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Existence and global attractivity of positive periodic solutions of functional differential equations with impulses. (English) Zbl 1061.34059
Existence and global attractivity of positive periodic solutions are obtained for the following nonlinear delay differential equation with impulses $$ \dot{y}(t)=y(t)F(t, y(t-\tau_1(t)),\dots,y(t-\tau_n(t))),\quad y(t_k^{+})=b_ky(t_k),\quad k=1,2\dots. $$ Applications to logistic equations with several delays are presented. For the proof the authors apply coincidence degree theory and the method of Lyapunov functionals.

34K45Functional-differential equations with impulses
34K13Periodic solutions of functional differential equations
34K20Stability theory of functional-differential equations
34K60Qualitative investigation and simulation of models
Full Text: DOI
[1] Barbalat, I.: Systems d’equations differentielle d’oscillations nonlineaires. Rev. roumaine math. Pures appl. 4, 267-270 (1959)
[2] Gaines, R. E.; Mawhin, J. L.: Coincidence degree and nonlinear differential equations. (1977) · Zbl 0339.47031
[3] Gopalsamy, K.: Stability and oscillations in delay differential equations of population dynamics. (1992) · Zbl 0752.34039
[4] Gopalsamy, K.; Kulenović, M. R. S.; Ladas, G.: Enrivonmental periodicity and time delays in a ”food-limited” population model. J. math. Anal. appl. 147, 545-555 (1990) · Zbl 0701.92021
[5] Gopalsamy, K.; Lalli, B. S.: Oscillatory and asymptotic behavior of a multiplicative delay logistic equation. Dynamics stability systems 7, 35-42 (1992) · Zbl 0764.34049
[6] Gopalsamy, K.; Zhang, B. G.: On the delay differential equations with impulses. J. math. Anal. appl. 139, 110-112 (1989) · Zbl 0687.34065
[7] Grace, S. R.; Györi, I.; Lalli, B. S.: Necessary and sufficient conditions for the oscillations of a multiplicative delay logistic equation. Quart. appl. Math. 53, 69-79 (1995) · Zbl 0837.34073
[8] Greaf, J. R.; Qian, C.; Spikes, P. W.: Oscillation and global attractivity in a periodic delay equation. Canad. math. Bull. 38, 275-283 (1996) · Zbl 0870.34073
[9] Györi, I.; Ladas, G.: Oscillation theorem of delay differential equations with applications. (1991) · Zbl 0780.34048
[10] H.F. Huo, W.T. Li, X. Liu, Existence and global attractivity of positive periodic solution of an impulsive delay differential equation, Appl. Anal., in press. · Zbl 1065.34068
[11] Krasnoselskii, M. A.: Positive solutions of operator equations. (1964)
[12] Kuang, Y.: Delay differential equations with application in population dynamics. (1993) · Zbl 0777.34002
[13] Lalli, B. S.; Zhang, B. G.: On a periodic delay population model. Quart. appl. Math. 52, 35-42 (1994) · Zbl 0788.92022
[14] Li, Y. K.: Existence and global attractivity of a positive periodic solution of a class of delay differential equation. Sci. China ser. A 41, 273-284 (1998) · Zbl 0955.34057
[15] Nicholson, A. J.: The balance of animal population. J. animal ecol. 2, 132-178 (1933)
[16] Nisbet, R. M.; Gurney, W. S. C.: Population dynamics in a periodically varying environment. J. theor. Biol. 56, 459-475 (1976)
[17] Pianka, E. R.: Evolutionary ecology. (1974)
[18] Rosen, G.: Time delays produced by essential nonlinearity in population growth models. Bull. math. Biol. 28, 253-256 (1987) · Zbl 0614.92015
[19] S. H. Saker, S. Agarwal, Oscillation and global attractivity of a periodic survival red blood cells model, Comput. Math. Appl., in press. · Zbl 1078.34062
[20] Saker, S. H.; Agarwal, S.: Oscillation and global attractivity in a periodic nichlson’s blowflies model. Math. comp. Modelling 35, 719-731 (2002) · Zbl 1012.34067
[21] Shen, J. H.: Global existence and uniqueness, oscillation and nonoscillation of impulsive delay differential equations. Acta math. Sinica 40, No. 1, 53-59 (1997) · Zbl 0881.34074
[22] Yan, J. R.: Existence and global attractivity of positive periodic solution for an impulsive lasota -- wazewska model. J. math. Anal. appl. 279, 111-120 (2003) · Zbl 1032.34077
[23] Yan, J.; Feng, Q.: Global attractivity and oscillation in a nonlinear delay equation. Nonlinear anal. TMA 43, 101-108 (2001) · Zbl 0987.34065
[24] Yan, J. R.; Zhao, A. M.: Oscillation and stability of linear impulsive delay differential equations. J. math. Anal. appl. 227, 187-194 (1998) · Zbl 0917.34060
[25] Zhang, B. G.; Gopalsamy, K.: Global attractivity and oscillations in a periodic delay-logistic equation. J. math. Anal. appl. 150, 274-283 (1990) · Zbl 0711.34090
[26] Zhang, B. G.; Liu, Y. J.: Global attractivity for certain impulsive delay differential equations. Nonlinear anal. 35, 1019-1030 (1999)
[27] Zhang, X. S.; Yan, J. Y.: Global attractivity in impulsive functional differential equation. Indian J. Pure appl. Math. 29, 871-878 (1998) · Zbl 0917.34062