Existence of positive solutions for generalized Lyapunov equations via a coupled fixed point theorem

TL;DR

This paper proves the existence and uniqueness of positive definite solutions for a generalized Lyapunov equation using a coupled fixed point theorem, and proposes an iterative method for finding the solution.

Contribution

It introduces a novel application of a coupled fixed point theorem to establish solution existence and uniqueness for generalized Lyapunov equations, along with an iterative solution method.

Findings

Proved existence and uniqueness of positive definite solutions.

Developed an iterative method for solution approximation.

Provided convergence conditions for the iterative method.

Abstract

We consider the generalized continuous-time Lyapunov equation: $$ A^*XB + B^*XA =-Q, $$ where $Q$ is an $N\times N$ Hermitian positive definite matrix and $A,B$ are arbitrary $N\times N$ matrices. Under some conditions, using the coupled fixed point theorem of Bhaskar and Lakshmikantham, we establish the existence and uniqueness of Hermitian positive definite solution for such equation. Moreover, we provide an iteration method to find convergent sequences which converge to the solution if one exists.