Electronic Journal of Differential Equations,
Vol. 2007(2007), No. 27, pp. 1-12.

Title: Periodicity and stability in neutral nonlinear dynamic equations
       with functional delay on a time scale

Authors: Eric R. Kaufmann (Univ. of Arkansas at Little Rock, AR, USA)
         Youssef N. Raffoul (Univ. of Dayton,  OH, USA)
 
Abstract:
 Let $\mathbb{T}$ be a periodic time scale. We use a
 fixed point theorem due to Krasnosel'skii to show that the
 nonlinear neutral dynamic equation with delay
 $$
 x^{\Delta}(t) = -a(t)x^{\sigma}(t)
 + \left(Q(t,x(t), x(t-g(t))))\right)^{\Delta}
 + G\big(t,x(t), x(t-g(t))\big), t \in \mathbb{T},
 $$
 has a periodic solution.  Under a slightly
 more stringent inequality we show that the periodic solution is
 unique using the contraction mapping principle. Also, by the aid
 of the contraction mapping principle we study the asymptotic stability of
 the zero solution provided that $Q(t,0,0)= G(t,0,0) = 0$.
 
Submitted July 11, 2006. Published February 12, 2007.
Math Subject Classifications: 34K13, 34C25, 34G20.
Key Words: Krasnosel'skii; contraction mapping; neutral;
           nonlinear; Delay; time scales;  periodic solution; 
	   unique solution; stability.