Positive definite solutions of certain nonlinear matrix equations

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 $\\Phi_i ~~(1\\leq i\\leq m)$ 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.