Stability in totally nonlinear neutral differential equations with variable delay using fixed point theory
DOI:
https://doi.org/10.4067/S071609172015000100003Keywords:
Fixed points, Stability, Neutral differential equations, Variable delays.Abstract
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 S^{0}ψ ; 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 [6]. An example is given to illustrate our main result.
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