
44. [44] U. Yamancı, M. Gürdal, I-statistical convergence in 2-normed space, Arab J. Math. Sci., 20(1) (2014), 41-47.
45. 45] U. Yamancı, M. Gürdal, I-statistically preCauchy double sequences, Glob. J. Math. Anal., 2(4) (2014), 297-303.
46. [46] E. Dündar, M. R. Türkmen, On I2-Cauchy double sequences in fuzzy normed spaces, Commun. Adv. Math. Sci., 2(2) (2019), 154-160.
47. [47] Ö. Ki¸si, E. Güler, I-Cesáro Summability of a Sequence of Order a of Random Variables in Probability, Fundam. J. Math. Appl., 1(2) (2018), 157-161.
48. [48] S. A. Mohiuddine, B. Hazarika, M. A. Alghamdi, Ideal relatively uniform convergence with Korovkin and Voronovskaya types approximation theorems, Filomat, 33(14) (2019), 4549-4560.
49. [49] S. A. Mohiuddine, B. Hazarika, Some classes of ideal convergent sequences and generalized difference matrix operator, Filomat, 31(6) (2017), 1827-1834.
50. [50] K. Raj, S. A. Mohiuddine, Applications of lacunary sequences to develop fuzzy sequence spaces for ideal convergence and orlicz function, Eur. J. Pure Appl. Math., 13(5) (2020), 1131-1148.
51. [51] V.A. Khan, S.A.A. Abdulla, K.M.A.S. Alshlool, Paranorm ideal convergent fibonacci difference sequence spaces, Commun. Adv. Math. Sci., 2(4) (2019), 293-302.
52. [52] M. Mursaleen, S.A. Mohiuddine, O.H.H. Edely, On the ideal convergence of double sequences in intuitionistic fuzzy normed spaces, Comput. Math. Appl., 59 (2010), 603-611.
53. [53] K. Menger, Statistical metrics, Proc. Nat. Acad. Sci., 28(12) (1942), 535-537.