Document Type : Research Paper

Authors

Department of Mathematics, Tripura University (A Central University), Suryamaninagar-799022, Agartala, India.

Abstract

In this paper, we introduce the notion of deferred statistical convergence in the neutrosophic normed spaces as an extension of statistical convergence, $\lambda$-statistical convergence, and lacunary statistical convergence. We investigate a few fundamental properties of the newly introduced notion. Finally, we introduce the concept of deferred statistical Cauchy sequence and show that deferred statistical Cauchy sequences are equivalent to deferred statistical convergent sequences in the neutrosophic normed spaces.

Keywords

###### ##### References
[1] R.P. Agnew, On deferred Cesaro mean, Comm. Ann. Math., 33 (1932), pp. 413-421.
[2] M. Altinok, M. Küçükaslan and U. Kaya, Statistical extension of bounded sequence space, Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat., 70(1) (2021), pp. 82-99.
[3] K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20(1) (1986), pp. 87-96.
[4] J.S. Connor, The statistical and strong p-Cesaro convergence of sequences, Analysis, 8(1-2) (1988), pp. 47-63.
[5] S. Debnath, V.N. Mishra and J. Debnath, On statistical convergent sequence spaces of intuitionistic fuzzy numbers, Bol. Soc. Parana. Mat., 36(1) (2018), pp. 235-242.
[6] M. Et, M. Cinar and H. Sengul, On $\triangle^m-$asymptotically deferred statistical equivalent sequences of order $\alpha$, Filomat, 33(7) (2019), pp. 1999-2007.

[7] M. Et and H. Sengul, Some Cesaro-type summability spaces of order $\alpha$ and lacunary statistical convergence of order $\alpha$, Filomat, 28(8) (2014), pp. 1593-1602.
[8] M. Et and H. Sengul, On pointwise lacunary statistical convergence of order $\alpha$ of sequence of function, Proc. Nat. Acad. Sci. India Sect. A, 85(2) (2015), pp. 253-258.
[9] H. Fast, Sur la convergence statistique, Cooloq. Math., 2(3-4) (1951), pp. 241-244.
[10] C. Felbin, Finite dimensional fuzzy normed linear space, Fuzzy Sets and Systems, 48(2) (1992), pp. 239-248.
[11] A.R. Freedman and J.J. Sember, Densities and summability, Pacific J. Math., 95(2) (1981), pp. 293-305.
[12] J.A. Fridy, On statistical convergence, Analysis, 5(4) (1985), pp. 301-313.
[13] B. Hazarika and A. Esi, On asymptotically Wijsman lacunary statistical convergence of set sequences in ideal context, Filomat, 31(9) (2017), pp. 2691-2703.
[14] S. Karakus, K. Demirci and O. Duman, Statistical convergence on intuitionistic fuzzy normed spaces, Chaos Solitons Fractals., 35(4) (2008), pp. 763-769.
[15] V.A. Khan, H. Fatima, M.D. Khan and A. Ahamd, Spaces of neutrosophic $\lambda-$statistical convergence sequences and their properties, J. Math. Comput. Sci., 23(1) (2021), pp. 1-9.
[16] V.A. Khan, M.D. Khan and M. Ahmad, Some new type of lacunary statistically convergent sequences in neutrosophic normed space, neutrosophic Sets Syst., 42(14) (2021), pp. 240-252.
[17] V.A. Khan, M.D. Khan and M. Ahmad, Some results of neutrosophic normed spaces via Fibonacci matrix, U.P.B. Sci. Bull. Series A, 83(2) (2021), pp. 99-110.
[18] M. Kirisci, Fibonacci statistical convergence on intuitionistic fuzzy normed spaces, Fuzzy Syst., 36(1) (2019), pp. 1–8.
[19] M. Kirisci and N. Simsek, neutrosophic normed space and statistical convergence, J. Anal., 28(4) (2020), pp. 1059–1073.
[20] M. Küçükaslan and M. Yilmaztürk, On deferred statistical convergence of sequences, Kyungpook Math. J., 56(2) (2016), pp. 357–366.
[21] S. Melliani, M. Küçükaslan, H. Sadiki and L.S. Chadli, Deferred statistical convergence of sequences in intuitionistic fuzzy normed spaces, Notes on Intuitionistic Fuzzy Sets, 24(3) (2018), pp. 64-78.
[22] K. Menger, Statistical metrics, Sigma J. Eng. Nat. Sci., 28(12) (1942), pp. 535-537.
[23] S.A. Mohiuddine and Q. M. Danish Lohani, On generalized statistical convergence in intuitionistic fuzzy normed space, Chaos Solitons Fractals., 42(3) (2009), pp. 1731-1737.
[24] M. Mursaleen, $\lambda-$statistical convergence, Math. Slovaca, 50(1) (2000), pp. 111-115.
[25] M. Mursaleen and S.A. Mohiuddine, Statistical convergence of double sequences in intuitionistic fuzzy normed spaces, Chaos Solitons Fractals., 41(5) (2009), pp. 2414-2421.
[26] R. Saadati and J.H. Park, On the intuitionistic fuzzy topological spaces, Chaos Solitons Fractals., 27(2) (2006), pp. 331-344.
[27] T. Salat, On statistically convergent sequences of real numbers, Math. Slovaca., 30(2) (1980), pp. 139-150.
[28] E. Savas and S. Debnath, Lacunary statistically $\phi-$convergence, Note Mat., 39(2) (2019), pp. 111-119.
[29] E. Savas and M. Gurdal, A generalized statistical convergence in intuitionistic fuzzy normed spaces, Sci. Asia, 41 (2015), pp. 289-294.
[30] H. Sengul and M. Et, On lacunary statistical convergence of order $\alpha$, Acta Math. Sci. Ser. B (Engl. Ed.), 34(2) (2014), pp. 473-482.
[31] F. Smarandache, neutrosophic set a generalization of the intuitionistic fuzzy sets, Int. J. Pure Appl. Math., 24(3) (2005), pp. 287-297.
[32] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math., 2 (1951), pp. 73-74.
[33] B.C. Tripathy and M. Sen, On generalized statistically convergent sequences, Indian J. Pure. Appl.Math., 32(11) (2001), pp. 1689-1694.
[34] M. Yilmaztürk and M. Küçükaslan, On strongly deferred Cesaro summability and deferred statistical convergence of the sequences, Bitlis Eren Univ. J. Sci. and Technol., 3(1) (2013), pp. 22-25.
[35] L.A. Zadeh, Fuzzy sets, Inf. Control, 8(3) (1965), pp. 338-353.