An application of Lyapunov–Razumikhin method to behaviors of Volterra integro-differential equations

This work presents some extensions and improvements of former results that allow proving asymptotic stability, uniform stability and global uniform asymptotic stability of zero solution to a class of non-linear Volterra integro-differential equations (VIDEs). Via the Lyapunov–Krasovskiĭ and the Lyapunov–Razumikhin methods, three new results are proved on the mentioned concepts. These results are proved using Lyapunov functional and quadratic Lyapunov function. The results of this paper improve and extend the known ones in the literature. Some examples are given to validate these results and the concepts introduced.