Electron. J. Diff. Eqns., Vol. 2007(2007), No. 27, pp. 1-12.

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

Eric R. Kaufmann, Youssef N. Raffoul

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.

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Eric R. Kaufmann
Department of Mathematics and Statistics
University of Arkansas at Little Rock
Little Rock, Arkansas 72204-1099, USA
email: erkaufmann@ualr.edu
Youssef N. Raffoul
Department of Mathematics, University of Dayton
Dayton, OH 45469-2316, USA
e-mail: youssef.raffoul@notes.udayton.edu

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