On the Behaviors of Solutions of Systems of Non-Linear Differential Equations With Multiple Constant Delays

Abstract

In this paper, non-perturbed and perturbed systems of non-linear differential equations with multiple constant delays are considered. Five new theorems on the qualitative properties of solutions, uniform asymptotic stability (UAS) and instability of trivial solution, boundedness and integrability of solutions, are obtained. The technique of the proofs is based on the construction of two new Lyapunov-Krasovskii functionals (LKFs). An advantage of the new LKFs used here is that they allow to eliminate the Gronwall's inequality and to obtain more convenient conditions. When we compare our results with the related results in the literature, the established conditions here are new, more convenient and general, less conservative, and they are more suitable for applications. We provide three examples to show the applications of the results of this paper.