Prolongation of solutions and Lyapunov stability for Stieltjes dynamical systems

In this article, we present Lyapunov-type results to study the stability of an equilibrium 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 stability (resp. uniform stability) and asymptotic stability (resp. asymptotic uniform stability). Finally, we present examples and real-life applications to study asymptotic stability of equilibria in two population dynamics models.