节点文献
几类p-Laplacian多点边值问题及非线性微分方程组解的存在性的研究
【作者】 王宇晴；
【导师】 贺小明；
【作者基本信息】 中央民族大学 ， 基础数学， 2009， 硕士
【摘要】 本论文主要应用Legget-Williams不动点定理、双锥不动点定理、Avery-Peterson不动点定理以及锥压缩与锥拉伸不动点定理等非线性分析的理论和方法,研究了一类具有反馈控制的非线性泛函微分系统正周期解的存在性,以及带有p-Laplace算子的多点边值问题正解的存在性和多重性问题.全文共分为三章:第一章主要介绍了具有反馈控制的非线性泛函微分系统以及带有p-Laplace算子的多点边值问题的应用背景和国内外关于这些问题的研究现状与成果,并简述本文的主要研究成果.第二章主要利用Avery-Peterson不动点定理,研究了具有反馈控制的非线性非自控泛函微分方程组的多重正周期解的存在性.我们证明了,对非线性项加以适当的增长性条件,所研究的方程组至少存在三个正周期解.第三章研究带有p-Laplace算子的三阶三点边值问题。我们分别运用双锥不动点定理、Legget-Williams不动点定理、锥压缩与锥拉伸不动点定理讨论了这些问题正解的存在性,多重性以及不存在性.
【Abstract】 This dissertation deals with the existence of positive periodic solutions to one kind of nonlinear functional differential system with feedback control, as well as the existence and multiplicity of positive solutions to p - Laplacian multi-point boundary value problems. The methods and techniques employed here are involved in nonlinear functional analysis, such as Avery-Peterson fixed-point theorem、cone tensile and compression fixed-point theorem、Leggett-Williams fixed-point theorem and double-cone fixed-point theorem. This dissertation consists of three chapters.In Chapter 1, we introduce the background and research status on the nonlinear functional differential system with feedback control and p- Laplacian multi-point boundary value problems at home and abroad. We also present a brief survey of our results.In Chapter 2, by using Avery-Peterson fixed-point theorem, we investigate the existence of multiple positive periodic solutions to the nonlinear non-autonomous functional differential system with feedback control, and prove that this system admits at least three positive periodic solutions under certain growth conditions imposed on the nonlinearity.In Chapter 3 we investigate several kinds of p- Laplacian third-order three-point boundary value problems, we discuss the existence, multiplicity and no-existence of positive solution to these problems by using double-cone fixed-point theorem, Leggett-Williams fixed-point theorem and cone-tensile and cone-compression fixed-point theorem, separately.