Hostname: page-component-7479d7b7d-68ccn Total loading time: 0 Render date: 2024-07-15T17:28:15.387Z Has data issue: false hasContentIssue false
  • English
  • Français

BOUNDEDNESS OF IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS WITH VARIABLE IMPULSIVE PERTURBATIONS

Part of: Functional-differential and differential-difference equations

Published online by Cambridge University Press:  01 April 2008

I. M. Stamova
I. M. Stamova*
Affiliation:
Bourgas Free University, 8000 Burgas, Bulgaria (email: stamova@bfu.bg)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In the present paper an initial value problem for impulsive functional differential equations with variable impulsive perturbations is considered. By means of piecewise continuous functions coupled with the Razumikhin technique, sufficient conditions for boundedness of solutions of such equations are found.

Keywords

boundednessimpulsive functional differential equationsvariable impulsive perturbations

MSC classification

Secondary: 34K45: Equations with impulses
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 77 , Issue 2 , April 2008 , pp. 331 - 345
Copyright
Copyright © 2008 Australian Mathematical Society

References

[1]Anokhin, A. V., ‘On linear impulsive systems for functional differential equations’, Soviet Math. Dokl. 33 (1986), 220–223.Google Scholar
[2]Bainov, D. D. and Dishliev, A. B., ‘The phenomenon “beating” of the solutions of impulsive functional differential equations’, Commun. Appl. Anal. 1 (1997), 435–441.Google Scholar
[3]Bainov, D. D. and Simeonov, P. S., Systems with impulse effect: stability, theory and applications (Ellis Horwood, Chichester, 1989).Google Scholar
[4]Bainov, D. D. and Simeonov, P. S., Theory of impulsive differential equations: periodic solutions and applications (Longman, Harlow, 1993).Google Scholar
[5]Bainov, D. D. and Stamova, I. M., ‘Lypschitz stability of impulsive functional differential equations’, ANZIAM J. 42 (2001), 504–515.CrossRefGoogle Scholar
[6]Bainov, D. D. and Stamova, I. M., ‘Global stability of the solutions of impulsive functional differential equations’, Kyungpook Math. J. 39 (1999), 239–249.Google Scholar
[7]Chen, M.-P., Yu, J. S. and Shen, J. H., ‘The persistence of nonoscillatory solutions of delay differential equations under impulsive perturbations’, Comput. Math. Appl. 27 (1994), 1–6.CrossRefGoogle Scholar
[8]Xilin, F. and Liqin, Z., ‘On boundedness of solutions of impulsive integro-differential systems with fixed moments of impulse effects’, Acta Math. Sci. 17 (1997), 219–229.Google Scholar
[9]Hale, J. K., Theory of functional differential equations (Springer, New York, 1977).CrossRefGoogle Scholar
[10]Hale, J. K. and Lunel, V., Introduction to functional differential equations (Springer, Berlin, 1993).CrossRefGoogle Scholar
[11]Lakshmikantham, V., Bainov, D. D. and Simeonov, P. S., Theory of impulsive differential equations (World Scientific, Singapore, 1989).CrossRefGoogle Scholar
[12]Lakshmikantham, V., Leela, S. and Martynyuk, A. A., Practical stability analysis of nonlinear systems (World Scientific, Singapore, 1990).CrossRefGoogle Scholar
[13]Luo, Z. and Shen, J., ‘Stability and boundedness for impulsive functional differential equations with infinite delays’, Nonlinear Anal. 46 (2001), 475–493.CrossRefGoogle Scholar
[14]Razumikhin, B. S., Stability of systems with retardation (Nauka, Moscow, 1988) (in Russian).Google Scholar
[15]Samoilenko, A. M. and Perestyuk, N. A., Differential equations with impulse effect (Visca Skola, Kiev, 1987) (in Russian).Google Scholar
[16]Shen, J. and Yan, J., ‘Razumikhin type stability theorems for impulsive functional differential equations’, Nonlinear Anal. 33 (1998), 519–531.CrossRefGoogle Scholar
[17]Stamova, I. M. and Stamov, G. T., ‘Lyapunov–Razumikhin method for impulsive functional differential equations and applications to the population dynamics’, J. Comput. Appl. Math. 130 (2001), 163–171.CrossRefGoogle Scholar
[18]Yan, J. and Shen, J., ‘Impulsive stabilization of impulsive functional differential equations by Lyapunov–Razumikhin functions’, Nonlinear Anal. 37 (1999), 245–255.CrossRefGoogle Scholar