Stability in Nonlinear Neutral Differential Equations with Infinite Delay

Abdelouaheb Ardjouni, Ahcene Djoudi

In this paper we use the contraction mapping theorem to obtain asymptotic stability results of the nonlinear neutral differential equation with infinite delay $\frac{\mathrm{d}}{\mathrm{d}t}x(t) = -a(t) x(t-\tau_{1}(t)) + \frac{\mathrm{d}}{\mathrm{d}t} Q(t, x(t-\tau_{2}(t))) + \int_{-\infty}^{t} D(t,s) f(x(s))\mathrm{d}s$. An asymptotic stability theorem with a necessary and sufficient condition is proved, which improves and generalizes some results due to Burton [6], Zhang [17], Althubiti, Makhzoum, Raffoul [1].