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

Abstract

The nonlinear neutral integro-differential equation

x′(t)=-∫_{t-τ(t)}^{t}a(t,s)g(x(s))ds+c(t)x′(t-τ(t)),

with variable delays τ(t)≥0 is investigated. We find suitable conditions for τ, a, c and g so that for a given continuous initial function ψ a mapping P for the above equation can be defined on a carefully chosen complete metric space S_{ψ}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 <cite>b2</cite>, Becker and Burton <cite>b1</cite> and Jin and Luo <cite>j1</cite>.