A CRITICAL STUDY OF STABILITY OF NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS

Abstract

This chapter presents a critical study of stability of neutral functional differential equations. The chapter defines a functional differential equation. It discusses the relationship among different concepts of stability in the sense of Lyapunov for the solutions of functional differential equations. The chapter emphasizes that uniform equiasymptotic stability does not imply uniform asymptotic stability even for scalar linear ordinary differential equations, and uniform stability and quasi-asymptotic stability do not imply uniform asymptotic stability. The chapter shows that for retarded equations, uniform stability and quasi-asymptotic stability imply equiasymptotic stability. For a general class of periodic neutral functional differential equations, quasi-asymptotic stability and uniform stability imply uniform asymptotic stability. The chapter also highlights that for functional differential equations, stability at a point t0 does not imply stability at a point t >t0 as is shown by an example given by Zverkin. It discusses the existence of almost periodic solutions of retarded functional differential equations connecting with boundedness.