On equations that are equivalent to the nonlinear matrix equation

The nonlinear matrix equation X − 1 + A ∗ X α A = Q ( 0 < α ≤ 1 ) is equivalent to the nonlinear matrix equation X + A ∗ X − α A = Q ( 0 < α ≤ 1 ) . The nonlinear matrix equation X − 1 + ( A X A ∗ ) 1 / α = Q ( 1 < α ) is equivalent to the nonlinear matrix equation X − 1 + A ∗ X α A = Q ( 1 < α ) . The necessary and sufficient conditions for the existence of a positive definite solution of X − 1 + A ∗ X α A = Q ( 0 < α ≤ 1 ) and X − 1 + ( A X A ∗ ) 1 / α = Q ( 1 < α ) are given. In the process, two iterative algorithms are obtained. Estimations of the errors of the iterative algorithms are derived. Two numerical examples are given that demonstrate that the iterative algorithms are applicable.