Existence of positive solutions for generalized Lyapunov equations via a coupled fixed point theorem
Provides theoretical guarantees for solving a class of matrix equations, but the result is incremental as it extends existing fixed-point techniques to a specific equation form.
The paper proves existence and uniqueness of Hermitian positive definite solutions for a generalized Lyapunov equation using a coupled fixed point theorem, and provides an iterative method to compute them.
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.