Asymptotic stability for impulsive functional differential equation

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,xt),t⩾t0,Δx=Ik(t,x(t−)),t=tk,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].