Solving a class of nonlinear matrix equations via the coupled fixed point theorem
We consider a class of nonlinear matrix equations of the type(1) X = Q + ? i = 1 m A i ? G ( X ) A i - ? j = 1 k B j ? K ( X ) B j , where Q is a positive definite matrix, A i , B j are arbitrary n i? n matrices and G , K are two order-preserving or order-reversing continuous maps from H ( n ) into 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.