On triple difference sequences of real numbers in probabilistic normed spaces
DOI:
https://doi.org/10.4067/S0716-09172014000200003Keywords:
Triple sequence, t-norm, probabilistic norm, cluster point, difference operator, secuencia triple, norma t, norma probabilística, punto de cluster.Abstract
In this paper we define concept of triple Δ-statistical convergent sequences in probabilistic normed space and give some results. Also we introduce the notions of Δ-statistical limit point and Δ-statistical cluster point and investigate their different properties.References
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[28] B.C. Tripathy and A. Baruah, Lacunary statistically convergent and lacunary strongly convergent generalized difference sequences of fuzzy real numbers, Kyungpook Math. J., 50(4), pp. 565-574, (2010).
[29] B.C. Tripathy, A. Baruah, M.Et and M. Gungor, On almost statistical convergence of new type of generalized difference sequence of fuzzy numbers, Iranian Jour. Sci. Tech., Trans. A : Sci., 36(2), pp. 147-155, (2012).
[30] B.C. Tripathy and S. Borgogain, Some classes of difference sequence spaces of fuzzy real numbers defined by Orlicz function, Advances Fuzzy Syst., 2011, Article ID216414, 6 pages, (2011).
[31] B.C. Tripathy and S. Debnath, On generalized difference sequence spaces of fuzzy numbers, Acta Scientiarum Technology, 35(1), pp. 117-121, (2013).
[32] B.C. Tripathy and H. Dutta, On some lacunary difference sequence spaces defined by a sequence of Orlicz functions and q-lacunary Δnm-statistical convergence, Anal. Stiintifice ale Universitatii Ovidius, Seria Mat., 20(1), pp. 417-430, (2012).
[33] B.C. Tripathy and B. Sarma, Statistically convergent difference double sequence spaces, Acta Math. Sinica(Eng. Ser.), 24(5), pp. 737-742, (2008).
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[2] C. Alsina, B. Schweizer and A. Sklar, On the definition of a probabilistic normed space, Aequat. Math., 46, pp. 91-98, (1993).
[3] C. Alsina, B. Schweizer and A. Sklar, Continuity properties of probabilistic norms, J. Math. Anal. Appl., 208, pp. 446-452, (1997).
[4] J. S. Connor, The statistical and strong p-Cesaro convergence of sequences; Analysis, 8, pp. 47-63, (1988).
[5] G. Constantin and I. Istratescu, Elements of Probabilistic Analysis with Applications; vol.36 Mathematics and Its Applications (East European Series), Kluwer Academic Publishers, Dordrecht, Netherlands, (1989).
[6] A. Esi, The A-statistical and strongly (A — p)-Cesaro convergence of sequences, Pure Appl. Math. Sci., XLIII(1-2), pp. 89-93, (1996).
[7] A. Esi and M. K. Ozdemir, Generalized m-statistical convergence in probabilistic normed space, J. Comput. Anal. Appl., 13(5), pp. 923-932, (2011).
[8] A.Esi, Statistical convergence of triple sequences in topological groups, Annals Univ. Craiova, Math. Comput. Sci. Ser., 40(1), pp. 29-33, (2013).
[9] H. Fast, Sur la convergence statistique, Colloq. Math., 2, pp. 241-244, (1995).
[10] J. A. Fridy, On statistical convergence, Analysis, 5, pp. 301-313, (1985).
[11] S. Karakus, Statistical convergence on probabilistic normed space, Math. Commun., 12, pp. 11-23, (2007).
[12] S. Karakus and K. Demirci, Statistical convergence of double sequences on probabilistic normed spaces, Int. J. Math. Math. Sci., (2007), 11 pages, (2007).
[13] E. Kolk, Statistically convergent sequences in normed spaces, Tartu, pp. 63-66, (1988).
[14] B. Lafuerza-Guillen, J. Lallena and C. Sempi, Some classes of probabilistic normed spaces, Rend. Mat. Appl., 17(7), pp. 237-252, (1997).
[15] B. Lafuerza-Guillen, J. Lallena and C. Sempi, A study of boundedness in probabilistic normed spaces, J. Math. Anal. Appl., 232, pp. 183-196, (1999).
[16] B. Lafuerza-Guillen and C. Sempi, Probabilistic norms and convergence of random variables, J. Math. Anal. Appl., 280, pp. 9-16, (2003).
[17] I. J. Maddox, Statistical convergence in a locally convex space, Math. Proc. Camb. Phil. Soc., 104, pp. 141-145, (1988).
[18] K. Menger, Statistical metrics, Proceedings of the National Academy of Sciences of the United States of America, 28(12), pp. 535-537, (1942).
[19] A. Pringsheim, Zur theorie der zweifach unendlichen Zahlenfolgen, Math. Anna., 53, pp. 289-321, (1900).
[20] T. Salat, On statistically convergent sequences of real numbers, Math. Slovaca, 30, pp. 139-150, (1980).
[21] E. Savas and A. Esi, Statistical convergence of triple sequences on probabilistic normed space, Annals Univ. Craiova, Math. Comput. Sci. Ser., 39(2), pp. 226-236, (2012).
[22] E. Savas and M. Mursaleen, On statistically convergent double sequences of fuzzy numbers, Inform. Sci., 162(3-4), pp. 183-192, (2004).
[23] B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North-Holland Series in Probability and Applied Mathematics, North-Holland, New York, NY,USA, (1983).
[24] B. Schweizer and A. Sklar, Statistical metric spaces, Pacic Jour. Math., 10, pp. 313-334, (1960).
[25] A. N. Serstnev, On the notion of a random normed space, Dokl. Akad. Nauk SSSR, 149, pp. 280-283, (1963).
[26] B. C. Tripathy, Statistically convergent double sequences, Tamkang Jour. Math., 34(3), pp. 231-237, (2003).
[27] B.C. Tripathy, On generalized difference paranormed statistically convergent sequences, Indian J. Pure Appl. Math.,35(5), pp. 655-663, (2004).
[28] B.C. Tripathy and A. Baruah, Lacunary statistically convergent and lacunary strongly convergent generalized difference sequences of fuzzy real numbers, Kyungpook Math. J., 50(4), pp. 565-574, (2010).
[29] B.C. Tripathy, A. Baruah, M.Et and M. Gungor, On almost statistical convergence of new type of generalized difference sequence of fuzzy numbers, Iranian Jour. Sci. Tech., Trans. A : Sci., 36(2), pp. 147-155, (2012).
[30] B.C. Tripathy and S. Borgogain, Some classes of difference sequence spaces of fuzzy real numbers defined by Orlicz function, Advances Fuzzy Syst., 2011, Article ID216414, 6 pages, (2011).
[31] B.C. Tripathy and S. Debnath, On generalized difference sequence spaces of fuzzy numbers, Acta Scientiarum Technology, 35(1), pp. 117-121, (2013).
[32] B.C. Tripathy and H. Dutta, On some lacunary difference sequence spaces defined by a sequence of Orlicz functions and q-lacunary Δnm-statistical convergence, Anal. Stiintifice ale Universitatii Ovidius, Seria Mat., 20(1), pp. 417-430, (2012).
[33] B.C. Tripathy and B. Sarma, Statistically convergent difference double sequence spaces, Acta Math. Sinica(Eng. Ser.), 24(5), pp. 737-742, (2008).
[34] B.C. Tripathy, M. Sen and S. Nath, I-convergence in probabilistic n-normed space, Soft Comput., 16, pp. 1021-1027, (2012) DOI 10.1007/s00500-011-0799-8.
Published
2017-03-23
How to Cite
[1]
B. C. Tripathy and R. Goswami, “On triple difference sequences of real numbers in probabilistic normed spaces”, Proyecciones (Antofagasta, On line), vol. 33, no. 2, pp. 157-174, Mar. 2017.
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