On the positive definite solution of a class of pair of nonlinear matrix equations

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: Xs1+A∗X-t1A+B∗Y-p1B=Q1Ys2+A∗Y-t2A+B∗X-p2B=Q2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} X^{s_1}+A^*X^{-t_1}A+B^*Y^{-p_1}B=Q_1\\ Y^{s_2}+A^*Y^{-t_2}A+B^*X^{-p_2}B=Q_2, \end{aligned}$$\end{document}and provide some algorithms for finding solutions. Finally, we give some numerical examples and study the convergence history of the iterations.