SOLUTION OF A PAIR OF NONLINEAR MATRIX EQUATIONS

In this paper we consider a pair of nonlinear matrix equations of the form X = Q(1) + (Y* XY)(r1), Y = Q(2) + (X*YX)(r2), where Q(1), Q(2) are n x n Hermitian positive definite matrices, r(1), r(2)is an element of R and prove the existence and uniqueness of positive definite solutions of these equations. We provide an algorithm to approach the solution. We present a coupled fixed point theorem for non-decreasing mapping and show that a particular case of our nonlinear matrix equations also can be solved by using the derived coupled fixed point theorem. Also we show that by replacing Y with Y-1 in first equation and X with X-1 in second equation and taking Q(1) = Q(2) and r(1) = r(2), the reduced system can be solved using the coupled fixed point theorem of Berinde [5].