Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 32119
ψ-exponential Stability for Non-linear Impulsive Differential Equations

Authors: Bhanu Gupta, Sanjay K. Srivastava


In this paper, we shall present sufficient conditions for the ψ-exponential stability of a class of nonlinear impulsive differential equations. We use the Lyapunov method with functions that are not necessarily differentiable. In the last section, we give some examples to support our theoretical results.

Keywords: Exponential stability, globally exponential stability, impulsive differential equations, Lyapunov function, ψ-stability.

Digital Object Identifier (DOI):

Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 3701


[1] A. Dimandescu; On the ¤ê-stability of nonlinear voltera integrodifferential systems, Electronic Journal of differential equations, 56: 1-14 (2005).
[2] B. Liu, X. Z. Liu, K. Teo, Q. Wang, Razumikhin-type theorems on exponential stability of impulsive delay systems, IMA J. Appl. Math. 71: 47 - 61 (2006).
[3] D. D. Bainov , P. S. Simeonov , Systems with impulse effect : Stability, Theory and Applications, Ellis Horwood, Chichester, UK, 1989.
[4] I. M. Stamova, G. T. Stamov, LyapunovRazumikhin method for impulsive functional equations and applications to the population dynamics, J. Comput. Appl. Math. 130: 163-171 (2001).
[5] J. Shen, J. Yan, Razumikhin type stability theorems for impulsive functional differential equations, Nonlinear Anal. 33: 519-531 (1998).
[6] J. Morchalo; On (¤ê − Lp)-stability of nonlinear systems of differential equations, Analele Stiintifice ale Universitatii Al. I. Cuza Iasi, Tomul XXXVI, s. I-a, Matematica, f.4: 353-360 (1990).
[7] L. Berezansky, L. Idels, Exponential stability of some scalar impulsive delay differential equation, Commun. Appl. Math. Anal. 2: 301-309 (1998).
[8] N. M. Linh and V. N. Phat, Exponential stability of nonlinear time-varying differential equations and applications, Electronic Journal of differential equations 34: 1-13(2001).
[9] O. Akinyele; On partial stability and boundedness of degree k, Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur.,(8), 65: 259-264(1978).
[10] Q. Wang, X. Z. Liu, Exponential stability for impulsive delay differential equations by Razumikhin method, J. Math. Anal. Appl. 309: 462-473 (2005).
[11] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov,"Theory of Impulsive Differential Equations." World Scientific, Singepore / New Jersey /London , 1989.
[12] X. Z. Liu, G. Ballinger, Uniform asymptotic stability of impulsive delay differential equations, Comput. Math. Appl. 41: 903-915 (2001).