In this paper we introduce the notion of convex property in cone metric spaces. We show that a cone convex metric space has convex property. We also show that there are cone metric spaces with convex property, which are not cone convex metric spaces. We prove a common fixed point theorem for two self maps on a cone metric space with convex property and obtain the corresponding result on cone convex metric spaces as a corollary. An open problem is also given at the end of the paper.
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