Stability in totally nonlinear neutral differential equations with variable delay using fixed point theory

Authors

  • Abdelouaheb Ardjouni University Souk Ahras.
  • Ahcene Djoudi UBMA.

DOI:

https://doi.org/10.4067/S0716-09172015000100003

Keywords:

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 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 [6]. An example is given to illustrate our main result.

Author Biographies

Abdelouaheb Ardjouni, University Souk Ahras.

Faculty of Sciences and Technology, Department of Mathematics and Informatics.

Ahcene Djoudi, UBMA.

Applied Mathematics Lab, Faculty of Sciences, Department of Mathematics.

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How to Cite

[1]
A. Ardjouni and A. Djoudi, “Stability in totally nonlinear neutral differential equations with variable delay using fixed point theory”, Proyecciones (Antofagasta, On line), vol. 34, no. 1, pp. 25-44, 1.

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