On the Stability and Instability of Functional Volterra Integro-Differential Equations of First Order

Abstract

This paper is concerned with non-linear Volterra integro-differential equation (VIDE) with constant time-lag, tau : x'(t) = P(t)f(x(t)) - integral(t)(t) (tau) K(t, s)f(x(s))ds. Via Lyapunov functionals and basic inequalities, sufficient conditions are given for the exponential stability (ES) and instability (I) of the trivial solution of the former (VIDE). We introduce two new results for the above topics for the trivial solution of that (VIDE). Our conditions involve the nonlinear generalization and extensions of those found in the literature. The results to be obtained are new and complements that in the literature.