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 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 Existence and Uniqueness of the Positive Definite Solution for the Matrix EquationX=Q+A∗(X^−C)−1A

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Dongjie Gao
Keyword(s):  
Fixed Point ◽  
Existence And Uniqueness ◽  
Positive Semidefinite ◽  
Positive Definite ◽  
Nonlinear Matrix Equation ◽  
Block Diagonal Matrix ◽  
Positive Definite Solution ◽  
The Matrix ◽  
Nonlinear Matrix ◽  
Definite Solution

We consider the nonlinear matrix equationX=Q+A∗(X^−C)−1A, whereQis positive definite,Cis positive semidefinite, andX^is the block diagonal matrix defined byX^=diag(X,X,…,X). We prove that the equation has a unique positive definite solution via variable replacement and fixed point theorem. The basic fixed point iteration for the equation is given.

Positive Definite Sequence of Operators and a Fixed Point Theorem

1972 ◽  
Vol 15 (2) ◽  
pp. 295-295
Author(s):  
A. T. Dash
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Positive Definite ◽  
Definite Sequence ◽  
Positive Definite Sequence ◽  
Image Position

The purpose of this note is to prove the following:Theorem. Let {An} be a positive definite sequence of operators on a Hilbert space H with A0=1. If A1f=f for some f in H, then Anf=f for all n.Note that a bilateral sequence of operators {An:n = 0, ±1, ±2,…} on H is positive definite iffor every finitely nonzero sequence {fn} of vectors in H [1].

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.

Functional delay fractional equations

2016 ◽  
Vol 19 (4) ◽  
Author(s):  
Tadeusz Jankowski
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Uniqueness Theorem ◽  
Point Theorem ◽  
Banach Fixed Point Theorem ◽  
Delay Argument ◽  
Fractional Equations ◽  
Functional Delay

AbstractIn this paper we discuss functional delay fractional equations. A Banach fixed point theorem is applied to obtain the existence (uniqueness) theorem. We also discuss such problems when a delay argument has a form

scholarly journals Common Positive Solution of Two Nonlinear Matrix Equations Using Fixed Point Results

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2199
Author(s):  
Hemant Kumar Nashine ◽  
Rajendra Pant ◽  
Reny George
Keyword(s):  
Fixed Point ◽  
Positive Definite ◽  
Matrix Equations ◽  
Trace Norm ◽  
Time Analysis ◽  
Positive Definite Solution ◽  
Cpu Time ◽  
Nonlinear Matrix ◽  
The Given ◽  
Definite Solution

We discuss a pair of nonlinear matrix equations (NMEs) of the form X=R1+∑i=1kAi*F(X)Ai, X=R2+∑i=1kBi*G(X)Bi, where R1,R2∈P(n), Ai,Bi∈M(n), i=1,⋯,k, and the operators F,G:P(n)→P(n) are continuous in the trace norm. We go through the necessary criteria for a common positive definite solution of the given NME to exist. We develop the concept of a joint Suzuki-implicit type pair of mappings to meet the requirement and achieve certain existence findings under weaker assumptions. Some concrete instances are provided to show the validity of our findings. An example is provided that contains a randomly generated matrix as well as convergence and error analysis. Furthermore, we offer graphical representations of average CPU time analysis for various initializations.

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