On the nonlinear matrix equation $X^s+A^{H}F(X)A=Q$

Nonlinear matrix equation often arises in control theory, statistics, dynamic programming, ladder networks, and so on, so it has widely applied background. In this paper, the nonlinear matrix equation $X^s+A^{H}F(X)A=Q$ are discussed, where operator $F$ are defined in the set of all $n\times n$ positive semi-definite matrices, and $Q$ is a positive definite matrix. Sufficient conditions for the existence and uniqueness of a positive semi-definite solution are derived based on some fixed point theorems. It is shown that under suitable conditions an iteration method converges to a positive semi-definite solution. Moreover, we consider the perturbation analysis for the solution of this class of nonlinear matrix equations, and obtain a perturbation bound of the solution. Finally, we give several examples to show how this works in particular cases, and some numerical results to specify the rationality of the results we have obtain.