Positive definite solution of a nonlinear matrix equation

Using fixed point theory, we present a sufficient condition for the existence of a positive definite solution of the nonlinear matrix equation X = Q ± ∑ i = 1 m 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 ( ψ , ϕ )-weak contractivity conditions in partially ordered G -metric spaces, which generalize some existing results related to ( ψ , ϕ )-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].