On the existence of extremal positive definite solutions of the nonlinear matrix equation Xr+∑i=1mAi∗XδiAi=I

In the present   paper, a necessary condition for the existence of positive definite solutions of the nonlinear matrix equation Xr+∑i=1mAi∗XδiAi=I is derived, where −1<δi<0−1<δi<0, II is an n×nn×n identity matrix, AiAi are n×nn×n nonsingular complex matrices and r,mr,m are positive integers. Based on the Banach fixed point theorem, a sufficient condition for the existence of a unique positive definite solution of this equation is also derived. Iterative methods for obtaining the extremal (maximal–minimal) positive definite solutions of this equation are proposed. Furthermore, the rate of convergence of some proposed algorithms is proved. Finally, numerical examples are given to illustrate the performance and effectiveness of the proposed algorithms.