Electron. J. Diff. Eqns., Vol. 2008(2008), No. 67, pp. 1-9.

Impulsive dynamic equations on a time scale

Eric R. Kaufmann, Nickolai Kosmatov, Youssef N. Raffoul

Abstract:
Let $\mathbb{T}$ be a time scale such that $0, t_i, T \in \mathbb{T}$, $i = 1, 2, \dots, n$, and $0 < t_i < t_{i+1}$. Assume each $t_i$ is dense. Using a fixed point theorem due to Krasnosel'ski\i}, we show that the impulsive dynamic equation
$$\displaylines{ y^{\Delta}(t) = -a(t)y^{\sigma}(t)+ f ( t, y(t) ),\quad t \in (0, T],\cr y(0) = 0,\cr y(t_i^+) = y(t_i^-) + I (t_i, y(t_i) ), \quad i = 1, 2, \dots, n, }$$
where $y(t_i^\pm) = \lim_{t \to t_i^\pm} y(t)$, and $y^\Delta$ is the $\Delta$-derivative on $\mathbb{T}$, has a solution. Under a slightly more stringent inequality we show that the solution is unique using the contraction mapping principle. Finally, with the aid of the contraction mapping principle we study the stability of the zero solution on an unbounded time scale.

Submitted November 20, 2007. Published May 1, 2008.
Math Subject Classifications: 34A37, 34A12, 39A05.
Key Words: Fixed point theory; nonlinear dynamic equation; stability; impulses.

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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
Nickolai Kosmatov
Department of Mathematics and Statistics
University of Arkansas at Little Rock
Little Rock, Arkansas 72204-1099, USA
email: nxkosmatov@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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