Electronic Journal of Differential Equations,
Vol. 2011 (2011), No. 128, pp. 1-7.

Title: Periodic solutions for a second-order nonlinear neutral
       differential equation with variable delay

Authors: Abdelouaheb Ardjouni (Univ. of Annaba, Algeria)
         Ahcene Djoudi  (Univ. of Annaba, Algeria)

Abstract:
 In this work, the hybrid fixed point theorem of Krasnoselskii
 is used to prove the existence of periodic solutions of the
 second-order nonlinear neutral differential equation
 $$
 \frac{d^2}{dt^2}x(t)+p(t)\frac{d}{dt}x(t)+q(t)x(t)
 =\frac{d}{dt}g(t,x(t-\tau(t)))+f(t,x(t),x(t-\tau(t))).
 $$
 We transform the problem into an integral equation and
 uniqueness of the periodic solution, by means of the
 contraction mapping principle.

Submitted March 2, 2011. Published October 11, 2011.
Math Subject Classifications: 34K13, 34A34, 34K30, 34L30.
Key Words: Periodic solution; neutral differential equation; 
           fixed point theorem.