Eric R. Kaufmann, Nickolai Kosmatov, Youssef N. Raffoul
Abstract:
Let
be a time scale such that
,
,
and
. Assume each
is dense.
Using a fixed point theorem due to Krasnosel'ski\i}, we show that the
impulsive dynamic equation
where
, and
is
the
-derivative
on
,
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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