微分方程边值问题的若干研究

2019-02-24 21:31:20

算子 Sturm Liouville Laplacian 边值问题









中文题名微分方程边值问题的若干研究

 





副题名 





外文题名 Study of the boundary value problems of differential equations 





论文作者孙伟平   





导师葛渭高教授   





学科专业应用数学   





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76页   





关键词微分方程  拓展定理  边值问题   





馆藏号BSLW

/2001

/O175

/136 





【中文摘要】

 
摘要
   拓扑方法自创立以来,一直被各国数学工作者广泛引用。本论文主要研究拓扑方法在常微分方程中的应用;此外,我们学详尽讨论了在流体力学、天体物理中有着广泛应用背景的p-Laplacian算子和Laplacian-型算子的各类边值问题解的存在性、多重性。这些问题都是目前讨论较少或尚未涉及的,结果均是新的。
   第一章主要介绍了拓展定理(continuation theorems)思想的起源,介绍了p-Laplacian算子各Laplacian-型算子边值问题产生的具体背景知识;简要介绍了到目前为止国内外专家、学者在这两个领域取得的研究成果。在这一章里还总结了本文的主要结论。
   缺失先验界情况下的拓展定理直到二十世纪九十年代以后才得到发展,在第二章里,我们利用它讨论了一类二阶超线性非齐次Sturm-Liouville边值问题,给出了存在多个解的充分条件。
   第三章研究的是p-Laplacian算子和Laplacian-型算子的边值问题。在第二节,我们定义了一种新的坐标变换-广义极坐标,并利用它讨论了p-Laplacian算子和Laplacian-型算子的Sturm-Liouville边值问题,分别得到了存在一个解、多个解、无穷多个解的多个充分条件;第三节研究p-Laplacian算子的Sturm-Liouville边值问题正解的存在性与多重性,采用的是锥上的不动点定理,全面推广了这一方面已有的结果;对目前研究较少的高维Laplacian-型算子及带有Laplacian-型算子的泛函微分方程的周期解问题,我们在第四节做了一些研究,这也是拓扑方法的一个应用。
   将拓扑方法应用于讨论微分方程模型(如Lotka-Voltcrra系统)正周期解的存在性非常有效。第四章里,我们应用拓展定理证明了一类捕食者-食饵模型正周期解的存在性,并用Lyapunov泛函方法讨论了周期解的全局渐近稳定性。











【外文摘要】

 
ABSTRACT
   This paper presents an investigation of applications of topological methods in ordinarydifferential equations and an exhaustive study of all kinds of boundary value problems(bvps)with p-Laplacian and Laplacian-like operators which arise in a multitude of appliedareas such as porous media,clasticity theorem and astrophysics.
In the first chapter,we introduce the background of continuation theorems and bvpsof p-Laplacian and Laplacian-like operators.The results in these fields that researchersboth at home and abroad have obtained till now are also presented.
In Chapter 2,by using of a continuation theorem in the absence of a priori bounds,we show the existence of multiple solutions for Sturm-Liouville bvps of a second-ordersuperlinear ordinary differential equation.
Chapter 3 is devoted to the bvps of p-Laplacian and Laplacian-like operators.InSection 2,we give several sufficient conditions for the existence of one solution,multiplesolutions and infinitely many solutions of the Sturm-Liouville bvps via the generalizedpolar coordinates.Then in Section 3,the existence of positive solutions are proved undersuperlinearity,sublinearity and many other conditions by a fixed point theorem in cones.Our results have generalized those in many articles.A detailed discussion of periodic solu-tions of a kind of functional differential equations with high-order Laplacian-like operatorcan be found in Section 4 and this subject has not been studied before.
In the last chapter,we show the validity of topological methods in studying bio-logical models.The existence and global attractivity of positive periodic solutions of aPredator-Prey system are investigated by applying a continuation theorem and Lyapunovfunctional.