Oscillation of impulisve delay differential equations

具滯後脈衝微分方程的振動性

黃少遠(Shao-Yuan Huang)

指導教授 ： 鄭穗生

國立清華大學/理學院/數學系/博士(2011年)

摘要

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.

關鍵字

振動 ； 脈衝 ； 正解 ； 微分方程

並列摘要

無資料

並列關鍵字

oscillation ； impulsive ； positive solution ； differential equation

參考文獻

[26] S. Y. Huang and S. S. Cheng, Comparison theorems for positive increasing solutions of second order differential equations with delays and impulses, preprint.

[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