×

Lyapunov-Razumikhin method for impulsive functional differential equations and applications to the population dynamics. (English) Zbl 1022.34070

The authors consider the following system of functional-differential equations \[ \dot{x}(t)=f(t,x_t),\quad t>t_0,\;t\neq t_k;\qquad x(t_k+0)-x(t_k-0)=I_k(x(t_k-0)),\quad t_k>t_0. \] They obtain sufficient conditions for stability, uniform stability and asymptotic stability. As an application they investigate an impulsive delay logistic equation. Stability conditions obtained for this equation are not explicit.

MSC:

34K20 Stability theory of functional-differential equations
92D25 Population dynamics (general)
34K45 Functional-differential equations with impulses
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bainov, D. D.; Covachev, V.; Stamova, I. M., Estimates of the solutions of impulsive quasilinear functional differential equations, Ann. Fac. Sci. Toulouse, 2, 149-161 (1991) · Zbl 0749.34039
[2] Bainov, D. D.; Kulev, G. K.; Stamova, I. M., Global stability of the solutions of impulsive differential-difference equations, SUT J. Math., 31, 55-71 (1995) · Zbl 0833.34070
[3] Bainov, D. D.; Stamova, I. M., Uniform asymptotic stability of impulsive differential-difference equations of neutral type by Lyapunov’s direct method, J. Comput. Appl. Math., 62, 359-369 (1995) · Zbl 0848.34057
[4] Bainov, D. D.; 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, 272-288 (1996) · Zbl 0848.34058
[5] Bainov, D. D.; Stamova, I. M., Second method of Lyapunov and existence of periodic solutions of linear impulsive differential-difference equations, PanAmer. Math. J., 7, 27-35 (1997) · Zbl 0870.34008
[6] Hale, J. K., Theory of Functional Differential Equations (1977), Springer: Springer New York · Zbl 0425.34048
[7] Hale, J. K.; Lunel, V., Introduction to Functional Differential Equations (1993), Springer: Springer Berlin · Zbl 0787.34002
[8] Kolmanovskii, V. B.; Nosov, V. R., Stability of Functional Differential Equations (1986), Academic Press: Academic Press New York · Zbl 0593.34070
[9] Lakshmikantham, V.; Leela, S.; Martynyuk, A. A., Stability Analysis of Nonlinear Systems (1989), Marcel Dekker: Marcel Dekker New York · Zbl 0676.34003
[10] B.S. Razumikhin, Stability of Hereditary Systems, Nauka, Moscow, 1988 (in Russian).; B.S. Razumikhin, Stability of Hereditary Systems, Nauka, Moscow, 1988 (in Russian).
[11] Simeonov, P. S.; Bainov, D. D., Stability with respect to part of the variables in systems with impulse effect, J. Math. Anal. Appl., 117, 247-263 (1986) · Zbl 0588.34044
[12] Zhang, B. G.; Gopalsamy, K., Oscillation and nonoscillation in a nonautonomous delay logistic equation, Quart. Appl. Math., 46, 267-273 (1988) · Zbl 0648.34078
[13] Zhang, B. G.; Gopalsamy, K., Global attractivity in the delay logistic equation with variable parameter, Math. Proc. Cambridge Philos. Soc., 107, 579-590 (1990) · Zbl 0708.34069
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.