A note about the relation between fixed point theory on cone metric spaces and fixed point theory on metric spaces

Abstract: Let Y be a locally convex Hausdorff space, K \subset E a cone and \leq_K the partial order defined by K. Let (X, p) be a TV S- cone metric space, {\phi} : K \rightarrow K a vectorial comparison function and f : X \rightarrow X such that p(f(x), f(y)) \leq_K {\phi}(p(x, y)), for all x, y \in X. We shall show that there exists a scalar comparison function {\psi} : R+ \rightarrow R+ and a metric d_p(in usual sense) on X such that d_p(f(x), f(y)) \leq {\psi}(d_p(x, y)), for all x, y \in X. Our results extend the results of Du (2010) [Wei-Shih Du, A note on cone metric fixed point theory and its equivalence, Nonlinear Anal. 72 (2010), 2259-2261].