Lacunary generalized difference statistical convergence in random 2-normed spaces
DOI:
https://doi.org/10.4067/S0716-09172012000400006Keywords:
Statistical convergence, lacunary sequence, difference sequence, t-norm, 2-norm, random 2-normed space, convergencia estadística, secuencia lagunaria, secuencia diferencial, t-normadoM 2-normado, espacio aleatorio 2-normado.Abstract
Recently in [22], Mursaleen introduced the concept of statistical convergence in random 2-normed spaces. In this paper, we define and study the notion of lacunary An-statistical convergence and lacunary An-statistical Cauchy sequences in random 2-normed spaces using la-cunary density and prove some interesting theorems.References
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[2] H. Cakalli, Lacunary statistical convergence in topological groups, Indian J. Pure Appl. Math. 26 (2), pp. 113-119, (1995), MR 95m:40016.
[3] J.Connor, M.A. Swardson, Measures and ideals of C*(X) , Ann. N.Y. Acad.Sci. 704,, pp. 80-91, (1993).
[4] A. Esi, M. K. Ozdemir, Generalized Am-Statistical convergence in probabilistic normed space, Jour. Comput. Anal. Appl., 13(5), pp. 923-932, (2011).
[5] A.Esi, M. K. Ozdemir, On lacunary statistical convergence in random n-normed space, Annals of Fuzzy Mathematics and Informatics, (To appear).
[6] M.Et and F.Nuray, Am-Statistical convergence, Indian J.Pure Appl.Math., 32(6), pp. 961-969, (2001).
[7] H. Fast, Sur la convergence statistique, Colloq. Math. 2, pp. 241-244, (1951).
[8] J. A. Fridy, On statistical convergence, Analysis, 5, pp. 301-313, (1985).
[9] J. A. Fridy, C. Orhan, Lacunary statistical convergence, Pacific J. Math., 160, No. 1, pp. 43-51, (1993), MR 94j:40014.
[10] J. A. Fridy, C. Orhan, Lacunary statistical summability, J. Math. Anal. Appl., 173, pp. 497-504, (1993), MR 95f :40004.
[11] S. Gahler, 2-metrische Raume and ihre topologische Struktur, Math. Nachr. 26, pp. 115-148, (1963).
[12] S. Gahler, Linear 2-normietre Raume, Math. Nachr. 28, pp. 1-43, (1965).
[13] I . Golet¸ , On probabilistic 2-normed spaces, Novi Sad J. Math. 35, pp. 95102, (2006).
[14] M.Gurdal and S. Pehlivan, Statistical convergence in 2-normed spaces, South. Asian Bull. Math.33, pp. 257-264, (2009).
[15] M. Gurdal, S. Pehlivan, The statistical convergence in 2-Banach spaces, Thai J. Math. 2 (1), pp. 107113, (2004).
[16] B. Hazarika, E. Savas, Lacunary statistical convergence in random 2-normed space, (communicatted).
[17] S. Karakus, Statistical convergence on probabilistic normed spaces, Math. Commun. 12, pp. 1123, (2007).
[18] S. Karakus, K. Demirci and O. Duman, Statistical convergence on intuitionistic fuzzy normed spaces, Chaos, Solitons and Fractals, 35, pp. 763-769, (2008).
[19] J. Li, Lacunary statistical convergence and inclusion properties between lacunary methods, Internat. J. Math. Math. Sci. 23(3), pp. 175180, (2000), S0161171200001964.
[20] K. Menger, Statistical metrics, Proc. Natl. Acad. Sci. USA 28, pp. 535537, (1942).
[21] H. I. Miller, A measure theoretical subsequence characterization of statistical convergence, Trans. Amer. Math. Soc., 347(5), pp. 1811-1819, (1995).
[22] M. Mursaleen, Statistical convergence in random 2-normed spaces, Acta Sci. Math.(Szeged), 76(1-2), pp. 101-109, (2010).
[23] M. Mursaleen and S.A. Mohiuddine, On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space, Jour. Comput. Appl. Math., 233(2), pp. 142-149, (2009).
[24] S.A.Mohiuddine, M. Aiyub, Lacunary statistical convergence in random 2-normed space, Appl.Math.Inf.Sci., 6(3), pp. 581-585, (2012).
[25] M. R. S. Rahmat, Lacunary Statistical Convergence on Probabilistic Normed Spaces, Int. J. Open Problems Compt. Math., 2(2), pp. 285-292, (2009).
[26] T. Salat, On statistical convergence of real numbers, Math. Slovaca, 30, pp. 139-150, (1980).
[27] I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly 66, pp. 361-375, (1959).
[28] B. Schweizer, A. Sklar, Statistical metric spaces, Pacific J. Math. 10, pp. 313334, (1960).
[29] B. Schweizer, A. Sklar, Probabilistic Metric Spaces, North Holland, New York- Amsterdam-Oxford, (1983).
[30] C. Sempi, A short and partial history of probabilistic normed spaces,Mediterr. J. Math. 3, pp. 283-300, (2006).
[31] A. N. Serstnev, On the notion of a random normed space, Dokl. Akad. Nauk SSSR 149(, pp. 280-283, (1963).
Published
2013-02-19
How to Cite
[1]
B. Hazarika, “Lacunary generalized difference statistical convergence in random 2-normed spaces”, Proyecciones (Antofagasta, On line), vol. 31, no. 4, pp. 373-390, Feb. 2013.
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