The authors use fixed-point theory to present a sufficient condition for the existence of a positive definite solution of the nonlinear matrix equation \(X=Q\pm\sum_{i1}^m A_i^* F(X)A_i\), where \(Q\) is a positive definite matrix, \(A_i\)'s are arbitrary \(n\times n\) matrices and \(F\) is a monotone map from the set of positive definite matrices to itself. They show that the presented condition is weaker than that presented by \textit{A. C. M. Ran} and \textit{M. C. B. Reurings} [Proc. Am. Math. Soc. 132, No. 5, 1435--1443 (2004; Zbl 1060.47056)]. To this end, they 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 \(f\). They conclude, by presenting an example, that their fixed-point theorem cannot be obtained from any existing fixed-point theorem using the process of \textit{M. Jleli} et al. [J. Fixed Point Theory Appl. 12, No. 1--2, 175--192 (2012; Zbl 1266.54089)].