Rough Convergence of Bernstein Fuzzy Triple Sequences

Subject Areas : Transactions on Fuzzy Sets and Systems

Ayhan Esi ¹ (Department of Basic Eng. Sci. (Math. Sect.), Malatya Turgut Ozal University, Malatya, Turkey.)
Subramanian Nagarajan ² (Department of Mathematics, SASTRA University, Thanjavur, India.)

Received: 2022-02-18 Accepted : 2022-02-18 Published : 2022-05-07

Keywords: Triple sequences, Rough convergence, Convergence almost surely, Convergence in probability, Convergence in mean, Convergence in distribution.,

Abstract :

The aim of this paper is to introduce and study a new concept of convergence almost surely (a.s.), convergence in probability, convergence in mean, and convergence in distribution are four important convergence concepts of random sequence and also discusses some convergence concepts of the fuzzy sequence: convergence almost surely, convergence in credibility, convergence in mean, and convergence in distribution.

References:

[1] S. Aytar, Rough statistical convergence, Numer. Funct. Anal. Optimiz., 29(3-4) (2008), 291-303.
[2] S. Aytar, The rough limit set and the core of a real sequence, Numer. Funct. Anal. Optimiz., 29(3-4) (2008), 283-290.
[3] S. Debnath, B. Sarma and B.C. Das, Some generalized triple sequence spaces of real numbers, Journal of Nonlinear Analysis and Optimization, 6(1) (2015), 71-79.
[4] E. Dundar and C. Cakan, Rough I−convergence, Demonstratio Math., 47(3) (2014), 638-651. 
[5] A. J. Dutta A. Esi and B. C. Tripathy, Statistically convergent triple sequence spaces defined by Orlicz function, Journal of Mathematical Analysis, 4(2) (2013), 16-22.
[6] A. Esi, On some triple almost lacunary sequence spaces defined by Orlicz functions, Research and Reviews: Discrete Mathematical Structures, 1(2) (2014), 16-25.
[7] A. Esi, S. Araci and M. Acikgoz, Statistical Convergence of Bernstein Operators, Appl. Math. and Inf. Sci., 10(6) (2016), 2083-2086.
[8] A. Esi and M. Necdet Catalbas, Almost convergence of triple sequences, Global Journal of Mathematical Analysis, 2(1) (2014), 6-10.
[9] A. Esi and E. Savas, On lacunary statistically convergent triple sequences in probabilistic normed space, Appl. Math. and Inf. Sci., 9(5) (2015), 2529-2534.
[10] P. K. Kamthan and M. Gupta, Sequence spaces and series, Lecture notes, Pure and Applied Mathematics, 65 Marcel Dekker, Inc. New York, (1981).
[11] J. Lindenstrauss and L. Tzafriri, On Orlicz sequence spaces, Israel J. Math., 10 (1971), 379-390.
[12] J. Musielak, Orlicz Spaces, Springer, Berlin, Heidelberg, (1983).
[13] H. X. Phu, Rough convergence in normed linear spaces, Numer. Funct. Anal. Optimiz., 22 (2001), 199-222.
[14] H. X. Phu, Rough continuity of linear operators, Numer. Funct. Anal. Optimiz., 23 (2002), 139-146.
[15] H. X. Phu, Rough convergence in infinite dimensional normed spaces, Numer. Funct. Anal. Optimiz., 24 (2003), 285-301.
[16] A. Sahiner, M. Gurdal and F. K. Duden, Triple sequences and their statistical convergence, Selcuk J. Appl. Math., 8(2) (2007), 49-55.
[17] A. Sahiner and B. C. Tripathy, Some I related properties of triple sequences, Selcuk J. Appl. Math., 9(2) (2008), 9-18.
[18] N. Subramanian and A. Esi, The generalized tripled difference of χ3 sequence spaces, Global Journal of Mathematical Analysis, 3(2) (2015), 54-60.