《几类非线性微分方程边值问题的解》由会员分享，可在线阅读，更多相关《几类非线性微分方程边值问题的解（51页珍藏版）》请在金锄头文库上搜索。
1、分类号：O175.8密级：公开U D C ：单位代码： 10424学位论文几类非线性微分方程边值问题的解黄玉梅申请学位级别：硕士学位专业名称：应用数学指导教师姓名：高德智职称：教授山东科技大学二零零八年五月论文题目：几类非线性微分方程边值问题的解作者姓名：黄玉梅入学时间：2005年 9月专业名称：应用数学研究方向：函数论与应用泛函分析指导教师：高德智职称：教授论文提交日期：2008年 5月论文答辩日期：2008年 5月授予学位日期：EXISTENCE OF SOLUTIONS FOR SOME NONLINEARDIFFERTIAL EQUATIONS BOUNDARY VALUEPROBLE
2、NSA Dissertation submitted in fulfillment of the requirements of the degree ofMASTER OF SCIENCEfromShandong University of Science and TechnologybyHuang YumeiSupervisor: Professor Gao DezhiCollege of information science and EngineeringMay 2008声明本人呈交给山东科技大学的这篇硕士学位论文，除了所列参考文献和世所公认的文献外，全部是本人在导师指导下的研究成果。
3、该论文资料尚没有呈交于其它任何学术机关作鉴定。硕士生签名：日期：AFFIRMATIONI declare that this dissertation, submitted in fulfillment of the requirementsfor the award of Master of Philosophy in Shandong University of Science andTechnology, is wholly my own work unless referenced of acknowledge. Thedocument has not been submitted f
4、or qualification at any other academicinstitute.Signature:Date:山东科技大学硕士论文摘要摘要非线性泛函分析是现代分析数学的一个重要分支，它是人们在研究古典和现代物理学、医学、生物学的过程中发展起来的，为解决当今科技领域中出现的各种非线性问题提供了丰富的理论依据。近年来，非线性泛函分析已经成为研究物理、航空航天技术、生物技术中非线性问题的一个重要的工具，所以，非线性泛函分析理论的研究及完备化具有非常重要的意义。国内外许多研究学者都对非线性问题的研究做了大量的工作，1921年，L. E. J. Brouwer对有限维空间建立了拓扑度概念
5、。1934年，J.Leray和 J.Schauder将这一概念推广到 Banach空间的全连续场。后来，E. Rothe，M. A. Krasnoselskii，P. H. Rabinowitz, H. Amann，V.Lakshmikantham，K. Deimling等对拓扑度理论，锥理论及其应用进行了深入的研究。国内许多著名的数学家，如张恭庆教授、陈文源教授、郭大钧教授、孙经先教授等在这一领域都做过很深刻的工作。本文主要是运用非线性泛函分析中的拓扑度理论和上下解方法研究了几类非线性微分方程边值问题解的存在性。第一章，阐述了非线性泛函方法的发展历程及其现阶段的研究进程，给出了本文中要用到的
6、一些与上下解方法及迭合度方法有关的概念及定理，最后介绍了本文的主要工作。第二章，运用上下解方法和 Leray-Schauder度理论讨论了一类二阶非线性三点边值问题三个解的存在性结果。第三章，讨论了一类二阶四点边值问题，运用上下解方法和不动点理论得到问题三个正解的存在性结果，并利用拓扑度理论对解的存在区域进行了估计。第四章，运用上下解方法和极大值原理讨论了一类四阶奇异边值问题的正解的存在性。关键词：边值问题，上下解，不动点，拓扑度，边界条件，多解，奇异边值问题，极大值原理。1山东科技大学硕士论文AbstractAbstractNonlinear functional analysis is a
7、n important branch of modern mathematics, it isdeveloped by people in the study of classical and modern physics, medicine, biology etc.Nonlinear functional analysis provides an effect theoretical tool for studying many nonlinearproblems.In recent years, the nonlinear functional analysis has become a
8、n important tool in theresearch of nonlinear problems of physical, aerospace technology and biological technology.Therefore, the research of nonlinear functional analysis theory has extremely importantsignificance. Some domestic and foreign scholars have done massive work to the nonlinearanalysis. L
9、. E. J. Brouwer had established the conception of topological degree for finitedimensional space in 1912. J.Leray and J.Schauder had extended the conception to completelycontinuous field of Banach space in 1934. Afrer, E. Rothe, M. A. Krasnoselskii, P.H.Rabinowitz, H. Amann, V. Lakshmikantham, K. De
10、imling had carried on research ontopological degree and cone theory. Same well known mathematicians, Zhang Gongqing, ChenWenyuan, Guo Dajun, Ma Ruyun, Sun Jingxian etc, have great works in various fields ofnonlinear functional analysis.This article studies some multi-point boundary value problems by
11、 topological degreetheory and method of upper and lower solutions.In chapter 1, we introduce the development of nonlinear functional method and theadvancement of present stage, give some related conceptions and theorems, finally, weintroduce the main work of this master thesis.In chapter 2, via the
12、method of upper and lower solution and the Leray-Schauder theory,we show the existence of triple positive solutions for a kind of second-order three pointboundary value problem.In chapter 3, we discuss a kind of second-order four point boundary value problem , viathe fixed-point theorem, we obtain the existence of at least three solutions of the four pointboundary value problem in the presence of two upper and lower solutions, and estimate theexistence regions of the solutions.2山东科技大学硕士论文AbstractIn chapter 4, we use
