【作者】 王宁；

【导师】 周宗福；

【作者基本信息】 安徽大学 ， 应用数学， 2023， 硕士

【摘要】 作为微分方程定性理论中的一个重要分支,非线性常微分方程边值问题具有广泛的应用背景.近年来,随着分数阶微积分理论的发展,分数阶微分方程在许多领域被广泛地应用,如:反常扩散、力学、生物医学、自动控制等.因此分数阶微分方程边值问题的研究受到学者们的关注,得到了许多深刻的结果.本文在已有工作的基础上运用一些泛函方法来研究几类分数阶微分方程边值问题解的存在性.这些分数阶微分方程中的导数涉及到Riemann-Liouville分数阶导数和Caputo分数阶导数.所用的泛函方法主要是利用不动点理论,包括Banach压缩映像原理,Avery-Peterson不动点定理等.第一章,绪论主要介绍分数阶微积分,分数阶微分方程以及其边值问题的研究背景.给出几种分数阶导数和积分的定义,及其相关的重要引理.概述本文要研究的几类分数阶微分方程边值问题.第二章,考虑一类具有p-Laplacian算子和无穷多点边界条件的分数阶微分耦合系统,利用Avery-Peterson不动点定理得到该系统至少三个正解的存在性,扩展了已有的结果.第三章,研究一类具有多点及泛函边值条件的Caputo分数阶微分方程的可解性问题.利用Banach压缩映像原理和Burton-Kirk不动点定理得到此边值问题解的存在性和唯一性的结果,所得定理推广了已有的结论.第四章,讨论一类具有Stieltjes积分边界条件和高阶混合分数阶导数的边值问题的正解.与已有的研究工作相比,本章所讨论边值问题中的积分条件更为广泛.通过Avery-Peterson不动点定理得到了该类分数阶微分方程边值问题的正解的多重性,并给出例子来说明结果的适用性.第五章,通过Avery-Peterson不动点定理,得到了一类无穷区间上的带有无穷积分的分数阶微分方程边值问题正解的多重性.更多还原

【Abstract】 As an important branch in the qualitative theory of differential equations,the boundary value problems of nonlinear ordinary differential equation have a wide application.In recent years,with the development of fractional calculus theory,fractional differential equations have been widely used in many fields,such as: anomalous diffusion,mechanics,biomedicine,and automatic control,etc.Therefore the study of the boundary value problem for fractional differential equations has received much attention from scholars and many profound results have been obtained.This paper builds on existing work to investigate the existence of solutions to several classes of boundary value problems for fractional differential equations using some generalized function methods.The derivatives in these fractional differential equations involve the Riemann-Liouville fractional order derivatives and the Caputo fractional order derivatives.The functional methods used are mainly based on the fixed point theory,including the Banach contraction mapping principle,the AveryPeterson fixed point theorem and so on.In Chapter 1,the background to the study of fractional calculus,fractional differential equations and their boundary problems is introduced.Definitions of several fractional order derivatives and integrals are given,together with their related important lemmas.Several classes of boundary value problems for fractional differential equation are presented in this paper.In Chapter 2,a class of coupled fractional differential systems with p-Laplacian operators and infinite multipoint boundary conditions is considered,and the existence of at least three positive solutions for this system is obtained using the Avery-Peterson fixed point theorem,extending the existing results.In Chapter 3,the problem of solvability of a class of Caputo fractional differential equations with multiple points and functional boundary conditions is investigated.Results on the existence and uniqueness of solutions to this boundary value problem are obtained by the Banach contracting mapping principle and the Burton-Kirk fixed point theorem,and the resulting theorem extends the existing conclusions.In Chapter 4,the positive solutions of a class of boundary value problems with Stieltjes integral boundary conditions and the high order mixed fractional derivatives are discussed.The integral conditions in the boundary value problems discussed in this chapter are more general than in the existing research work.The multiplicity of positive solutions to this class of boundary value problems for fractional differential equations is obtained via the Avery-Peterson boundary point theorem,and examples are given to illustrate the applicability of the results.In Chapter 5,the multiplicity of positive solutions of a class of boundary value problems of fractional differential equations with infinite integrals on the infinite interval is obtained by means of the Avery-Peterson fixed point theorem.更多还原