Stability for nonlinear neutral integro-differential equations with variable delay

In this paper we use the contraction mapping principle to obtain asymptotic stability results of a nonlinear neutral integro-differential equation with variable delay. An asymptotic stability theorem with a necessary and sufficient condition is proved, which improves and generalizes some previous results due to Burton [7], Becker and Burton [6] and Jin and Luo [17]. In the end we provide an example to illustrate our claim