Stability in totally nonlinear neutral differential equations with variable delay using fixed point theory
The totally nonlinear neutral differential equation(d/ dt) (x(t))=−a(t)g(x(t−τ (t))) + (d/ dt)( G(t,x(t−τ (t)))),with variable delay τ(t) ≥ 0 is investigated. We find suitable conditions for t, a, g and G so that for a given continuous initial function 0 a mapping P for the above equation can be defined on a carefully chosen complete metric space S0ψ ; and 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 condition. The obtained theorem improves and generalizes previous results due to Becker and Burton . An example is given to illustrate our main result.