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 F the matrix equation X − ∑ i = 1 m A i ⋆ F ( X ) A i = Q , where A is an n × n matrix and Q 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 = 1 m A i ⋆ F ( X ) A i + B i ⋆ G ( X ) B i = Q , where A is an n × n matrix, Q is a Hermitian positive definite matrix, F is order preserving and G is order reversing.