On the nonlinear matrix equation X+A∗X−qA=Q(q≥1)X+A∗X−qA=Q(q≥1)

The general nonlinear matrix equation X+A∗X−qA=Q(q≥1)X+A∗X−qA=Q(q≥1) is investigated. Some sufficient conditions and necessary conditions for the existence of a (unique) positive definite solution are given. An iterative method is proposed to compute positive definite solutions.