Positive Definite Solutions Of One Type Of Nonlinear Matrix Equation

Nonlinear matrix equation is one of the important issues in the field of numerical algebra and nonlinear analysis,Which has very important applications in control theory, Dynamic programming,statistics,random inflow,queuing theory,Trapezoidal mesh,and other fields.In this paper,the Hermite positive definite solutions of the nonlinear matrix equation (?) are studied,where Aj,Bi are n×n-order complex matrices,Q is n×n-order Hermite positive definite matrix.This paper is structured as follows: The introduction describes the background of the nonlinear matrix equation and research results.ChapterⅡfirstly the particular cir-cumstance is studied,where m=k=1,Q=I,which are the Hermite positive definite solutions of the matrix equation X+B*X-1B-A*X-1A=I.Using an important matrix inequality,we obtain a necessary condition for the existence of positive definite solutions of this matrix equation,And then using the fixed point theorem,we obtain two sufficient conditions for the existence of positive definite solutions of this matrix equation.In the case of the matrix equation having positive definite solutions,the ranges of solutions are given and a sufficient condition for uniqueness of solution is obtained,and using the contraction mapping principle that is proved under certain conditions specified inter-val having a unique solution.Then the conclusions obtained are extended to the matrix equation (?),some necessary and sufficient conditions for the existence of positive definite solutions are given. Followed by a transforma-tion (?),the problem of solving the matrix equation (?) is transformed into the problem of solving the matrix equation (?).Thus the purpose of studing the Her-mite positive definite solutions of the matrix equation (?) is reached.ChapterⅢusing the fixed point theorem constructes two iteration methods for finding the positive definite solutions of these equations studied in this paper and the corresponding convergence analysis are given；Finally,numerical examples are given to illustrate the obtained conclusions.