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Vector Lyapunov functions and conditional stability for systems of impulsive differential-difference equations

Published online by Cambridge University Press:  17 February 2009

D. D. Bainov and
I. M. Stamova
D. D. Bainov
Affiliation:
Medical University of Sofia, P. O. Box 45, Sofia - 1504, Bulgaria.
I. M. Stamova
Affiliation:
Technical University of Sliven, Sliven, Bulgaria.
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Abstract

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By means of piecewise continuous vector functions, which are analogues of the classical Lyapunov functions and via the comparison method, sufficient conditions are found for conditional, stability of the zero solution of a system of impulsive differential-difference equations.

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 42 , Issue 3 , January 2001, pp. 341 - 353
Copyright
Copyright © Australian Mathematical Society 2001

References

[1] Bainov, D. D., Minchev, E. and Stamova, I. M., “Present state of the stability theory for impulsive differential equations”. Commun. Appl. Anal. 2 (1998) 197–226.Google Scholar
[2] Bainov, D. D. and Simeonov, P. S., Systems with impulse effect: stability, theory and applications (Ellis Horwood, Chichester, 1989).Google Scholar
[3] Bainov, D. D. and Simeonov, P. S., Theory of impulsive differential equations: asymptotic properties of the solutions (World Scientific, Singapore, 1995).CrossRefGoogle Scholar
[4] Bainov, D. D. and Stamova, I. M., “On the practical stability of the solutions of impulsive systems of differential-difference equations with variable impulsive perturbations”, J. Math. Anal. Appl. 200 (1996) 272–288.Google Scholar
[5] Bainov, D. D. and Stamova, I. M., “Global stability of the solutions of impulsive differential-difference equations with variable impulsive perturbations”, COMPEL, Internal. J. Comput. Math. in Electrical and Electronic Eng. 1 (1997) 3–16.Google Scholar
[6] Bainov, D. D. and Stamova, I. M., “Second method of Lyapunov and comparison principle for impulsive differential-difference equations”, J. Austral Math. Soc. Ser. B 38 (1997) 489–505.CrossRefGoogle Scholar
[7] Bainov, D. D. and Stamova, I. M., “Stability of sets for impulsive differential-difference equations with variable impulsive perturbations”, Commun. Appl. Nonlinear Anal. 5 (1998) 69–81.Google Scholar
[8] Cunningham, W. J., “A nonlinear differential-difference equation of growth”, Proc. National Acad. Sci. USA 40 (1954) 708–713.CrossRefGoogle ScholarPubMed
[9] Gopalsamy, K., Stability and oscillation in delay differential equations of population dynamics (Kluwer, Boston, 1992).CrossRefGoogle Scholar
[10] Hutchinson, G. E., “Circular causal systems in ecology”, Ann. New York Acad. Sci. 50 (1948) 221–246.CrossRefGoogle ScholarPubMed
[11] Kuang, Y., Delay differential equations with applications in population dynamics (Academic Press, New York, 1993).Google Scholar
[12] Kulev, G. K., “Vector functions of Lyapunov and conditional stability of systems with impulse effect”, Univ. de Plovdiv, Travaux scientifiques 25 (1987) 47–62.Google Scholar
[13] Lakshmikantham, V., Bainov, D. D. and Simeonov, P. S., Theory of impulsive differential equations (World Scientific, Singapore, 1989).CrossRefGoogle Scholar
[14] Lakshmikantham, V., Leela, S. and Martynyuk, A. A., Stability analysis of nonlinear systems (Marcel Dekker, New York, 1989).Google Scholar
[15] Lakshmikantham, V., Matrosov, V. M. and Sivasundaram, S., Vector functions and stability analysis of nonlinear systems (Kluwer, The Netherlands, 1991).CrossRefGoogle Scholar
[16] Razumikhin, B. S., Stability of systems with retardation, (in Russian) (Nauka, Moscow, 1988).Google Scholar
[17] Simeonov, P. S. and Bainov, D. D., “Stability with respect to part of the variables in systems with impulse effect”, J. Math. Anal. Appl. 117 (1986) 247–263.CrossRefGoogle Scholar
[18] Zhang, B. G. and Gopalsamy, K., “Oscillation and nonoscillation in a nonautonomous delay logistic equation”, Quart. Appl. Math. 46 (1988) 267–273.CrossRefGoogle Scholar
[19] Zhang, B. G. and Gopalsamy, K., “Global attractivity in the delay logistic equation with variable parameter”, Math. Proc. Cambridge Philos. Soc. 107 (1990) 579–590.Google Scholar