Acta Mathematica Hungarica

, Volume 130, Issue 1–2, pp 167–187

Two valued measure and summability of double sequences in asymmetric context

Article

Abstract

We extend the ideas of convergence and Cauchy condition of double sequences extended by a two valued measure (called μ-statistical convergence/Cauchy condition and convergence/Cauchy condition in μ-density, studied for real numbers in our recent paper [7]) to a very general structure like an asymmetric (quasi) metric space. In this context it should be noted that the above convergence ideas naturally extend the idea of statistical convergence of double sequences studied by Móricz [15] and Mursaleen and Edely [17]. We also apply the same methods to introduce, for the first time, certain ideas of divergence of double sequences in these abstract spaces. The asymmetry (or rather, absence of symmetry) of asymmetric metric spaces not only makes the whole treatment different from the real case [7] but at the same time, like [3], shows that symmetry is not essential for any result of [7] and in certain cases to get the results, we can replace symmetry by a genuinely asymmetric condition called (AMA).

Key words and phrases

asymmetric metric space approximate metric axiom (AMA) double sequences forward and backward μ-statistical convergence/divergence/Cauchy condition convergence/divergence/Cauchy condition in μ-density condition (APO2

2000 Mathematics Subject Classification

40A30 40A05 

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References

  1. [1]
    J. Connor, Two valued measure and summability, Analysis, 10 (1990), 373–385. MATHMathSciNetGoogle Scholar
  2. [2]
    J. Connor, R-type summability methods, Cauchy criterion, P-sets and statistical convergence, Proc. Amer. Math. Soc., 115 (1992), 319–327. MATHMathSciNetGoogle Scholar
  3. [3]
    J. Collins and J. Zimmer, An asymmetric Arzela–Ascoli theorem, Top. Appl., 154 (2007), 2312–2322. MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    Pratulananda Das and P. Malik, On the statistical and I variation of double sequences, Real Anal. Exchange, 33 (2008), 351–364. MATHMathSciNetGoogle Scholar
  5. [5]
    Pratulananda Das, P. Kostyrko, W. Wilczyński and P. Malik, I and I -convergence of double sequences, Math. Slovaca, 58 (2008), 605–620. MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    Pratulananda Das, P. Malik and E. Savas, On statistical limit points of double sequences, Appl. Math. Computation, 215 (2009), 1030–1034. MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    Pratulananda Das and S. Bhunia, Two valued measure and summability of double sequences, Czechoslovak Math. J., 59 (2009), 1141–1155. CrossRefMathSciNetGoogle Scholar
  8. [8]
    H. Fast, Sur la convergence statistique, Colloq. Math., 2 (1951), 241–244. MATHMathSciNetGoogle Scholar
  9. [9]
    J. A. Fridy, On statistical convergence, Analysis, 5 (1985), 301–313. MATHMathSciNetGoogle Scholar
  10. [10]
    H. P. A. Künzi, A note on sequentially compact quasipseudometric spaces, Monatsh. Math., 95 (1983), 219–220. MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    A. Mainik and A. Mielke, Existence results for energetic models for rate-independent system, Cale. Var. Partial Differential Equetion, 22 (2005), 73–99. MATHMathSciNetGoogle Scholar
  12. [12]
    A. Mennucci, On asymmetric distances, Technical report, Scuola Superiore (Pisa, 1959). Google Scholar
  13. [13]
    A. Mielke and T. Roubicek, A rate independent model for inelastic behavior of shaoe-memory alloys, Multiscale Model, Simul, 1 (2003), 571–597 (electronic). MATHMathSciNetGoogle Scholar
  14. [14]
    H. I. Miller and R. F. Patterson, Core theorems for double subsequences and rearrangements, Acta Math. Hungar., 119 (2008), 71–80. MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    F. Móricz, Statistical convergence of multiple sequences, Arch. Math. (Basel), 81 (2003), 82–89. MATHMathSciNetGoogle Scholar
  16. [16]
    F. Móricz, Regular statistical convergence of double sequences, Colloq. Math., 102 (2005), 217–227. MATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    M. Mursaleen and Osama H. H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl., 288 (2003), 223–231. MATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    A. Pringsheim, Zur Theorie der zweifach unendlichen Zahlenfolgen, Math. Ann., 53 (1900), 289–321. MATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    M. O. Rienger and J. Zimmer, Young measure flow as a model for damage, Preprint 11/05, Bath Institute for Complex System, Bath, UK (1959). Google Scholar
  20. [20]
    T. Šalát, On statistically convegent sequences of real numbers, Math. Slovaca, 30 (1980), 139–150. MATHMathSciNetGoogle Scholar
  21. [21]
    I. J. Scholenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66 (1959), 361–375. CrossRefMathSciNetGoogle Scholar
  22. [22]
    W. A. Wilson, On quasi-metric spaces, Amer. J. Math., 53 (1991), 675–684. CrossRefGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2010

Authors and Affiliations

  1. 1.Department of MathematicsF. C. CollegePinIndia
  2. 2.Department of MathematicsJadavpur UniversityKolkataIndia

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