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

• Ardjouni, Abdelouaheb ^[2] ; Djoudi, Ahcene ^[1]
  1. [1] Badji Mokhtar University

     Badji Mokhtar University

     Argelia

     
  2. [2] University Souk Ahras.
• Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 34, Nº. 1, 2015, págs. 25-44
• Idioma: inglés
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• Resumen

  • 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.