A Fixed Point Theorem for Monotone Maps and Its Applications to Nonlinear Matrix Equations

Dongjie Gao

doi:10.1155/2015/167049

Abstract

By using the fixed point theorem for monotone maps in a normal cone, we prove a uniqueness theorem for the positive definite solution of the matrix equationX=Q+A⁎f(X)A, wherefis a monotone map on the set of positive definite matrices. Then we apply the uniqueness theorem to a special equationX=kQ+A⁎(X^-C)qAand prove that the equation has a unique positive definite solution whenQ^≥Candk>1and0<q<1. For this equation the basic fixed point iteration is discussed. Numerical examples show that the iterative method is feasible and effective.

Highlights

We consider the matrix equationX = Q + A∗f (X) A, (1)where Q is an n × n positive definite matrix, A is arbitrary m × n matrix, and f is a monotone map on P(m).The study of matrix equation has a long history, involving in particular the study of algebraic Riccati equations for discrete time optimal control and for the stochastic realization problem
By using the fixed point theorem for monotone maps in a normal cone, we prove a uniqueness theorem for the positive definite solution of the matrix equation X = Q + A∗f(X)A, where f is a monotone map on the set of positive definite matrices
In [14], the author provided a new proof for the uniqueness of the positive definite solution of this equation using a change of variable and a fixed point theorem, which is an easier argument than the one used in [12]

Summary

Introduction

Where Q is an n × n positive definite matrix, A is arbitrary m × n matrix, and f is a monotone map on P(m). In [14], the author provided a new proof for the uniqueness of the positive definite solution of this equation using a change of variable and a fixed point theorem, which is an easier argument than the one used in [12]. This development leads to consideration of a general class of matrix equations, which started with the paper by El-Sayed and Ran [15], and was developed further by Ran and Reurings [16,17,18,19]. A cone is said to be a solid cone if P0 ≠ φ

Preliminaries

Numerical Examples