Prolongation of solutions and Lyapunov stability for Stieltjes dynamical systems

TL;DR

This paper develops Lyapunov-based methods to analyze stability and prolongation of solutions in Stieltjes dynamical systems, with applications to population models.

Contribution

It introduces Lyapunov-type stability results and prolongation techniques specifically for Stieltjes dynamical systems, extending existing stability theory.

Findings

Established conditions for stability and asymptotic stability using Lyapunov functions.

Proved global existence of maximal solutions via prolongation results.

Applied the theory to real-life population dynamics models.

Abstract

In this article, we introduce Lyapunov-type results to investigate the stability of the trivial solution of a Stieltjes dynamical system. We utilize prolongation results to establish the global existence of the maximal solution. Using Lyapunov's second method, we establish results of (uniform) stability and (uniform) asymptotic stability by employing a Lyapunov function. Additionally, we present examples and real-life applications to study asymptotic stability of equilibria in two population dynamics models.