微分方程概周期解的研究

2019-02-24 21:36:25

Equations solutions 周期 微分方程 periodic









中文题名微分方程概周期解的研究

 





副题名 





外文题名 Study of the almost periodic solutions for differential equations 





论文作者冯春华   





导师葛渭高教授   





学科专业应用数学  应用微分方程   





研究领域\研究方向 





学位级别博士 





学位授予单位北京理工大学   





学位授予日期2001   





论文页码总数91页   





关键词微分方程  概周期泛函微分方程   





馆藏号BSLW

/2001

/O175

/135 





【中文摘要】

 
摘要
   众所周知,研究概周期微分方程解的存在性、唯一性、稳定性诸问题,不仅有着重要的理论意义,也有着深刻的实际意义,同时也是定性理论研究的重要课题。特别是概周期泛函微分方程,其理论体系尚未完善,留待解决的问题不少,在实际中则有着十分广阔的应用前景。
   本文结合运用Liapunov函数,Liapunov泛函,指数二分法,渐近概周期函数,不动点定理等方法,研究了概周期常微分方程和概周期泛函微分方程概周期解的存在性、唯一性、稳定性等问题,这些结果均是新的。
   第一章绪论综述了概周期微分方程研究的历史和现状,以及到目前为止国内外专家、学者在这一研究领域取得的研究成果,同时介绍有关概周期函数的基本定义、引理以及有关的主要结论。
   第二章研究概周期常微分方程。应用Liapunov函数,我们给出了概周期常微分方程存在唯一概周期解的一个判别准则,应用不动点定理和指数二分法研究了一类n维概周期系统概周期解的存在性,另外还给出了n维Duffing方程概周期解存在性的一组充分条件。
   第三章研究概周期泛函微分方程。应用Lipunov泛函,我们给出了概周期泛函微分方程存在唯一概周期解的一个判别准则。对带有概周期强迫项的时滞Lienard方程,我们给出了存在唯一的一致渐近稳定概周期解的一组充分条件。我们还利用壳(hull)方程研究概周期解的存在唯一性。运用不动点定理和指数二分法,研究了具有反射变元的中立型方程概周期解的存在性。最后,作为应用,我们对生态系统中出现的一类种群动力学模型以及具有无穷时滞的Logistic方程,研究了其概周期解的存在唯一性。
   第四章提出了未来研究的一些设想。











【外文摘要】

 
ABSTRACT
   It is well known that such problems on the existence,uniqueness and stability of thesolutions for differential equations not only have important theoretical significance,but alsohave profound practical significance.It is also principal topic of the stable theory.For thealmost periodic functional differential equations,especially,the theoretical system is notperfected,and many problems are to be solved.There is a wide prospect in respect of theapplication.
In the present paper,we investigate the existence,uniqueness and stability of the almostperiodic solutions both of the ordinal differential equations and functional differentialequations.Those are all new results.
The dissertation consists of four chapters.In the first chapter,we review the history andthe current situation of studies on the almost periodic differential equations,and the resultsthat researchers both at home and abroad have obtained till now.We also introduce thebasic concepts and principal results.
In the second chapter,we investigate the almost periodic solutions of the ordinaldifferential equations.A criterion on the existence and uniqueness of the almost periodicsolutions was obtained.We study some n dimensional almost periodic differential equationsby using the method of the fixed point theory and a sufficient condition was obtained on theexistence of the n dimensional Duffing equation.
In the third chapter,we study the almost periodic solutions of the functional differentialequations.A criterion on the existence and uniqueness of the almost periodic solutions wasobtained by using Liapunov functional.A sufficient condition on the uniformly asymptoticstability of the almost periodic solutions was obtained for the delay Lienard equationshaving almost periodic force term.By using Hull equation,we investigate the existence anduniqueness of the almost periodic solutions.A sufficient condition on the existence anduniqueness of the almost periodic solutions of the functional differential equations wasobtained.We also investigate the existence and uniqueness of the almost periodic solutions ofthe functional differential equations with reflection of the argument and the model in thepopulation dynamics differential equations and with infinite delay Logistic equations.
In the forth chapter,we present some ideas for the future researches.