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International Journal of Differential Equations
Volume 2011, Article ID 703189, 13 pages
http://dx.doi.org/10.1155/2011/703189
Research Article

Practical Stability in terms of Two Measures for Impulsive Differential Equations with “Supremum”

Plovdiv University "Paisii Hiledarski", 4000 Plovdiv, Bulgaria

Received 19 May 2011; Accepted 6 August 2011

Academic Editor: Jianshe Yu

Copyright © 2011 S. G. Hristova and A. Georgiev. 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.

Abstract

The object of investigations is a system of impulsive differential equations with “supremum.” These equations are not widely studied yet, and at the same time they are adequate mathematical model of many real world processes in which the present state depends significantly on its maximal value on a past time interval. Practical stability for a nonlinear system of impulsive differential equations with “supremum” is defined and studied. It is applied Razumikhin method with piecewise continuous scalar Lyapunov functions and comparison results for scalar impulsive differential equations. To unify a variety of stability concepts and to offer a general framework for the investigation of the stability theory, the notion of stability in terms of two measures has been applied to both the given system and the comparison scalar equation. An example illustrates the usefulness of the obtained sufficient conditions.