On the Positive Definite Solutions of the Nonlinear Matrix Equation $ X-A^{*}X^{-s}A-B^{*}X^{-t}B=I$

In the present paper, the positive definite solutions for the nonlinear matrix equation $X-A^{*}X^{-s}A-B^{*}X^{-t}B=I$, where $s,tin(0,1]$ are studied. The convergence of the proposed iterative methods are discussed. The necessary and sufficient conditions of the existence of positive definite solutions are derived. Finally, the results are illustrated by some numerical examples.