Abstract

‎We investigate positive definite solutions of nonlinear matrix equations $X-f(Phi(X))=Q$ and $X-sum_{i=1}^{m}f(Phi_i(X))=Q$‎, ‎where $Q$ is a positive definite matrix‎, ‎$Phi$ and $Phi_i‎ ~‎~(1leq ileq m)$   are positive linear maps on $mathbb{M}_n(mathbb{C})$ and $f$ is a nonnegative matrix monotone or matrix anti-monotone function on $[0,infty)$‎. ‎In this article‎, ‎using appropriate inequalities and some fixed point results‎, ‎we prove the existence of unique positive definite solutions for the mentioned above equations‎.