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

The nonlinear neutral integro-differential equation x' (t) = - integral(t)(t-tau(t)) a(t, s)g(x(s))ds + c(t)x' (t - tau(t)), with variable delay tau(t) >= 0 is investigated. We find suitable conditions for tau, a, c and g so that for a given continuous initial function psi mapping P for the above equation can be defined on a carefully chosen complete metric space S-psi(0) 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].