Electronic Journal of Differential Equations,
Vol. 2008(2008), No. 67, pp. 1-9.
Title: Impulsive dynamic equations on a time scale
Authors: Eric R. Kaufmann (Univ. of Arkansas, Little Rock, AR, USA)
Nickolai Kosmatov (Univ. of Arkansas, Little Rock, AR, USA)
Youssef N. Raffoul (Univ. of Dayton, Dayton, OH, USA)
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'skii, 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 01, 2008.
Math Subject Classifications: 34A37, 34A12, 39A05.
Key Words: Fixed point theory; nonlinear dynamic equation;
stability; impulses.