## Linearized oscillation theory for a nonlinear delay impulsive equation.(English)Zbl 1045.34039

The authors provide sufficient conditions for the oscillatory and nonoscillatory of the solutions of the following impulsive delay nonlinear differential equation $\dot x(t)+\sum_{k=1}^{m}r_{k}(t)f_{k}[x(h_{k}(t))]=0, \quad t\not=\tau_{j},\quad x(\tau_{j})=I_{j}x(\tau_{j}^{-})), j=1,2,\dots,$ where $$r_{k}(t)\geq 0$$, $$h_{k}(t)\leq t$$, $$\lim_{j\to\infty}\tau_{j}=\infty$$. It is shown that oscillatory and nonoscillatory properties of the solutions of the above nonlinear equation are deduced from the following auxiliary linear impulsive differential equation $\dot x(t)+\sum_{k=1}^{m}r_{k}(t)a_{k}(t)x(h_{k}(t))=0, \quad t\not=\tau_{j},\quad x(\tau_{j})=b_{j}x(\tau_{j}^{-}),\quad j=1,2,\dots\;.$ Some applications for impulsive models of mathematical biology and numerical simulations for impulsive logistic equations are presented. The results of the present paper continue recent ones of the authors on the linearized theory for nonimpulsive delay equations in [J. Comput. Appl. Math. 151, No. 1, 119–127 (2003; Zbl 1025.34069)].

### MSC:

 34K11 Oscillation theory of functional-differential equations 34K45 Functional-differential equations with impulses

### Keywords:

oscillation; delay impulsive equations; linearization

Zbl 1025.34069
Full Text:

### References:

 [1] Arino, O.; Györi, I., Qualitative properties of the solutions of a delay differential equation with impulses. II. oscillations, Differential equations dyn. systems, 7, 2, 161-179, (1999) · Zbl 0989.34054 [2] Berezansky, L.; Braverman, E., Oscillations of a linear delay impulsive differential equation, Comm. appl. nonlinear anal., 3, 1, 61-77, (1996) · Zbl 0858.34056 [3] Berezansky, L.; Braverman, E., On oscillation of a generalized logistic equation with several delays, J. math. anal. appl., 253, 389-405, (2001) · Zbl 1024.34059 [4] Berezansky, L.; Braverman, E., On oscillation of an impulsive logistic equation, Dynamics of continuous discrete impulsive systems, 9, 3, 377-396, (2002) · Zbl 1015.34055 [5] Berezansky, L.; Braverman, E., Linearized oscillation theory for a nonlinear nonautonomous delay differential equation, J. comput. appl. math., 151, 119-127, (2003) · Zbl 1025.34069 [6] Duan, Y.; Feng, W.; Yan, J., Linearized oscillation of nonlinear impulsive delay differential equations, Comput. math. appl., 44, 1267-1274, (2002) · Zbl 1045.34041 [7] Erbe, L.N.; Kong, Q.; Zhang, B.G., Oscillation theory for functional differential equations, (1995), Marcel Dekker New York, Basel [8] Fishman, S.; Marcus, R., A model for spread of plant disease with periodic removals, J. math. biol., 21, 2, 149-158, (1984) · Zbl 0548.92015 [9] Gopalsamy, K., Stability and oscillation in delay differential equations of population dynamics, (1992), Kluwer Academic Publishers Dordrecht, Boston, London · Zbl 0752.34039 [10] Györi, I.; Ladas, G., Oscillation theory of delay differential equations, (1991), Clarendon Press Oxford · Zbl 0780.34048 [11] Kocic, V.L.; Ladas, G.; Qian, C., Linearized oscillations in nonautonomous delay differential equations, Differential integral equations, 6, 671-683, (1993) · Zbl 0779.34057 [12] Kuang, Y.; Zhang, B.G.; Zhao, T., Qualitative analysis of a nonautonomous nonlinear delay differential equations, Tohoku math. J., 43, 509-528, (1991) · Zbl 0727.34063 [13] Ladas, G.; Qian, C., Linearized oscillations for odd-order neutral delay differential equations, J. differential equations, 88, 238-247, (1990) · Zbl 0717.34074 [14] Ladde, G.S.; Lackshmikantham, V.; Zhang, B.G., Oscillation theory of delay differential equations with deviating arguments, (1987), Marcel Dekker New York, Basel · Zbl 0622.34071 [15] Li, B.; Kuang, Y., Sharp conditions for oscillations in some nonlinear nonautonomous delay equations, Nonlinear anal., 29, 1265-1276, (1997) · Zbl 0887.34068 [16] Liu, X.; Ballinger, G., Uniform asymptotic stability of impulsive delay differential equations, Comput. math. appl., 41, 7-8, 903-915, (2001) · Zbl 0989.34061 [17] Luo, Z.; Shen, J., Oscillation for solutions of nonlinear neutral differential equations with impulses, Comput. math. appl., 42, 10-11, 1285-1292, (2001) · Zbl 1005.34060 [18] d’Onofrio, A., Stability properties of pulse vaccination strategy in SEIR epidemic models, Math. biosci., 179, 57-72, (2002) · Zbl 0991.92025 [19] Peng, M.; Ge, W., Oscillation criteria for second order nonlinear differential equations with impulses, Comput. math. appl., 39, 5-6, 217-225, (2000) · Zbl 0948.34044 [20] Sficas, Y.G.; Staikos, V.A., The effect of retarded actions on nonlinear oscillations, Proc. amer. math. soc., 46, 259-264, (1974) · Zbl 0263.34075 [21] Shen, J.H.; Yu, J.S.; Qian, X.Z., A linearized oscillation result for odd-order neutral delay differential equations, J. math. anal. appl., 186, 365-374, (1994) · Zbl 0814.34060 [22] Wang, Q., Oscillation theorems for first order nonlinear neutral functional differential equations, Comput. math. appl., 39, 19-28, (2000) · Zbl 0954.34058 [23] Yu, J., Oscillation of nonlinear delay impulsive differential equations and inequalities, J. math. anal. appl., 265, 332-342, (2002) · Zbl 1001.34059 [24] Yu, J.; Yan, J., Positive solutions and asymptotic behavior of delay differential equations with nonlinear impulses, J. math. anal. appl., 207, 388-396, (1997) · Zbl 0877.34054
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.