Positive definite solutions of the matrix equations X ± A*X−qA = Q

In this paper we investigate nonlinear matrix equations X + A*X−qA = Q and X − A*X−qA = Q where q ∈ (0, 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 = Q to have two different positive definite solutions and a sufficient condition for the equation X − A*X−qA = Q to have a unique positive definite solution. We also propose iterative methods for obtaining positive definite solutions for these equations.