Asymptotic stability for impulsive functional differential equation

Abstract

In this paper, sufficient conditions are given for asymptotic stability and uniformly asymptotic stability of impulsive functional differential equation of the form { x ′ ( t ) = f ( t , x t ) , t ⩾ t 0 , Δ x = I k ( t , x ( t − ) ) , t = t k , k ∈ Z + , and the conditions in theorems extend or improve the corresponding ones in [J. Yan, J. Shen, Impulsive stabilization of functional differential equation by Liapunov–Razumikhin functions, Nonlinear Anal. 37 (1999) 245–255; J. Shen, Z. Luo, Impulsive stabilization of functional differential equations via Liapunov functionals, J. Math. Anal. Appl. 240 (1999) 1–15; L. Hatvani, On the asymptotic stability for nonautonomous functional differential equations by Liapunov functionals, Trans. Amer. Math. Soc. 354 (2002) 3555–3571].