三阶微分方程边值问题正解的存在性

发布时间：2018-03-08 22:14

  本文选题：三阶边值问题　切入点：格林函数　出处：《山东师范大学》2017年硕士论文　论文类型：学位论文

【摘要】：常微分方程边值问题是常微分方程理论研究中最为重要的课题之一.随着科学技术的进步与发展,工程、力学、天文学、经济学、控制论及生物学等自然学科和边缘学科领域中的许多实际问题都可归结为常微分方程的边值问题.我们知道,寻求微分方程的通解十分困难,故从理论上探讨解的存在性及其性态,一直是近年来研究的热点问题.本文主要研究非线性三阶边值问题的存在性.全文共分三章.第一章介绍了研究背景和预备知识及重要定理。第二章研究三阶两点边值问题一个正解、两个正解、三个正解的存在性.令(?),(?),若f_0=0,f_∞=∞(超线性),或者f_0=∞且f_∞=0(次线性),则边值问题(Ⅰ)至少存在一个正解.若f_0=f_∞=0,则边值问题(Ⅰ)至少有两个正解.令(?),(?).假设存在实数d_0,d_1和c且0d_0d_1d_1/γc 使得下列式子满足 f(x)d_0/D,x∈[0,d_0]f(x)d_1/C,x∈[d_1,d_1/γ],f(x)c/D,x∈[0,c],则边值问题(Ⅰ)至少有三个正解.第三章研究三阶三点边值问题一个正解、两个正解、三个正解的存在性.令(?),(?),若f_0=0,f_∞=∞(超线性),或者f_0=∞且f_∞=0(次线性),则边值问题(Ⅱ)至少存在一个正解.若f_0=f_∞=0,则边值问题(Ⅱ)至少有两个正解.令(?),(?).假设存在实数d_0,d_1和c且0d_0d_1d_1d_1/γc使得下列式子满足f(t,u)d_0/D,u∈[0,d_0],f(t,u)d_1/C,u∈[d_1,d_1/γ],f(t,u)c/D,u∈[0,c]边值问题(Ⅱ)至少有三个正解.
[Abstract]:The boundary value problem of ordinary differential equation is one of the most important subjects in the research of ordinary differential equation theory. With the progress and development of science and technology, engineering, mechanics, astronomy, economics, Many practical problems in natural and marginal disciplines such as cybernetics and biology can be reduced to boundary value problems of ordinary differential equations. Therefore, the existence and behavior of the solution are discussed theoretically. This paper mainly studies the existence of nonlinear third-order boundary value problems. The thesis is divided into three chapters. The first chapter introduces the research background, preparatory knowledge and important theorems. In Chapter 2, we study the third-order second-order problems. A positive solution to the point boundary value problem, Existence of two positive solutions, three positive solutions. What? If f0 _ 鈭，

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