Two valued measure and summability of double sequences in asymmetric context Article

First Online: 03 November 2010 Received: 01 February 2010 Revised: 20 April 2010 Accepted: 26 April 2010 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 space approximate metric axiom (AMA) double sequences forward and backward μ -statistical convergence/divergence/Cauchy condition convergence/divergence/Cauchy condition in μ -density condition (APO_{2} ) The second author is thankful to Department of Science and Technology, Govt. of India for granting a major research project NO. DST-PURSE/JU/MATH//P-7 during the tenure of which this work had been done.

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Authors and Affiliations 1. Department of Mathematics F. C. College Pin India 2. Department of Mathematics Jadavpur University Kolkata India