Solution of a class of cross-coupled nonlinear matrix equations

Abstract 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 form X = Q 1 + ∑ i = 1 m A i * F i ( X ) A i − ∑ j = 1 n B j * G j ( Y ) B j , Y = Q 2 + ∑ k = 1 p C k * F ˜ k ( Y ) C k − ∑ l = 1 q D l * G ˜ l ( X ) D l , where Q1, Q2 are n × n Hermitian positive definite matrices, Ai, Bj, Ck, Dl’s are n × n matrices, and F 1 , … , F m , F ˜ 1 , … , F ˜ p are order-preserving mappings and G 1 , … , G n , 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.

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