Acta Mathematica Hungarica

, Volume 130, Issue 1, pp 167–187

Two valued measure and summability of double sequences in asymmetric context

Article

DOI: 10.1007/s10474-010-0005-y

Cite this article as:
Bhunia, S. & Das, P. Acta Math Hung (2011) 130: 167. doi:10.1007/s10474-010-0005-y

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 spaceapproximate metric axiom (AMA)double sequencesforward and backward μ-statistical convergence/divergence/Cauchy conditionconvergence/divergence/Cauchy condition in μ-densitycondition (APO2)

2000 Mathematics Subject Classification

40A3040A05

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2010

Authors and Affiliations

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