Some Notes on Cone Metric Spaces

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, we aim in this paper to deal with some of the published articles on cone metric spaces by repairing some gaps, providing new proofs and extending their results to topological vector spaces. Several authors have worked with a class of special cones which known as strongly minhedral cones where the strongly minihedrality condition (that is, each nonempty bounded above subset has a least upper bound) is very restrictive. Another goal of this article is to eliminate or mitigate this condition. Furthermore, we present some examples in order to show that the imagination of many authors that the behavior of the ordering induced by a strongly minihedral cone is just as the behavior of the usual ordering on the real line, that has caused an error in their proofs, is not correct. We establish a relationship between strong minihedrality and total orderness. Finally, a fixed point theorem for a contractive mapping, which generalizes the corresponding result given in [1], is investigated. One can consider the results of this paper as a generalization and correction of some recent papers that have been written in this area.