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Oscillatory and asymptotic behavior of solutions for nonlinear impulsive delay differential equations. (English) Zbl 1110.34058
The oscillatory and asymptotic behavior of bounded solutions for the third-order nonlinear differential equation $$x'''(t)=f(t,x(t-\tau))$$ are studied. Criteria for all solutions to be oscillatory or be asymptotic to the piecewise continuous functions are established. Few examples are also given to illustrate the effectiveness of these criteria.

34K45Functional-differential equations with impulses
34K11Oscillation theory of functional-differential equations
34K25Asymptotic theory of functional-differential equations
Full Text: DOI
[1] Bainov, D.D., Simeonov, P.S. Systems with impulse effect, stability, theory and applications. Ellis Horwood Publishers, Chichester, 1989 · Zbl 0676.34035
[2] Bainov, D.D., Simeonov, P.S. Theory of impulsive differential equations: asymptotic properties of the solutions and applications. World Scientific Publishers, Singapore, 1995 · Zbl 0828.34002
[3] Berezansky, L., Braverman, E. On oscillation of second order impulsive linear delay differential equation. Journal of Mathematical Analysis and Applications, 133: 276--300 (1998) · Zbl 0926.34054
[4] Chen, Y.S., Feng, W.Z. Oscillations of second order nonlinear ode with impulses. Journal of Mathematical Analysis and Applications, 210: 150--169 (1997) · Zbl 0877.34014 · doi:10.1006/jmaa.1997.5378
[5] Gopalsamy, K. Stability and oscillation in delay differential equations of population dynamics. Kluwer Academic Publishers Dordrecht, Boston, London, 1992 · Zbl 0752.34039
[6] Gopalsamy, K., Zhang, B.G. On Delay Differential Equations with Impulses. Journal of Mathematical Analysis and Applications, 139(1): 110--122 (1989) · Zbl 0687.34065 · doi:10.1016/0022-247X(89)90232-1
[7] Lakshmikantham, V., Bainov, D.D., Simeonov, P.S. Theory of impulsive differential equations. World Scientific Publishers, Singapore, 1989 · Zbl 0719.34002
[8] Wen, L.Z., Chen, Y.S. Razumikhin type theorems for functional differential equations with impulses. Dynamics of Continuous, Discrete and Impulsive Systems, 6: 389--400 (1999) · Zbl 0936.34070
[9] Xu, W.J. Oscillation and Asymptotic Behavior of Third order Impulsive Differential Equation. Journal of South China Normal University (Natural Science Edition), 2: 59--64 (2001) · Zbl 0984.34060
[10] Yan, J.R., Zhao, A.M. Oscillation and stability of linear impulsive delay differential equation. Journal of Mathematical Analysis and Applications, 277: 187--194 (1998) · Zbl 0917.34060 · doi:10.1006/jmaa.1998.6093
[11] Yu, J., Yan, J.R. Positive solutions and asymptotic behavior of delay differential equations with nonlinear impulses. Journal of Mathematical Analysis and Applications, 207: 388--396 (1997) · Zbl 0877.34054 · doi:10.1006/jmaa.1997.5276