On the existence of Hermitian positive definite solutions of the matrix equation [formula omitted

Abstract

In this paper, the existence of Hermitian positive definite solutions of the general nonlinear matrix equation X s + A ∗ X - t A = 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.