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

TL;DR

This paper establishes a connection between fixed point theories in cone metric spaces and traditional metric spaces, showing that fixed point results in cone spaces can be translated into metric space results.

Contribution

It introduces a method to derive scalar comparison functions and metrics from cone metric spaces, extending prior fixed point theorems to a broader setting.

Findings

Existence of scalar comparison functions from vectorial ones

Equivalence of fixed point results between cone and metric spaces

Extension of Du's 2010 fixed point results

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].