On the positive definite solutions of nonlinear matrix equation X + A X−δA = Q

In this paper we consider the positive definite solutions of nonlinear matrix equation X + A X−δA = Q, where δ (0, 1], which appears for the first time in [S.M. El-Sayed, A.C.M. Ran, On an iteration methods for solving a class of nonlinear matrix equations, SIAM J. Matrix Anal. Appl. 23 (2001) 632–645]. The necessary and sufficient conditions for the existence of a solution are derived. An iterative algorithm for obtaining the positive definite solutions of the equation is discussed. The error estimations are found.