λ3--Statistical Convergence of Triple Sequences on Probabilistic Normed Space
Abstract
The idea of \(\lambda\)-statistical convergence of single sequences was studied by Alotaibi [39] and double sequences was studied by Savas and Mohiuddine [3] in probabilistic normed spaces. The purpose of this paper is to study statistical convergence of triple sequences in probabilistic normed spaces and give some important theorems.
About this article
Keywords
- \(\lambda\)-sequence
- probabilistic normed space
- statistical convergence
- t-norm.
References
[1] E. Sava¸s and A. Esi, Statistical convergence of triple sequences on probabilistic normed space, Annals of the University of Craiova, Mathematics and Computer Science Series, 39(2)(2012), 226-236.
[2] A. Esi and E. Sava¸s, On lacunary statistically convergent triple sequences in probabilistic normed space, submitted. Global Journal of Mathematical Analysis 35
[3] E. Sava¸s and S. A. Mohiuddine, λ-statistically convergent double sequences in probabilistic normed spaces, Math.Slovaca, 62(1)(2012), 99-108.
[4] C. Alsina, B. Schweizer, A. Sklar, Continuity properties of probabilistic norms, J. Math. Anal. Appl. 208 (1997), 446–452.
[5] H. C¸ akalli, A study on statistical convergence, Funct. Anal. Approx. Comput., 1(2)(2009), 19-24, MR2662887.
View more references (38)
[6] J. Connor, M.A. Swardson, Measures and ideals of C∗(X) , Ann. N.Y. Acad.Sci.704(1993), 80-91.
[7] A. Esi, M. K. Ozdemir, ”Generalized ∆ ¨ m-Statistical convergence in probabilistic normed space, Jour. Comput. Anal.Appl., 13(5)(2011), 923-932.
[8] H. Fast, Sur la convergence statistique, Colloq. Math. 2(1951) 241-244.
[9] J. A. Fridy, On statistical convergence, Analysis, 5(1985) 301-313.
[10] S. G¨ahler, 2-metrische Raume and ihre topologische Struktur, Math. Nachr. 26(1963) 115-148.
[11] I . Golet¸ , On probabilistic 2-normed spaces, Novi Sad J. Math. 35 (2006), 95–102.
[12] M. G¨urdal and S. Pehlivan, Statistical convergence in 2-normed spaces, South. Asian Bull. Math.33, (2009), 257-264.
[13] M. G¨urdal, S. Pehlivan, The statistical convergence in 2-Banach spaces, Thai J. Math. 2 (1), (2004), 107–113.
[14] S. Karakus, Statistical convergence on probabilistic normed spaces, Math. Commun. 12 (2007), 11–23.
[15] S. Karakus, K. Demirci and O. Duman, Statistical convergence on intuitionistic fuzzy normed spaces, Chaos, Solitons and Fractals, 35 (2008), 763-769.
[16] V. Kumar and M. Mursaleen, On (λ, µ)-statistical convergence of double sequences on intuitionistic fuzzy normed spaces,Filomat, 25(2) (2011) 109-120.
[17] L. Leindler, Uber die de la Vallee-Pousinsche Summierbarkeit allgenmeiner Othogonalreihen, Acta Math. Acad. Sci.Hungar, 16(1965), 375-387.
[18] I. J. Maddox, Statistical convergence in a locally convex space, Math. Proc. Cambridge Philos. Soc., 104(1)(1988),141-145.
[19] G. D. Maio, L. D. R. Kocinac, Statistical convergence in topology, Topology Appl. 156, (2008), 28-45.
[20] K. Menger, Statistical metrics, Proc. Natl. Acad. Sci. USA 28 (1942), 535–537.
[21] H. I. Miller, A measure theoretical subsequence characterization of statistical convergence, Trans. Amer. Math. Soc.,347(5)(1995), 1811-1819.
[22] M. Mursaleen, Statistical convergence in random 2-normed spaces, Acta Sci.Math.(Szeged), 76(1-2)(2010), 101-109.
[23] M. Mursaleen, λ−statistical convergence, Mathematica Slovaca, 50(1)(2000), 111-115.
[24] M. Mursaleen and A. Alotaibi, Statistical summability and approximation by de la Vallee-Pousin mean, Applied Math.Letters, 24 (2011) 320-324.
[25] M. Mursaleen, C. C¸ akan, S. A. Mohiuddine and E. Sava¸s, Generalized statistical convergence and statitical core of double sequences, Acta Math. Sinica, 26(11)(2010) 2131-2144.
[26] M. Mursaleen and Osama H. H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl., 288(2003) 223-231.
[27] M. Mursaleen and Osama H. H. Edely, Generalized statistical convergence, Information Sciences , 162(2004), 287-294.
[28] M. Mursaleen and Osama H. H. Edely, On the invariant mean and statistical convergence, Appl. Math. Letters,22(2009) 1700-1704.
[29] S. A. Mohiuddine and Q. M. Danish Lohani, On generalized statistical convergence in intuitionistic fuzzy normed space,Chaos, Solitons and Fractals, 42 (2009) 1731-1737.
[30] M. Mursaleen and S. A. Mohiuddine, On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space, Jour. Comput. Appl. Math., 233(2)(2009) 142-149.
[31] T. Sal´at, On statistical convergence of real numbers, Math. Slovaca, 30(1980), 139-150. ˘36 Global Journal of Mathematical Analysis
[32] I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly 66(1959)
361-375.
[33] B. Schweizer, A. Sklar, Statistical metric spaces, Pacific J. Math. 10 (1960), 313–334.
[34] B. Schweizer, A. Sklar, Probabilistic Metric Spaces, North Holland, New York- Amsterdam-Oxford, 1983.
[35] C. Sempi, A short and partial history of probabilistic normed spaces, Mediterr. J. Math. 3(2006), 283-300.
[36] C. Sencimen and S. Pehlivan, Statistical convergence in fuzzy normed linear spaces, Fuzzy Sets and Systems, 159 (2008),
361-370.
[37] A. N. Serstnev, On the notion of a random normed space, Dokl. Akad. Nauk SSSR 149(1963), 280-283.
[38] R. F. Patterson and E. Savas, Lacunary statistical convergence of double sequences, Math. Commun., 10(2005), no. 1,
55–61.
[39] A. Alotaibi, Generalized statistical convergence in probabilistic normed spaces, Open Math.J., 1(2008), 82-88.
[40] A. Asadollah and K. Nourouzi, Convex sets in probabilistic normed spaces, Chaos, Solitons and Fractals, 36(2)(2008),322-328.