µ-Statistically convergent function sequences in probabilistic normed linear spaces

• Sen, Mausumi ^[1] ; Haloi, Rupam ^[1] ; Tripathy, Binod Chandra
  1. [1] National Institute Of Technology

     National Institute Of Technology

     Japón

     
• Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 38, Nº. 5, 2019, págs. 1039-1056
• Idioma: inglés
• DOI: 10.22199/issn.0717-6279-2019-05-0067
• Enlaces
  • Texto completo
• Resumen

  • In this article, we introduce the concept of µ-statistical convergence and µ-density convergence of sequences of functions defined on a compact subset D of the probabilistic normed space (X, N, ∗), where µ is a finitely additive two valued measure. In particular, we introduce the notions of µ-statistical uniform convergence as well as µ-statistical point-wise convergence of sequences of functions in probabilistic normed space (in short PN-space) and we give some characterization results on these two convergences of sequences of functions in PN-space. We have also observed that µ-statistical uniform convergence of sequences of functions in PN-spaces inherits the basic properties of uniform convergence.

• Referencias bibliográficas
  • C. Alsina, B. Schweizer, and A. Sklar, “On the definition of a probabilistic normed space”, Aequationes mathematicae, vol. 46, no. 1-2, pp....
  • A. Asadollah and K. Nourouzi, “Convex sets in probabilistic normed spaces”, Chaos, solutions & fractals, vol. 36, no. 2, pp. 322-328,...
  • R. Bartle, Elements of real analysis, New York, NY: John Wiley and Sons Inc., 1964.
  • F. Bașar, Summability theory and its applications, BenthameBooks, 2012, doi: 10.2174/97816080545231120101.
  • J. Connor, “The statistical and strong p-Cesàro convergence of sequences”, Analysis, vol. 8, no. 1-2, pp. 47-63, 1988, doi: 10.1524/anly.1988.8.12.47.
  • J. Connor, “Two valued measure and summability”, Analysis, vol. 10, no. 4, pp. 373-385, 1990, doi: 10.1524/anly.1990.10.4.373.
  • J. Connor, “R-type summability methods, Cauchy criterion, P-sets and statistical convergence”, Proceedings of the American mathematical society,...
  • J. Connor, “A topological and functional analytic approach to statistical convergence”, in Analysis of divergence, W. Bray and Č. Stanojević,...
  • J. Connor and J. Kline, “On statistical limit points and the consistency of statistical convergence”, Journal of mathematical analysis and...
  • O. Duman and C. Orhan, “µ-statistically convergent function sequences”, Czechoslovak mathematical journal, vol. 54, no. 2, pp. 413-422, Jun....
  • H. Fast, “Sur la convergence statistique”, Colloquium mathematicae, vol. 2, no. 3-4 pp. 241-244, 1951. [On line]. Available: https://bit.ly/2M5zmod
  • J. Fridy, “On statistical convergence”, Analysis, vol. 5, no. 4, pp. 301-313, 1985, doi: 10.1524/anly.1985.5.4.301.
  • B. Lafuerza, J. Rodríguez and C. Sempi, “Some classes of probabilistic normed spaces”, Rendiconti di matematica, vol. 17. no. 7, pp. 237-252,...
  • P. Harikrishnan, B. Lafuerza and K. Ravindran, “Compactness and D-boundedness in Menger’s 2-probabilistic normed spaces”, Filomat, vol. 30,...
  • S. Karakus, “Statistical convergence on probalistic normed spaces”, Mathematical communications, vol. 12, no. 1 pp. 11-23, 2007. [On line]....
  • E. Kolk, “Convergence-preserving function sequences and uniform convergence”, Journal of mathematical analysis and applications, vol. 238,...
  • M. Mursaleen, Applied summability methods, Cham: Springer, 2014, doi: 10.1007/978-3-319-04609-9.
  • T. Šalát, “On statistically convergent sequences of real numbers”, Mathematica slovaca, vol. 30, no. 2, pp. 139-150, 1980. [On line]. Available:...
  • B. Schweizer and A. Sklar, “ Statistical metric spaces”, Pacific journal of mathematics, vol. 10, n. 1, pp. 313-334, Sep. 1960. [On line]....
  • B. Schweizer and A. Sklar, Probabilistic metric spaces, New York, NY: Dover, 1983.
  • A. Šerstnev, “On the concept of a stochastic normalized space”, Doklady akademii nauk, vol. 142, no. 2, pp. 280-283, 1963.
  • H. Steinhaus, “Sur la convergence ordinaire et la convergence asymptotique”, Colloquium mathematicum, vol. 2, pp. 73-74, 1951.
  • B. Tripathy and R. Goswami, “On triple difference sequences of real numbers in probabilistic normed spaces”, Proyecciones (Antofagasta, On...
  • B. Tripathy and R. Goswami, “Multiple sequences in probabilistic normed spaces”, Afrika matematika, vol. 26, no. 5-6, pp. 753-760, Sep. 2015,...
  • B. Tripathy. and R. Goswami, “Fuzzy real valued p-absolutely summable multiple sequences in probabilistic normed spaces”, Afrika matematika,...
  • B. Tripathy and R. Goswami, “Statistically convergent multiple sequences in probabilistic normed spaces”, Scientific Bulletin - "Politehnica"...
  • B. Tripathy and M. Sen, “On generalized statistically convergent sequences”, Indian journal of pure & applied mathematics, vol. 32, no....
  • B. Tripathy, M. Sen and S. Nath, “I-convergence in probabilistic n-normed space”, Soft computing, vol. 16, no. 6, pp. 1021-1027, Jun. 2012,...
  • W. Wilczyński, “Statistical convergence of sequences of functions”, Real analysis exchange, vol. 25, pp. 49-50, 2000.