On nonlinear matrix equations

A.C.M. Ran and M.C.B. Reurings [A.C.M. Ran, M.C.B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. American Mathematical Society 132 (2003) 1435–1443.] proved that under continuity of FF the matrix equationX−∑i=1mAi⋆F(X)Ai=Q, where AA is an n×nn×n matrix and QQ is a Hermitian positive definite matrix, has a unique solution. The purpose of this paper is to solve the above equation without the continuity condition. Also, we prove a coupled fixed point theorem and apply it to solve the equation X−∑i=1mAi⋆F(X)Ai+Bi⋆G(X)Bi=Q, where AA is an n×nn×n matrix, QQ is a Hermitian positive definite matrix, FF is order preserving and GG is order reversing.