Electronic Journal of Differential Equations,
Vol. 2010(2010), No. 127, pp. 1-8.

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

Authors: Deham Hafsia  (Univ. of Annaba, Algeria)
         Djoudi Ahcene (Univ. of Annaba, Algeria)

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 07, 2010.
Math Subject Classifications: 34K20, 45J05, 45D05.
Key Words: Periodic solution; nonlinear neutral differential equation;
           large contraction; integral equation.