摘要
本文利用锥理论,不动点理论,Krasnoselskii不动点定理、上下解方法等研究了有限区间和无穷区间上几类微分方程奇异和半正边值问题(组)解的存在性、和多解性等情况,同时建立了抽象无界函数族相对紧性的判定定理,最后我们研究了非线性二阶脉冲混合型积分-微分初值问题解的存在性.通过深入的研究,在较弱的条件下获得了一些新的有趣的成果.
全文分为六章.第一章,我们主要介绍了非线性泛函分析的历史背景和一些基本概念.第二章我们研究了无穷区间上一类奇异边值问题解的存在性、多解性,并建立了无穷区间上两类无界函数族相对紧性的两个判定定理.§2.2考察了无穷区间上包含一阶导数项的奇异二阶三点边值问题一个解和三个解的存在性,其非线形项可取负值.§2.3我们建立了判定无穷区间上抽象无界连续函数组和抽象无界可微函数组相对紧性的充分必要条件,并对无穷区间上抽象二阶微分方程两点边值问题无界解的存在性做了研究.第三章我们利用线性算子的第一特征值讨论了两类奇异三点边值问题和一类周期边值问题.§3.2我们建立了奇异三点边值问题正解的存在性结果,§3.3我们研究了具有一般形式即包含一阶导数的二阶三点边值问题.§3.4在没有非线性项非负的假设条件下,解决了一类周期边值问题非平凡解的存在性.第四章,我们把注意力放在两类半正边值问题和一类半正方程组上的研究上.§4.2我们得到了具有变号非线性项的奇异二阶M点边值问题非平凡解的存在性结果,其非线性项可以下无界.§4.3是对§4.2的深入,我们处理了具有变号非线性项的奇异高阶边值问题非平凡解的存在性结果.§4.4我们建立了共轭边值问题半正方程组正解的存在性定理.第五章我们讨论了一类奇异微分方程组正解的存在性.第六章,利用一个新的比较结果和M(o|¨)nch不动点定理,研究了Banach空间中二阶非线性混合型脉冲微分-积分方程初值问题解的存在性.
The present paper employs the cone theory,fixed point index theory,and Krasnoselskii fixed point theorem,the method of lower and upper solutions and so on,to investigate the existence of solutions,multiple solutions to several kinds of singular and semipositone boundary value problems(system) on finite intervals and on the half-line. Besides,we establish sufficient and necessary conditions for relative compactness of two abstract unbounded function groups.Finally we investigate the existence of solutions of initial value problem for nonlinear second order impulsive integro-differential equations of mixed type in Banach space.By deep study,we obtain some new interesting. results under weaker conditions.
The thesis is divided into six chapters.In ChapterⅠ,we mainly introduce the background of nonlinear functional analysis and some concepts.In ChapterⅡ,we are concerned with the existence of solution,multiple solutions of one kind of singular boundary value problems on the half-line and establish two decision theorems of relative compactness for unbounded function groups in an infinite interval.In§2.2,the conditions for the existence of at least one solution and three solutions are established for a class of singular second-order three-point boundary value problem on the halfline, where the nonlinear term can take a negative value and includes the first order derivative.In§2.3,sufficient and necessary conditions are given for relative compactness of an abstract unbounded continuous function group and an abstract differentiable function group.Furthermore,the existence of unbounded solutions for a second-order boundary value problem in infinite interval are obtained.In ChapterⅢ,we discuss two kinds of singular three-point boundary value problems and a periodic boundary value problem by the first eigenvalue corresponding to the relevant linear operator. In§3.2,the existence criteria of positive solutions for a singular three-point boundary value problem is established.What we study in§3.3 is a three-point boundary value problem with a more general form which possesses the first derivative.§3.4 deals with the existence of nontrivial solutions to a periodic boundary value problem without any nonnegative assumption on nonlinearity,by using the topological degree.ChapterⅣfocuses on the study of two kinds of semipositone boundary value problems and one semipositone system.§4.2 establishes the existence of nontrivial solutions for a singular second-order m-point boundary value problem with a sign-changing nonlinear term which may be unbounded from below.§4.3 deals with the existence of nontrivial solutions for a singular higher-order boundary value problem with a sign-changing nonlinear term.In§4.4,the existence of positive solutions for a semipositone system of conjugate boundary value problems is established.ChapterⅤdeals with the existence of positive solutions to a class of singular systems.In ChapterⅥ,by employing a new comparison result and the M(o|¨)nch fixed point theorem,the existence of solutions of initial value problem for second order impulsive integro-differential equations of mixed type in Banach space is investigated.
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