Solving a class of nonlinear matrix equations via the coupled fixed point theorem

We consider a class of nonlinear matrix equations of the typeequation(1)X=Q+∑i=1mAi∗G(X)Ai-∑j=1kBj∗K(X)Bj,where Q   is a positive definite matrix, Ai,BjAi,Bj are arbitrary n×nn×n matrices and G,KG,K are two order-preserving or order-reversing continuous maps from H(n)H(n) into P(n)P(n). In this paper we first discuss existence and uniqueness of coupled fixed points in a L-space endowed with reflexive relation. Next on the basis of the coupled fixed point theorems, we prove the existence and uniqueness of positive definite solutions to such equations.