Some Notes on Cone Metric Spaces

Abstract

Recently, several articles have been written on the cone metric spaces.Despite the fact that any cone metric space is equivalent to a usual metric space, weaim in this paper to deal with some of the published articles on cone metric spaces byrepairing some gaps, providing new proofs and extending their results to topologicalvector spaces. Several authors have worked with a class of special cones whichknown as strongly minhedral cones where the strongly minihedrality condition (that is, each nonempty bounded above subset has a least upper bound) is veryrestrictive. Another goal of this article is to eliminate or mitigate this condition.Furthermore, we present some examples in order to show that the imagination ofmany authors that the behavior of the ordering induced by a strongly minihedralcone is just as the behavior of the usual ordering on the real line, that has causedan error in their proofs, is not correct. We establish a relationship between strongminihedrality and total orderness. Finally, a xed point theorem for a contractivemapping, which generalizes the corresponding result given in [11], is investigated.One can consider the results of this paper as a generalization and correction of somerecent papers that have been written in this area.