Summary: BOUNDEDNESS AND VANISHING OF SOLUTIONS FOR
A FORCED DELAY DYNAMIC EQUATION
DOUGLAS R. ANDERSON
Received 30 March 2006; Revised 10 July 2006; Accepted 14 July 2006
We give conditions under which all solutions of a time-scale first-order nonlinear vari-
able-delay dynamic equation with forcing term are bounded and vanish at infinity, for
arbitrary time scales that are unbounded above. A nontrivial example illustrating an ap-
plication of the results is provided.
Copyright © 2006 Douglas R. Anderson. This 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.
1. Delay dynamic equation with forcing term
Following Hilger's landmark paper , a rapidly expanding body of literature has sought
to unify, extend, and generalize ideas from discrete calculus, quantum calculus, and con-
tinuous calculus to arbitrary time-scale calculus, where a time scale is simply any non-
empty closed set of real numbers. This paper illustrates this new understanding by ex-
tending some continuous results from differential equations to dynamic equations on
time scales, thus including as corollaries difference equations and q-difference equations.
Throughout this work, we consider the nonlinear forced delay dynamic equation