Positive definite solution of the matrix equation X = Q − A∗X −1A + B∗X −1B via Bhaskar-Lakshmikantham fixed point theorem

2012 ◽  
Vol 6 (1) ◽  
pp. 27 ◽  
Author(s):  
Maher Berzig ◽  
Xuefeng Duan ◽  
Bessem Samet
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Matrix Equation ◽  
Positive Definite ◽  
Positive Definite Solution ◽  
The Matrix ◽  
Definite Solution

scholarly journals Notes on the Hermitian Positive Definite Solutions of a Matrix Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Jing Li ◽  
Yuhai Zhang
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Matrix Equation ◽  
Positive Definite ◽  
Nonlinear Matrix Equation ◽  
Positive Definite Solution ◽  
Hermitian Positive Definite Solution ◽  
The Matrix ◽  
Definite Solution

The nonlinear matrix equation,X-∑i=1mAi*XδiAi=Q,with-1≤δi<0is investigated. A fixed point theorem in partially ordered sets is proved. And then, by means of this fixed point theorem, the existence of a unique Hermitian positive definite solution for the matrix equation is derived. Some properties of the unique Hermitian positive definite solution are obtained. A residual bound of an approximate solution to the equation is evaluated. The theoretical results are illustrated by numerical examples.

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

2015 ◽  
Vol 2015 ◽  
pp. 1-6
Author(s):  
Dongjie Gao
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Uniqueness Theorem ◽  
Point Theorem ◽  
Positive Definite ◽  
Positive Definite Solution ◽  
Monotone Map ◽  
Monotone Maps ◽  
The Matrix ◽  
Definite Solution

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.

scholarly journals Necessary and Sufficient Conditions for the Existence of a Positive Definite Solution for the Matrix Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Naglaa M. El-Shazly
Keyword(s):  
Matrix Equation ◽  
Sufficient Conditions ◽  
Positive Definite ◽  
Necessary And Sufficient Conditions ◽  
Identity Matrix ◽  
Positive Definite Solution ◽  
The Matrix ◽  
Necessary And Sufficient ◽  
Real Matrices ◽  
Definite Solution

In this paper necessary and sufficient conditions for the matrix equation to have a positive definite solution are derived, where , is an identity matrix, are nonsingular real matrices, and is an odd positive integer. These conditions are used to propose some properties on the matrices , . Moreover, relations between the solution and the matrices are derived.

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