Studied Hermitian Positive Definite Solutions Of Three Classes Of Nonlinear Matrix Equations

The problem of solving nonlinear matrix equations is one of important issues in thefields of numerical algebra and nonlinear analysis in recent years. Actually, it is widelyused in many fields such as control theory, dynamic programming, statistics, stochasticfiltering and ladder networks. In this paper. Hermitian positive definite solutions ofthe nonlinear matrix equations studied, where |δi| < 1, Ai(i = 1,2,...,m) is an n×ncomplex matrix, m,n is positive integer, Ci(i = 1,2,...,m) is an positive semidefinitematrix, A and Q are positive definite matrix. Here, we study three basic problems:(1) the problem of the existence of solutions; (2) the methods for solving the numericalsolution; (3) the perturbation analysis of the solution.The main results are as follows:1. The properties of solutions are studied. Firstly, based on fixed point theoremsfor m-onotone and mixed monotone operators in a normal cone, we prove that thenonlinear matrix equation must has a unique positivedefinite solution. The new upper boundary and lower boundary are derived withdi?erentδi. Secondly, by the fixed point theory, some new su?cient conditions andnecessary conditions are obtained for the existence of the positive definite solutionsof the nonlinear matrix equation . Thesu?cient conditions that the matrix equation has the unique positive definite solutionare given.2. The methods for solving the numerical solution are constructed. By the fixedpoint theory, a muti–step stationary iterative method is presented to compute theunique positive definite solution of matrix equation andthe fixed iterative methods to compute the positive definite solution of matrix equation. The corresponding convergence theoremsare proved.3. Some new perturbation bounds are derived.4. The numerical examples are given. The numerical examples are given to illus-trate the correctness of theoretical results and the e?ectiveness of iterative methods.