Abstract
We find some necessary and sufficient conditions for the existence of Hermitian positive definite solution of a pair of nonlinear matrix equations of the form: $$\begin{aligned} X^{s_1}+A^*X^{-t_1}A+B^*Y^{-p_1}B=Q_1\\ Y^{s_2}+A^*Y^{-t_2}A+B^*X^{-p_2}B=Q_2, \end{aligned}$$and provide some algorithms for finding solutions. Finally, we give some numerical examples and study the convergence history of the iterations.
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