Fixed Points and Stability in Nonlinear Neutral Integro-Differential Equations With Variable Delay

Imene Soualhia, Abdelouaheb Ardjouni, Ahcene Djoudi

The nonlinear neutral integro-differential equation \[ x'(t)=-\int^t_{\tau(t)}a(t,s)g(x(s))ds+c(t)x'(t-\tau(t)), \] with variable delay $\tau(t)\geq0$ is investigated. We find suitable conditions for $\tau$, $a$, $c$ and $g$ so that for a given continuous initial function$\psi$ a mapping $P$ for the above equation can be defined on a carefully chosen complete metric space $S^0_\psi$ in which $P$ possesses a unique fixed point. The final result is an asymptotic stability theorem for the zero solution with a necessary and sufficient conditions. The obtained theorem improves and generalizes previous results due to Burton [6], Becker and Burton [5] and Jin and Luo [16].