Stability, Instability, Boundedness and Integrability of Solutions of a Class of Integro-Delay Differential Equations

Abstract

This article is devoted to a class of systems of nonlinear Volterra integro-delay differential equations (IDDEs). Based upon the definition of two new Lyapunov -Krasovskii functionals (LKFs), four new results are obtained on uniformly asymptotic stability (UAS), instability of zero solution and integrability of solutions of the unperturbed IDDE and as well as on boundedness of solutions of the perturbed IDDE. When we have comparisons with some former results, it follows that the conditions of UAS here are more general, less restrictive and convenient to applications. Next, the instability result is a new contribution to the topic and literature. The integrability and boundedness results are different and more general than those available in the literature. Here, we also eliminate Gronwall's inequality as an advantage to investigate the boundedness of solutions as well as obtain some former results under more suitable and weaker conditions via the first LKF. To show the application of the new theorems, three examples are also provided on the mentioned results.