Electron. J. Diff. Equ., Vol. 2010(2010), No. 127, pp. 1-8.

Existence of periodic solutions for neutral nonlinear differential equations with variable delay

Deham Hafsia, Djoudi Ahcene

Abstract:
We use a variation of Krasnoselskii fixed point theorem introduced by Burton to show that the nonlinear neutral differential equation
$$
 x'(t)=-a(t)x^3(t)+c(t)x'(t-g(t))+G(t,x^3(t-g(t))
 $$
has a periodic solution. Since this equation is nonlinear, the variation of parameters can not be applied directly; we add and subtract a linear term to transform the differential into an equivalent integral equation suitable for applying a fixed point theorem. Our result is illustrated with an example.

Submitted April 15, 2010. Published September 7, 2010.
Math Subject Classifications: 34K20, 45J05, 45D05.
Key Words: Periodic solution; nonlinear neutral differential equation; large contraction, integral equation.

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Deham Hafsia
Department of Mathematics, Faculty of Sciences
University of Annaba, P.O. Box 12 Annaba, Algeria
email: deh_71@yahoo.fr
  Djoudi Ahcéne
Department of Mathematics, Faculty of Sciences
University of Annaba, P.O. Box 12 Annaba, Algeria
email: adjoudi@yahoo.com

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