Electronic Journal of Differential Equations,
Vol. 2005(2005), No. 142, pp. 1-11.

Title: Periodicity and stability in neutral nonlinear
       differential equations with functional delay
       
Authors: Youssef M. Dib (Univ. of Louisiana at Lafayette, LA, USA)
         Mariette R.  Maroun (Baylor Univ., Waco, TX, USA)
	 Youssef N. Raffoul (Univ. of Dayton, Dayton, OH, USA)

Abstract: 
 We study the existence and uniqueness of periodic
 solutions and the stability of the zero solution of the nonlinear
 neutral differential equation
 $$
 \frac{d}{dt}x(t) = -a(t)x(t)+ \frac{d}{dt}Q(t, x(t-g(t)))
 +G(t,x(t), x(t-g(t))).
 $$
 In the process we use  integrating factors and
 convert the given neutral differential equation into an equivalent
 integral equation. Then we construct appropriate mappings and
 employ Krasnoselskii's fixed point theorem to show the existence
 of a periodic solution of this neutral differential equation. We
 also use the contraction mapping principle to show the existence
 of a unique periodic solution and the asymptotic stability of the
 zero solution provided that $Q(0,0)= G(t, 0,0) = 0$.
 
Submitted November 30, 2004. Published December 6, 2005.
Math Subject Classifications: 34K20, 45J05, 45D05.
Key Words: Krasnoselskii; contraction; neutral differential equation;
           integral equation; periodic solution; asymptotic stability