Hindawi Publishing Corporation
International Journal of Diferential Equations
Volume 2013, Article ID 764389, 10 pages
http://dx.doi.org/10.1155/2013/764389
Research Article
Qualitative Analysis of Solutions of Nonlinear Delay
Dynamic Equations
Mehmet Ünal
1
and Youssef N. Raffoul
2
1
Department of Mathematics, Canakkale Onsekiz Mart University, 17020 Canakkale, Turkey
2
Department of Mathematics, University of Dayton, OH 45469-2316, USA
Correspondence should be addressed to Mehmet
¨
Unal; munal@comu.edu.tr
Received 21 October 2013; Revised 9 December 2013; Accepted 9 December 2013
Academic Editor: Tongxing Li
Copyright © 2013 M.
¨
Unal and Y. N. Rafoul. 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.
We use the fxed point theory to investigate the qualitative analysis of a nonlinear delay dynamic equation on an arbitrary time
scales. We illustrate our results by applying them to various kind of time scales.
1. Introduction
In this paper, we investigate the qualitative analysis of
solutions of nonlinear delay dynamic equation of the form
Δ
()=−()((()))
Δ
(), ∈[
0
,∞)
T
(1)
on an arbitrary time scale T which is unbounded above,
where the functions and are rd-continuous, the delay
function : [
0
,∞)
T
→ [(
0
),∞)
T
is strictly increasing,
invertible, and delta diferentiable such that ()<, |
Δ
()| <
∞ for ∈ T , and (
0
)∈ T .
Although it is assumed that the reader is already familiar
with the time scale calculus, for completeness, we will provide
some essential information about time scale calculus in the
Section 1.1. We should only mention here that this theory was
introduced in order to unify continuous and discrete analysis;
however it is not only unify the theories of diferential
equations and of diference equations, but also it is able to
extend these classical cases to cases “in between,” for example,
to so-called -diference equations. Also note that, when T =
R,(1) is reduced to the nonlinear delay diferential equation
()=−()((−)) (2)
and when T = Z, it becomes a nonlinear delay diference
equation
Δ()=−()((−)). (3)
In the case of quantum calculus which defned as T =
N
:=
{
:∈ N}, >1 is a real number, (1) leads to the nonlinear
delay -diference equation
Δ
()=−()((()))Δ
(), (4)
where Δ
()=(()−())/(−1).
Motivated by the papers [1, 2], in this paper we study the
qualitative properties of solution of nonlinear delay dynamic
equation (1) by means of fxed point theory. Te results of this
paper unify the results given by [1] for (2) and by [2] for (3).
Moreover, we obtain new results for the -diference equation
(4) and explicitly provide an example in which we show how
our conditions can be applied. Our technique in proving the
results naturally has some common features with the ones
employed in both [1] and [2] but it is actually quite diferent
due to difculties that are peculiar to the time scale calculus.
Also, our results may be considered as generalization of the
ones obtained in [3, 4] and [5] in which the authors studied
the stability of the delay dynamic equation
Δ
()=−()(())
Δ
(). (5)
In [6], the authors establish some sufcient conditions for the
uniform stability and the uniformly asymptotical stability of
the frst order delay dynamic equation
Δ
()=()(−()). (6)