#### Karima Bessioud, Abdelouaheb Ardjouni, Ahcene Djoudi, Asymptotic stability in nonlinear neutral Levin-Nohel integro-differential equations, Vol. 2017 (2017), Article ID 19, pp. 1-12

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DOI: 10.23952/jnfa.2017.19

Received November 11, 2016; Accepted February 16, 2017; Published February 26, 2017

Abstract. In this paper we use the contraction mapping theorem to obtain asymptotic stability results about the zero solution for the following nonlinear neutral Levin-Nohel integro-differential equation $\frac{d}{dt}x(t)+\int_{t-\tau (t)}^{t}a(t,s)x(s)ds+\frac{d}{dt}g\left(t,x(t-\tau (t))\right) =0.$ An asymptotic stability theorem with a necessary and sufficient condition is proved. In addition, the case of the equation with several delays is studied.