Positive definite solutions of the matrix equations X±A∗X−qA=Q(q≥1)

In this paper we investigate nonlinear matrix equations X±A∗X−qA=QX±A∗X−qA=Q where q≥1q≥1. We derive necessary conditions and sufficient conditions for the existence of positive definite solutions for these equations. We provide a sufficient condition for the equation X+A∗X−qA=QX+A∗X−qA=Q to have two different positive definite solutions and several sufficient conditions for the equation X−A∗X−qA=QX−A∗X−qA=Q to have a unique positive definite solution. We also propose iterative methods for obtaining positive definite solutions of these equations. Numerical examples are given to illustrate the effectiveness of the methods.