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.