Triple Lacunary $\Delta $-Statistical Convergence in Neutrosophic Normed Spaces

Year 2022, Volume: 10 Issue: 1, 127 - 133, 15.04.2022

Ömer Kişi , Verda Gürdal

https://izlik.org/JA65NF48SH

Abstract

The aim of this article is to investigate triple lacunary $\Delta $ -statistically convergent and triple lacunary $\Delta $-statistically Cauchy sequences in a neutrosophic normed space (NNS). Also, we present their feature utilizing triple lacunary density and derive the relationship between these notions.

Keywords

triple sequence , lacunary sequence , difference sequence , neutrosophic normed space

References

• [1] F. Smarandache, Neutrosophic set, a generalisation of the intuitionistic fuzzy sets, Int. J. Pure Appl. Math., 24(3) (2005), 287–297.
• [2] F. Smarandache, Neutrosophy. Neutrosophic Probability, Set, and Logic, ProQuest Information & Learning, Ann Arbor, Michigan, USA, 1998.
• [3] M. Kirisci, N. S¸ims¸ek, Neutrosophic metric spaces, Math. Sci., 14 (2020), 241–248.
• [4] M. Kirisci, N. S¸ims¸ek, Neutrosophic normed spaces and statistical convergence, J. Anal., 28 (2020), 1059–1073.
• [5] M. Kirisci, N. S¸ims¸ek, M. Akyi˘git, Fixed point results for a new metric space, Math. Methods Appl. Sci., 44(9) (2020), 7416–7422.
• [6] O¨ . Kis¸i, Lacunary statistical convergence of sequences in neutrosophic normed spaces, 4th International Conference on Mathematics: An Istanbul Meeting for World Mathematicians, Istanbul, 2020, 345–354.
• [7] O¨ . Kis¸i, Ideal convergence of sequences in neutrosophic normed spaces, J. Intell. Fuzzy Syst., 41(2) (2021), 2581–2590.
• [8] V.A. Khan, M.D. Khan, M. Ahmad, Some new type of lacunary statistically convergent sequences in neutrosophic normed space, Neutrosophic Sets Syst., 42 (2021), 239–252.
• [9] A. Zygmund, Trigonometric series, Cambridge University Press, Cambridge, UK, 1979.
• [10] H. Fast, Sur la convergence statistique, Colloq. Math., 2 (1951), 241–244.
• [11] J.A. Fridy, On statistical convergence, Analysis, 5 (1985), 301–313.
• [12] A.A. Nabiev, E. Savas¸, M. G¨urdal, Statistically localized sequences in metric spaces,J. Appl. Anal. Comput., 9(2) (2019), 739–746.
• [13] E. Savas¸, M. G¨urdal, Generalized statistically convergent sequences of functions in fuzzy 2-normed spaces, J. Intell. Fuzzy Systems, 27(4) (2014), 2067–2075.
• [14] M. Mursaleen, O.H.H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl., 288 (2003), 223–231.
• [15] B. Altay, F. Bas¸ar, Some new spaces of double sequences, J. Math. Anal. Appl., 309 (1) (2005), 70–90.
• [16] M. G¨urdal, A. S¸ ahiner, Extremal I-limit points of double sequences, Appl. Math. E-Notes, 8 (2008), 131–137.
• [17] A. S¸ ahiner, M. G¨urdal, F.K. D¨uden, Triple sequences and their statistical convergence, Selc¸uk J. Appl. Math., 8(2) (2007), 49–55.
• [18] A. Esi, E. Savas¸, On lacunary statistically convergent triple sequences in probabilistic normed space, Appl. Math. Inf. Sci., 9(5) (2015), 2529–2534.
• [19] M.B. Huban, M. G¨urdal, Wijsman lacunary invariant statistical convergence for triple sequences via Orlicz function, J. Classical Anal., 17(2) (2021), 119–128.
• [20] A. Esi, Statistical convergence of triple sequences in topological groups, Annals Univ. Craiova. Math. Comput. Sci. Ser., 10(1) (2013), 29–33.
• [21] B.C. Tripathy, R. Goswami, On triple difference sequences of real numbers in propobabilistic normed space, Proyecciones J. Math., 33(2) (2014), 157–174.
• [22] J.A. Fridy, C. Orhan, Lacunary statistical convergence, Pac. J. Math., 160(1) (1993), 43–51.
• [23] F. Nuray, Lacunary statistical convergence of sequences of fuzzy numbers, Fuzzy Sets Syst., 99(3) (1998), 353–355.
• [24] U. Yamanci , M. Gurdal, On lacunary ideal convergence in random n-normed space, J. Math., 2013, Article ID 868457, 8 pages.
• [25] H. Kızmaz,On certain sequence spaces, Canad. Math. Bull., 24 (1981), 169–176.
• [26] M. Bas¸arır, On the statistical convergence of sequences, Fırat Univ. Turk. J. Sci. Technol., 2 (1995), 1–6.
• [27] T. Bilgin, Lacunary strongly D-convergent sequences of fuzzy numbers, Inform. Sci., 160 (2004), 201–206.
• [28] B. Hazarika, Lacunary generalized difference statistical convergence in random 2-normed spaces, Proyecciones, 31 (2012), 373–390.
• [29] R. C¸ olak, H. Altınok, M. Et, Generalized difference sequences of fuzzy numbers, Chaos Solitons Fractals, 40(3) (2009), 1106–1117.
• [30] Y. Altın, M. Bas¸arır, M. Et, On some generalized difference sequences of fuzzy numbers, Kuwait J. Sci., 34(1A) (2007), 1–14.
• [31] S. Altunda˘g, E. Kamber, Lacunary D-statistical convergence in intuitionistic fuzzy n-normed space, J. Inequal. Appl., 2014(40) (2014), 1–12.
• [32] B. Hazarika, A. Alotaibi, S.A. Mohiudine, Statistical convergence in measure for double sequences of fuzzy-valued functions, Soft Comput., 24(9) (2020), 6613–6622.
• [33] F. Bas¸ar, Summability theory and its applications, Bentham Science Publishers, ˙Istanbul, 2012.
• [34] M. Mursaleen, F. Bas¸ar, Sequence Spaces: Topics in Modern Summability Theory, CRC Press, Taylor & Francis Group, Series: Mathematics and Its Applications, Boca Raton London New York, 2020.
• [35] K. Menger, Statistical metrics, Proc. Natl. Acad. Sci. USA 28(12) (1942), 535–537.