Positive definite solution of a nonlinear matrix equation

Abstract

Using fixed point theory, we present a sufficient condition for the existence of a positive definite solution of the nonlinear matrix equation ${X = Q \pm \sum^{m}_{i=1}{A_{i}}^*F(X)A_{i}}$, where Q is a positive definite matrix, A _i ’s are arbitrary n × n matrices and F is a monotone map from the set of positive definite matrices to itself. We show that the presented condition is weaker than that presented by Ran and Reurings [Proc. Amer. Math. Soc. 132 (2004), 1435–1443]. In order to do so, we establish some fixed point theorems for mappings satisfying (${\psi, \phi}$)-weak contractivity conditions in partially ordered G-metric spaces, which generalize some existing results related to (${\psi, \phi}$)-weak contractions in partially ordered metric spaces as well as in G-metric spaces for a given function f. We conclude, by presenting an example, that our fixed point theorem cannot be obtained from any existing fixed point theorem using the process of Jleli and Samet [Fixed Point Theory Appl. 2012 (2012), Article ID 210].