Abstract
In this paper we examine fixed point results for mappings satisfying certain contractive type conditions in semimetric spaces. In addition to the standard semimetric assumptions it is shown that in many instances some combination of the following three conditions often suffice: (i) uniqueness of limits; (ii) the Cauchy convergence criterion, which asserts that a sequence {x(n)} in a semi metric space X is Cauchy provided Sigma(infinity)(n=1) d (x(n)., x(n+1)) < infinity; and (iii) continuity of the distance function d. While these conditions all follow from the triangle inequality, each holds in broader classes of spaces. In particular, all three conditions hold in a semimetric space (X, d) which satisfies the following weakened version of the triangle inequality, which asserts that there exists k >= 1 such that given x, y, z is an element of X, vertical bar d (x, z) - d (z, y)vertical bar <= < kd (x, y). Among other things, it is shown that there are nontrivial semimetric spaces which satisfy this condition but which are not metric.