 
Summary: DELAY DYNAMIC EQUATIONS WITH STABILITY
DOUGLAS R. ANDERSON, ROBERT J. KRUEGER, AND ALLAN C. PETERSON
Received 13 August 2005; Accepted 23 October 2005
We first give conditions which guarantee that every solution of a first order linear delay
dynamic equation for isolated time scales vanishes at infinity. Several interesting examples
are given. In the last half of the paper, we give conditions under which the trivial solution
of a nonlinear delay dynamic equation is asymptotically stable, for arbitrary time scales.
Copyright © 2006 Douglas R. Anderson et al. This is an open access article distributed
under the Creative Commons Attribution License, which permits unrestricted use, dis
tribution, and reproduction in any medium, provided the original work is properly cited.
1. Preliminaries
The unification and extension of continuous calculus, discrete calculus, qcalculus, and
indeed arbitrary realnumber calculus to timescale calculus, where a time scale is sim
ply any nonempty closed set of real numbers, were first accomplished by Hilger in [4].
Since then, timescale calculus has made steady inroads in explaining the interconnec
tions that exist among the various calculuses, and in extending our understanding to a
new, more general and overarching theory. The purpose of this work is to illustrate this
new understanding by extending some continuous and discrete delay equations to cer
tain time scales. Examples will include specific cases in differential equations, difference
equations, qdifference equations, and harmonicnumber equations. The definitions that
