Research On The Existence Of Positive Definite Solutions For Two Types Of Nonlinear Matrix Equations

With the development of science and technology,nonlinear matrix equations themselves are the basic models in some areas such as circuit network,elastic mechanics,heat conduc-tion,vibration,etc.Also,they can be directly or indirectly converted into one of the parts of process of numerical methods.Thus,it has a good prospect to study nonlinear matrix equations and many scholars pay more and more attention to solve matrix equations.This thesis is mainly divided into three chapters:Chapter 1 is the preface.We simply introduce the studying background,current situ-ation,the main lemmas and definitions of this paper.In chapter 2,we use fixed point theorems for monotone operators and property of normal cone to show the existence and uniqueness of positive definite solutions for nonlinear matrix equations(?) and(?) where p,m,j are positive integers,1 ?j?m,A,B are n×n Hermitian positive definite matrices and Mi(i=1,2,…,m)are n×n nonsingular complex matrices.Furthermore,we use a muti-step stationary iterative method to approximate the unique positive definite solution of these equations.Finally,we give one example to illustrate efficiency of this iterative method.In chapter 3,we use a fixed point theorem in a cone and the Banach fixed point theorem to derive some necessary and sufficient conditions for the existence and uniqueness of positive definite solutions of equation(?) wherep,m are positive integers,Mi(i=1,2,…,m)are n×n nonsingular complex matrices,A and B are Hermitian positive definite matrices.Our results show that the unique solution exists in a partially order set and can be approximated by making an iterative sequence for any given initial point in this set.Furthermore,we analyze three algorithms and study the convergence history of the iterations of these algorithms.Finally,we give two examples to illustrate the efficiency of those algorithms.