Authors’ abstract: Based on the elegant properties of the Thompson metric, we prove that the following two kinds of nonlinear matrix equations \(X=\sum _{i=1}^{m} A_{i}^{*} X^{\delta_{i}}A_{i}\) and \(X=\sum_{i=1}^{m} (A_{i}^{*} XA_{i})^{\delta_{i}}\) \((0<|\delta_{i}|<1)\) always have a unique positive definite solution. Iterative methods are proposed to compute the unique positive definite solution. We show that the iterative methods are more effective as \(\delta=\max\{|\delta_i|, i=1,2,\dots,m\}\) decreases. Perturbation bounds for the unique positive definite solution are derived in the end.