On the existence of Hermitian positive definite solutions of the matrix equation Xs+A∗X-tA=Q

In this paper, the existence of Hermitian positive definite solutions of the general nonlinear matrix equation Xs+A∗X-tA=Q is studied systematically and deeply. A new estimate of Hermitian positive definite solutions is derived. Based on a fixed point theorem, some new sufficient conditions and new necessary conditions for the existence of Hermitian positive definite solutions are obtained. In the end, a necessary and sufficient condition for the existence of a Hermitian positive definite solution is proved.