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μ-Statistically Convergent Multiple Sequences in Probabilistic Normed Spaces

  • Rupam Haloi
  • Mausumi SenEmail author
Conference paper
Part of the Trends in Mathematics book series (TM)

Abstract

In this article, we introduce the notions of μ-statistically convergent and μ-statistically Cauchy multiple sequences in probabilistic normed spaces (in short PN-spaces). We also give a suitable characterization for μ-statistically convergent multiple sequences in PN-spaces. Moreover, we introduce the notion of μ-statistical limit points for multiple sequences in PN-spaces, and we give a relation between μ-statistical limit points and limit points of multiple sequences in PN-spaces.

Keywords

Probabilistic normed space μ-statistical convergence Multiple sequence Two-valued measure 

Notes

Acknowledgements

The work of the first author has been supported by the Research Project SB/S4/MS:887/14 of SERB - Department of Science and Technology, Govt. of India.

References

  1. 1.
    Alsina, C., Schweizer, B., Sklar, A.: On the definition of a probabilistic normed space. Aequationes Math. 46, 91–98 (1993)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Asadollah, A., Nourouzi, K.: Convex sets in probabilistic normed spaces. Chaos, Solutions & Fractals. 36, 322–328 (2008)Google Scholar
  3. 3.
    Connor, J.: The statistical and strong p-Cesàro convergence of sequences. Analysis. 8, 47–63 (1988)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Connor, J.: Two valued measure and summability. Analysis. 10, 373–385 (1990)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Connor, J.: R-type summability methods, Cauchy criterion, P-sets and statistical convergence. Proc. Amer. Math. Soc. 115, 319–327 (1992)zbMATHGoogle Scholar
  6. 6.
    Datta, A.J., Esi, A., Tripathy, B.C.: Statistically convergent triple sequence spaces defined by Orlicz function. Journal of Mathematical Analysis. 4 (2), 16–22 (2013)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Esi, A., Sharma, S.K.: Some paranormed sequence spaces defined by a Musielak-Orlicz function over n-normed spaces. Konuralp Journal of Mathematics. 3 (1), 16–28 (2015)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Fast, H.: Sur la convergence statistique. Colloq. Math. 2, 241–244 (1951)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Fridy, J.A.: On Statistical convergence. Analysis. 5, 301–313 (1985)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fridy, J.A., Orhan, C.: Lacunary Statistical convergence. Pacific J. Math. 160, 43–51 (1993)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Fridy, J.A., Orhan, C.: Lacunary statistical summability. J. Math. Anal. Appl. 173, 497–503 (1993)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Guillén, B., Lallena, J., Sempi, C.: Some classes of probabilistic normed spaces. Rend. Math. 17 (7), 237–252 (1997)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Hardy, G.H.: On the Convergence of Certain Multiple Series. Proceedings of the Cambridge Philosophical Society. 19 (3), 86–95 (1917)Google Scholar
  14. 14.
    Karakus, S.: Statistical Convergence on PN-spaces. Mathematical Communications. 12, 11–23 (2007)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Karakus, S., Demirci, K.: Statistical Convergence of Double Sequences on Probabilistic Normed Spaces. International Journal of Mathematics and Mathematical Sciences. (2007) https://doi.org/10.1155/2007/14737 MathSciNetCrossRefGoogle Scholar
  16. 16.
    Mohiuddine, S.A., Savaş, E.:, Lacunary statistically convergent double sequences in probabilistic normed spaces. Ann Univ Ferrara. 58 (2), 331–339 (2012)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Pringsheim, A.: Zur Theorie der zweifach unendlichen Zahlenfolgen. Mathematische Annalen. 53 (3), 289–321 (1900)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Savaş, E., Mohiuddine, S.A.: \(\overline {\lambda }\)-statistically convergent double sequences in probabilistic normed spaces. Mathematica Slovaca. 62 (1), 99–108 (2012)Google Scholar
  19. 19.
    Schweizer, B., Sklar, A.: Statistical metric spaces. Pacific J. Math. 10, 313–334 (1960)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Šerstnev, A.N.: On the notion of a random normed space. Dokl. Akad. Nauk. SSSR. 142 (2), 280–283 (1963)MathSciNetGoogle Scholar
  21. 21.
    Sharma, S.K., Esi, A.: Some \(\mathcal {I}\)-convergent sequence spaces defined by using sequence of moduli and n-normed space. Journal of the Egyptian Mathematical Society. 21, 29–33 (2013)Google Scholar
  22. 22.
    Sharma, S.K., Esi, A.: μ-statistical convergent double lacunary sequence spaces. Afrika Matematika. 26 (7–8), 1467–1481 (2015)Google Scholar
  23. 23.
    Steinhaus, H.: Sur la convergence ordinaire et la convergence asymptotique. Colloq. Math. 2, 73–74 (1951)CrossRefGoogle Scholar
  24. 24.
    Tripathy, B.C., Goswami, R.: Multiple sequences in probabilistic normed spaces. Afr. Mat. 26, 753–760 (2015)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Tripathy, B.C., Goswami, R.: Statistically Convergent Multiple Sequences in Probabilistic Normed Spaces. U.P.B. Sci. Bull. Series A. 78 (4), 83–94 (2016)Google Scholar
  26. 26.
    Tripathy, B.C., Sen, M., Nath, S.: I-convergence in probabilistic n-normed spaces. Soft Computing. 16 (6), 1021–1027 (2012)CrossRefGoogle Scholar
  27. 27.
    Tripathy, B.C., Sen, M., Nath, S.: Lacunary I-convergence in probabilistic n-normed spaces. IMBIC 6th International Conference on Mathematical Sciences for Advancement of Science and Technology (MSAST 2012), December 21–23, Salt Lake City, Kolkata, IndiaGoogle Scholar
  28. 28.
    Tripathy, B.C., Sen, M., Nath, S.: I-Limit Superior and I-Limit Inferior of Sequences in Probabilistic Normed Space. International Journal of Modern Mathematical Sciences. 7 (1), 1–11 (2013)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Mathematics, NIT SilcharSilcharIndia

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