The nonlinear matrix equation X^p = A + M^T (X # B)M is considered, where p ≥ 1 is a positive integer, M is an n × n nonsingular matrix, A is a positive semidefinite matrix and B is a positive definite matrix. We call C # D the geometric mean of positive definite matrices C and D.
Some properties of the positive definite solution of the nonlinear matrix equation is investigated, including the existence and uniqueness of the solution, a lower and an upper bound. Iteration method for finding the numerical solution is proposed. Perturbation bounds with respect to small perturbations on the coefficient matrices are obtained.