DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

BOUNDEDNESS OF SOLUTIONS OF FUNCTIONAL
DIFFERENTIAL EQUATIONS WITH
STATE-DEPENDENT IMPULSES
XINZHI LIU AND QING WANG
This paper studies the boundedness of functional differential equations with state-dependent
impulses. Razumikhin-type boundedness criteria are obtained by using Lyapunov
functions and Lyapunov functionals. Some examples are also given to illustrate
the effectiveness of our results.
Copyright © 2006 X. Liu and Q. Wang. 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. Introduction
Impulsive differential equations have attracted lots of interest in recent years due to their
important applications in many areas such as aircraft control, drug administration, and
threshold theory in biology [2, 3, 5, 7]. There has been a significant development in the
theory of impulsive differential equations in the past decade, especially in the area where
impulses are fixed. However, the corresponding theory of impulsive functional differential
has been less developed because of numerous theoretical and technical difficulties. Recently,
the existence and continuability results of solutions for differential equations with
delays and state-dependent impulses have been presented in [1, 4], while some stability
results of nontrivial solutions of delay differential equations with state-dependent impulses
have been stated in [6]. In this paper, we will establish some boundedness criteria
for the functional differential equations with state-dependent impulses. Some examples
are also discussed to illustrate the effectiveness of our results.
2. Preliminaries
Let R denote the set of real numbers, R + the set of nonnegative real numbers, and R n the
n-dimensional Euclidean linear space equipped with the Euclidean norm ‖·‖.
For a,b ∈ R with a