Abstract
There have been numerous attempts recently to extend many of the metric standard fixed point theorems to a more general semimetric context. In many instances a weakened form of the triangle inequality is involved and the space is assumed to be complete. Thus Cauchy sequences play a central role. One of the standard tests to determine when a sequence is Cauchy in a metric space (X, d) is the summation criterion: If \({\{ p_{n} \} \subset X}\) and \({{\sum_{i=1}^{\infty}} d(p_{i}, p_{i+1}) < \infty}\), then {p n } is Cauchy. In this note we examine instances in which this criterion plays a critical role.
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Kirk, W.A., Shahzad, N. Fixed points and Cauchy sequences in semimetric spaces. J. Fixed Point Theory Appl. 17, 541–555 (2015). https://doi.org/10.1007/s11784-015-0233-4
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DOI: https://doi.org/10.1007/s11784-015-0233-4