Practical and Lipschitz stability for generalized ordinary differential equations and functional Volterra–Stieltjes integral equations

This paper develops Lyapunov–type criteria for practical and Lipschitz stability of generalized ordinary differential equations (GODEs). After formulating these stability concepts within the GODE framework, we obtain new sufficient conditions for their validity and establish converse Lyapunov theorems that characterize both properties through appropriate Lyapunov functionals. The resulting criteria provide quantitative information on the boundedness of solutions and on their dependence on initial data, while accommodating systems that may display jumps, discontinuities, or memory effects generated by nonabsolute integrators. We further extend the analysis to functional Volterra–Stieltjes integral equations (FV–SIEs), which can be embedded into the GODE formalism, and demonstrate how the Lyapunov methods developed here apply in that setting. Illustrative examples are provided to clarify the implementation and scope of the theory.