Solution of a class of cross-coupled nonlinear matrix equations

摘要

The cross-coupled nonlinear matrix equations play an important role in decision making of a variety of dynamical systems and control theory [1]. In this paper we solve the cross-coupled nonlinear matrix equations of the formX=Q1+∑i=1mAi*Fi(X)Ai−∑j=1nBj*Gj(Y)Bj,Y=Q2+∑k=1pCk*F˜k(Y)Ck−∑l=1qDl*G˜l(X)Dl,where Q1, Q2 are n × n Hermitian positive definite matrices, Ai, Bj, Ck, Dl’s are n × n matrices, and F1,…,Fm,F˜1,…,F˜p are order-preserving mappings and G1,…,Gn,G˜1,…,G˜q are order-reversing mappings from the set of n × n Hermitian positive definite matrices to itself. Our approach is based on a new fixed point result discussed in the framework of G-metric spaces, followed by some examples, that distinguishes it from the previously used methods.