Many evolution processes are characterized by the fact that at certain moments of time they experience a change of state abruptly. These processes are subject to short-term perturbations whose duration are negligible in compari-
son with the duration of the processes. Consequently, it is natural to assume that these perturbations act instantaneously, that is, in the form of impulses.
It is known, for example, that many biological phenomena involving thresholds, bursting rhythm models in medicine and biology, optimal control models in economics, pharmacokinetics and frequency modulated systems, do exhibit
impulsive effects. Thus impulsive differential equations appear as natural descriptions of observed evolution phenomena of several real world problems.

Our major objective is to study the oscillatory properties of impulsive delay differential equations. When we introduce impulsive effects, however, the original oscillation concepts will need to be modified. For example, initial value problems of such equations may not, in general, possess any solutions at all even when the corresponding differential equations are ‘smooth’; fundamental properties such as continuous dependence relative to initial data may be violated,and qualitative properties such as oscillation may need a suitable new
interpretation. Moreover, even a very simple impulsive delay differential equation may exhibit new properties. Therefore, we need to be careful when oscillatory
criteria in for non-impulsive equations are applied to deal with the impulsive ones. Otherwise, we may be led to erroneous results.

In this thesis, we provide necessary and/or sufficient conditions such that first, second and higher order impulsive delay differential equations possess nonoscillatory solutions. In general, it is well known that ”Comparison Theorems” are essential because we can combine them with known oscillatory criteria to establish more oscillatory theories. To this end, we establish comparison theories in first order and second order impulsive delay differential equations. We will be concerned with impulsive delay differential equations with forcing terms. We obtain necessary and sufficient conditions for the existence of nonoscillatory solutions and also a comparison theorem which enables us to apply known oscillation results for impulsive equations without forcing terms to yield oscillation criteria for our equations.

In particular, we may relate the oscillatory properties of impulsive delay differential equations with the absence of real roots of the corresponding characteristic equations, which are quasi-polynomials. The problem of existence of real roots of quasi-polynomials can often be solved by the Cheng-Lin envelope method. So we can apply the Cheng-Lin envelope method and known results
to establish oscillatory criteria.

For higher order impulsive delay differential equation, there are relatively few studies. We provide necessary and/or sufficient conditions for higher order
impulsive delay differential equations to have nonoscillatory solutions.

As illustrations, we can apply our results to study the oscillatory properties of Hutchinson’s equation, pendulum’s equation, Nicholson’s equation,Wright’s equation, Lasota-Wazewska equation, Klein-Gordon equation and Em-
den Fowler equation, etc. Also, we point out some mistakes in published papers and provide some new results or remedies.

[1] R. P. Agarwal, M. Bohner and W. T. Li, Nonoscillation and Oscillation Theory for Functional Differential Equations, Marcel Dekker, 2004.
[2] R. P. Agarwal, S. R. Grace, and D. O’Regan, Oscillation Theory for Second Order Dynamic Equatopns,
Taylor & Francis, 2003.
[3] D. D. Bainov and M. B. Dimitrova, Oscillatory properties of the solutions of impulsive differential
equations with a deviating argument and nonconstant coefficients, Rocky Mountain J. Math., 27 (1997), 1027–1040.
[4] D. D. Bainov, M. B. Dimitrova and A. B. Dishliev, Oscillation of the solutions of impulsive differential equations and inequalities with a retarded argument, Rocky Mountain J. Math., 28(1998), 25–40.
[5] D. D. Bainov, M. B. Dimitrova and A. B. Dishliev, Oscillation of the bounded solutions of impulsive differential-difference equations of second order, Appl. Math. Comput., 114 (2000),61–68.
[6] D. D. Bainov and M. B. Dimitrova, Sufficient conditions for the oscillation of bounded solutions of a class of impulsive differential equations of second order with a constant delay, Georgian Math. J., 6 (1999), 99–106.
[7] L. Berezansky and E. Braverman, Linearized oscillation theory for a nonlinear delay impulsive equation, J. Comput. Appl. Math., 161 (2003), 477–495.
[8] L. Berezansky and E. Braverman, Oscillation of a linear delay impulsive differential equation,
Comm. Appl. Nonlinear Anal., 3 (1996), 61–77.
[9] L. Berezansky and E. Braverman, On oscillation of a second order impulsive linear delay differential
equation, J. Math. Anal. Appl., 233 (1999), 276–300.
[10] V. G. Boltyanskii, Envelopes, Popular Lectures in Mathematics, Vol. 12, Macmillian, New Youk, 1964.
[11] M. P. Chen, J. S. Yu, and J. H. Shen, The persistence of nonoscillatory solutions of delay differential equations under impulsive perturbations, Comput. Math. Appl., 27 (1994), 1–6.
[12] S. S. Cheng and Y. Z. Lin, Dual Sets of Envelopes and Characteristic Regions of Quasi-Polynomials, World Scientific, 2009.
[13] S. S. Cheng and Y. Z. Lin, Exact regions of oscillation for a neutral differential equation, Proc.
Roy. Soc. Edin., 130A (2000), 277–286. 149
[14] S. S. Cheng and Y. Z. Lin, The exact region of oscillation for first order neutral differential
equation with delays, Quart. Appl. Math., 64(3) (2006), 433–445.
[15] Y. G. Duan, W. Feng and J. R. Yan, Linearized oscillation of nonlinear impulsive delay differential
equations, Comput. Math., Appl., 44 (2002), 1267–1274.
[16] Y. G. Duan, W. P. Zhang, P. Tian and J. R. Yan, Generic oscillations of second order delay differential equations with impulses, J. Math. Anal. Appl., 277 (2003), 465–473.
[17] L. P. Eisenhart, A Treatise on the Differential Geometry of Curves and Surfaces, Dover, 2004.
[18] L. P. Gimenes and M. Fedderson, Oscillation by impulses for a second order delay differential
equation, Comput. Math. Appl., 52 (2006), 819–828.
[19] K. Gopalsamy and B. G. Zhang, On delay differential equations with impulses, J. Math. Anal. Appl., 139 (1989), 110–122.
[20] I. Gy¨ori and G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Clarendon Press, Oxford, 1991.
[21] J. K. Hale and S.M. Verduyn, Introduction to Functional Differential Equation, Springer Verlag,
1993.
[22] Z. M. He and W. G. Ge, Oscillation in second order linear delay differential equations with nonlinear impulses, Math. Slovaca, Vol. 52 (2002), No. 3, 331–341.
[23] M. G. Huang and W. Z. Feng, Forced oscillations for second order delay differential equations with impulses, Comput. Math. Appl., 59 (2010), 18–30.
[24] M. G. Huang and W. Z. Feng, Oscillation of second-order impulsive delay differential equations with forcing term, Heilongjiang Daxue Ziran Kexue Xuebao, 23(4) (2006), 452–456.
[25] S. Y. Huang and S. S. Cheng, Eventually positive solutions for nonlinear impulsive differential equations with Delays, accepted to appear in Ann. Polon. Math., (2011)
[26] S. Y. Huang and S. S. Cheng, Comparison theorems for positive increasing solutions of second order differential equations with delays and impulses, preprint.
[27] S. Y. Huang and S. S. Cheng, Existence of eventually positive solutions of higher order impulsive delay differential equations, preprint.
[28] S. Y. Huang and S. S. Cheng, Necessary and sufficient conditions for the existence of nonoscillatory solutions of impulsive delay differential equations with forcing, preprint.
[29] S. Y. Huang and S. S. Cheng, Absence of real roots of characteristic functions of functional differential equations with nine real parameters, Taiwanese J. Math., 15 (2011), 395–432.
[30] S. Y. Huang and S. S. Cheng, Absence of positive roots of sextic polynomials, accepted to appear in Taiwanese J. Math., (2011).
[31] G. Hutchinson, Circular causal systems in ecology, Ann. N. Y. Acad. Sci., 50 (1948), 221–246.
[32] I. O. Isaac and Z. Lipcsey, Linearized oscillations in nonlinear neutral delay impulsive differential
equations, J. Mod. Math., Stat. 3 (1) (2009), 1–7. 150
[33] Y. Jiang and J. R. Yan, Positive solutions and asymptotic behavior of delay differential equations
with nonlinear impulses, J. Math. Anal. Appl., 207 (1997), 388–396.
[34] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.
[35] X. Li, Oscillation properties of higher order impulsive delay differential equations, Int. J. Difference
Equ., 2 (2007), 209–219.
[36] W. D. Luo, J. W. Luo and L. Debnath, Oscillation of second order quasilinear delay differential equations with impulses, J. Appl. Math. & Comput., 13 (2003), 165–182.
[37] J. W. Luo, Second-order quasilinear oscillation with impulses, Comput. Math. Appl., 46 (2003), 279–291.
[38] A. H. Nasr, Necessary and sufficient conditions for the oscillation of forced nonlinear second order differential equations with delayed argument, J. Math. Anal. Appl., 212 (1997), 21–59.
[39] M. S. Peng, Oscillation caused by impulses, J. Math. Anal. Appl., 255 (2001), 163–176.
[40] M. S. Peng and W. G. Ge, Oscillation criteria for second order nonlinear differential equations with impulses, Comput. Math. Appl., 39 (2000) 217–225.
[41] M. S. Peng, W. G. Ge and Q. L. Xu, Observation of nonoscillatory behavior of solutions of second-order delay differential equations under impulsive perturbations, Appl. Math. Lett., 15(2002), 203–210.
[42] G. R¨ost, S. Y. Huang, and L. Sz´ekely, On a SEIR epidemic model with delay, accepted to appear in Dynam. Systems Appl.
[43] F. S. Saryal and A. Zafer, Oscillation of second order nonlinear impulsive delay differenital equations, Dynamics of Continuous, Discrete and Impulsive Systems, 16 (2009), 221–231.
[44] J. H. Shen, The nonoscillatory solutions of delay differential equations with impulses, Appl.
Math. Comput., 77 (1996), 153–165.
[45] I. Stamova, Stability Analysis of Impoulsive Functional Differential Equations, Walter De Gruyter Inc, (2009).
[46] Y. G. Sun, Necessary and sufficient condition for the oscillation of forced nonlinear differential equation with delay, Pure Appl. Math., 18(2) (2002), 170–173.
[47] X. L. Wu, S. Y. Chen and H. J. Tang, Oscillation of a class of second-order delay differential equation with impulses, Appl. Math. Comput., 145 (2003), 561–567.
[48] J. R. Yan, Oscillation of nonlinear delay impulsive differential equations and inequalities, J. Math. Anal. Appl., 265 (2002), 332–342.
[49] J. R. Yan, Oscillation properties of a second order impulsive delay differential equation, Comput.
Math. Appl., 47 (2004), 253–258.
[50] J. R. Yan and C. H. Kou, Oscillation of solutions of impulsive delay differential equations, J.
Math. Anal. Appl., 254 (2001), 358–370.
151
[51] J. R. Yan and A. M. Zhao, Oscillation and stability of linear impulsive delay differential equations, J. Math. Anal. Appl., 227 (1998), 187–194.
[52] J. R. Yan, A. M. Zhao and Q. X. Zhang, Oscillation properties of nonlinear impulsive delay differential equations and applications to population models, J. Math. Anal. Appl., 322 (2006), 359–370.
[53] G. B. Ye and X. Q. Zhou, Oscillation of a class of second-order impulsive delay differential equations with forcing term, Math. Theory Appl., 29(1) (2009), 37–40.
[54] Y. Z. Zhang, A. M. Zhao and J. R. Yan, Oscillation criteria for impulsive delay differential equations, J. Math. Anal. Appl., 205 (1997), 461–470.
[55] A. M. Zhao and J. R. Yan, Necessary and sufficient conditions for oscillations of delay equations with impulses, Appl. Math. Lett., 10 (1997), 23–29.