Necessary and sufficient conditions for the existence of a positive definite solution of a nonlinear matrix equation

Some necessary and sufficient conditions for the existence of a positive definite solution of the nonlinear matrix equation X+A*X ^?α A=Q (0<α≤1) are given. By the way an iterative method is presented. Furthermore, the convergence and error estimation of the iterative algorithm are derived. The illustrative numerical examples due to Peng are worked out.