Perron’s theorem for linear impulsive differential equations with distributed delay. (English) Zbl 1101.34065

Akhmet, M. U.; Alzabut, J.; Zafer, A.

J. Comput. Appl. Math. 193, No. 1, 204-218 (2006).

The authors show that under a Perron condition, the trivial solution of a linear impulsive differential equation with distributed delay is uniformly asymptotically stable.

Reviewer: Gani T. Stamov (Sliven)

MSC:

34K45	Functional-differential equations with impulses
34K20	Stability theory of functional-differential equations
34K06	Linear functional-differential equations

Keywords:

Perron condition; Stability; Impulse; Distributed delay

References:

[1]	Angelova, J.; Dishliev, A.; Nenov, S., I-optimal curve for impulsive lotka – volterra predator – prey model, Comput. math. applic., 43, 10-11, 1203-1218, (2002) · Zbl 1007.34008
[2]	Anokhin, A.; Berezansky, L.; Braverman, E., Exponential stability of linear delay impulsive differential equations, J. math. anal. appl., 193, 923-941, (1995) · Zbl 0837.34076
[3]	Azbelev, N.V.; Berezanskii, L.M.; Simonov, P.M.; Chistyakov, A.V., Stability of linear systems with time lag, Differential equations, 23, 493-500, (1987) · Zbl 0652.34079
[4]	Bellman, R.; Cooke, K.L., Differential-difference equations, (1963), Academic Press Inc. New York · Zbl 0115.30102
[5]	Berezansky, L.; Braverman, E., Boundedness and stability of impulsively perturbed systems in a Banach space, Int. J. theoret. phys., 33, 2075-2091, (1994) · Zbl 0814.34047
[6]	Bohl, P., Riene angew math., 144, 284-318, (1913)
[7]	Gusarenko, S.A.; Domoshnitskii, A.I., Asymptotic and oscillational properties of first order linear scalar functional differential equations, Differential equations, 25, 1480-1491, (1989) · Zbl 0726.45011
[8]	Halanay, A., Differential equationsstability, oscillations, time lags, (1966), Academic Press Inc. New York
[9]	Halanay, A.; Wexler, D., Qualitative theory of impulsive systems, (1968), Editura Academiei Republicii Socialiste Romania Bucharest
[10]	Hale, J.K.; Lunel, R.M.V., Introduction to functional differential equations, (1966), Springer New York
[11]	Kreyszig, E., Introductory functional analysis with applications, (1989), Wiley New York · Zbl 0706.46001
[12]	Liu, X.; Ballinger, G., Boundedness for impulsive delay differential equations and applications to population growth models, Nonlinear anal., 53, 7-8, 1041-1062, (2003) · Zbl 1037.34061
[13]	Nieto, J.J., Impulsive resonance periodic problems of first order, Appl. math. lett., 15, 4, 489-493, (2002) · Zbl 1022.34025
[14]	Perron, O., Die stabilitatsfrage bei differentialgleichungen, Math. Z., 32, 703-728, (1930) · JFM 56.1040.01
[15]	Samoilenko, A.M.; Perestyuk, N.A., Impulsive differential equations, (1995), World Scientific Singapore · Zbl 0837.34003
[16]	Sun, J.; Zhang, Y.; Wu, Q., Less conservative conditions for asymptotic stability of impulsive control systems, IEEE trans. automat. control, 48, 5, 829-831, (2003) · Zbl 1364.93691
[17]	Tang, S.; Chen, L., Global attractivity in a food-limited population model with impulsive effects, J. math. anal. appl., 292, 1, 211-221, (2004) · Zbl 1062.34055
[18]	Tian, Y.P.; Yu, X.; Chua, O.L., Time-delayed impulsive control of chaotic hybrid systems, Internat. J. bifur. chaos appl. sci. eng., 14, 3, 1091-1104, (2004) · Zbl 1129.93515
[19]	Tyshkevich, V.A., A perturbations – accumulation problem for linear differential equation with time lag, Differential equations, 14, 177-186, (1978) · Zbl 0409.34073
[20]	Zhang, S.; Dong, L.; Chen, L., The study of predator – prey system with defensive ability of prey and impulsive perturbations on the predator, Chaos solitons fractals, 23, 2, 631-643, (2005) · Zbl 1081.34041