Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 103894, 5 pages
http://dx.doi.org/10.1155/2013/103894
Research Article
-Exponential Stability of Nonlinear Impulsive
Dynamic Equations on Time Scales
Veysel Fuat HatipoLlu,
1
Deniz Uçar,
2
and Zeynep Fidan Koçak
1
1
Department of Mathematics, Faculty of Science, Mu˘ gla University, K¨ otekli Campus,
48000 Mu˘ gla, Turkey
2
Department of Mathematics, Faculty of Sciences and Arts, Usak University, 1 Eylul Campus,
64200 Usak, Turkey
Correspondence should be addressed to Veysel Fuat Hatipo˘ glu; veyselfuat.hatipoglu@mu.edu.tr
Received 26 November 2012; Accepted 15 March 2013
Academic Editor: Stefan Siegmund
Copyright © 2013 Veysel Fuat Hatipo˘ glu et al. Tis is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
Te purpose of this paper is to present the sufcient -exponential, uniform exponential, and global exponential stability conditions
for nonlinear impulsive dynamic systems on time scales.
1. Introduction
In recent years, a signifcant progress has been made in the
stability theory of impulsive systems [1, 2], and in [3] authors
studied the -exponential stability for nonlinear impulsive
diferential equations. Tere are various types of stability of
dynamic systems on time scales such as asymptotic stability
[4, 5], exponential and uniform exponential stability [6–8],
and ℎ-stability [9]. In the past decade, many authors studied
impulsive dynamic systems on time scales [10–14]. Tere
are some papers on the theory of the stability of impulsive
dynamic systems on time scales. In [15], stability criteria for
impulsive systems are given and in [16], authors studied -
uniform stability of linear impulsive dynamic systems.
In this paper, we consider the -exponential stability
of the zero solution of the frst-order nonlinear impulsive
dynamic system
Δ
()=(,()), ∈ T
+
0
,̸ =
,
(
+
)−(
−
)=
((
−
)), =
,=1,2,...,,
(
+
0
)=
0
,
(1)
where T is a time scale which has at least fnitely many right-
dense points of impulsive
, : [0,∞) × R
→ R
is
a nonlinear function and rd continuous in (
−1
,
]× R
,
∈
rd
[R
, R
], and 0≤
0
<
1
<
2
<⋅⋅⋅<
< are
fxed moments of impulsive efect. Let
: T → (0,∞),
= 1,2,...,, be rd continuous functions and let =
diag[
1
,
2
,...,
]. Troughout the paper, we assume that
(,0)=0, for all in the time scale interval [0,∞), and call
the zero function the trivial solution of (1) and we consider
T
+
0
={∈ T :≥
0
}. Existence and uniqueness of solutions
of (1) have been studied in [10].
In the following part we present some basic concepts
about time scale calculus and we refer the reader to resource
[17] for more detailed information on dynamic equations on
time scales.
2. Preliminaries
A time scale T is an arbitrary nonempty closed subset of
the real numbers R. For ∈ T we defne the forward jump
operator : T → T by
():= inf {∈ T :>} (2)
while the backward jump operator : T → T is defned by
():= sup {∈ T :<}. (3)