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 and , then {p (n) } is Cauchy. In this note we examine instances in which this criterion plays a critical role.