分数阶微分方程反周期边值问题解的存在性

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分数阶微分方程反周期边值问题

解的存在性

摘要

本文研究两类分数阶微分方程反周期边值问题解的存在性. 一类是在非线性

项中含有未知函数分数导数的分数阶微分方程反周期边值问题, 通过构造 Banach

空间, 利用 Schauder 不动点定理及压缩映射原理进行研究. 另一类是脉冲点与边界

条件都涉及到分数阶导数的分数阶脉冲微分方程反周期边值问题, 利用 Schaefer 定

理, Krasnoselskii’s 不动点定理及压缩映射原理进行研究. 本文对两类问题的多解性

及有唯一解分别进行阐述, 使得已有的结果得到推广, 并有所创新. 本文内容分为

四章:

第一章绪论, 首先是有关分数阶微分方程的一些简介, 然后介绍了本课题研究

的意义及国内外关于本课题研究的现状, 同时给出了本文主要研究内容及结构.

第二章是为下面的研究做一些准备工作, 主要介绍了分数阶积分微分, 不动点

定理, 以及一些相关引理的知识.

第三章研究了一类在非线性项中含有未知函数分数导数的分数阶微分方程反

周期边值问题

解的存在性, 利用 Schauder 不动点定理及压缩映射原理, 在非线性项有界和无界的

情况下, 分别研究了反周期边值问题解存在的条件. 得到了关于分数微分方程反周

期边值问题解的多个存在性定理.

第四章研究了一类脉冲点与边界条件都涉及到分数阶导数的分数阶脉冲微分

方程反周期边值问题

解的存在性, 运用 Schaefer 定理, Krasnoselskii’s 不动点定理及压缩映射原理, 分别

得出了反周期边值问题存在唯一解和多解性的几个条件.

关键词：分数阶微分方程　反周期边值问题　不动点定理　压缩映射

原理　存在性和唯一性　Green 函数

ABSTRACT

In this paper, we study the existence of solutions for the anti-periodic boundary

value problems of the fractional differential equations. At first, we study the fractional

differential equation which includes fractional derivatives of unknown function in the

nonlinear term. By using the Schauder fixed point theorem and contraction mapping

principle on a special Banach space under some suitable conditions. Then, we study the

fractional equation which includes fractional derivatives in the pulse point and boundary

conditions. By using the Schaefer fixed point theorem, Krasnoselskii’s fixed point

theorem and contraction mapping principle. This paper discusses the multiplicity and

uniqueness of solution of the anti-periodic boundary value problems of the fractional

differential equations. What’s more, this paper makes the results already have been

promoted and innovation. According to the content divided into the following four

chapters:

In Chapter 1, to begin with the paper, we give some introductions about the

fractional differential equations. In addition, we introduce some backgrounds and

researches in domestic and overseas. And the end, this paper makes a study of the

content and structure.

For convenience, in Chapter 2, we are readying for the next chapters. In the chapter

we give the integro-differential equations of fractional order and some fixed point

theorems, which include some important theorems that will be used in our paper.

Besides, this chapter introduces some this paper research institute of the use of

nonlinear functional analysis. Meanwhile, the arrangements of our paper have also been

given.

Inspired by related works, in Chapter 3, we studies the existence of solutions for

the anti-periodic boundary value problems of the fractional differential equations which

include fractional derivatives of unknown function in the nonlinear term.

In the case of nonlinear term bounded and unbounded, we study the existence con-

ditions of anti-periodic boundary value problems by means of Schauder fixed point

theorem and contraction mapping principle. Some existence results for the anti-periodic

boundary value problems are obtained.

In Chapter 4, we discuss the existence of solution for impulsive anti-periodic

boundary value problem for fractional differential equation which includes fractional

derivatives in the pulse point and boundary conditions.

By using Schaefer fixed point theorem, Krasnoselskii’s fixed point theorem and

contraction mapping principle, some sufficient conditions for the existence of anti-

periodic boundary value problem’s solution was obtained.

Key Words: Fractional differential equations, Anti-periodic boundary

value problems, Fixed point theorem, Contraction map-

ping principle, Existence and uniqueness, Green's func-

tion

中文摘要

ABSTRACT

第一章绪论.....................................................................................................................1

1.1 研究意义及研究现状...........................................................................................1

1.2 主要研究内容.......................................................................................................3

第二章预备知识.............................................................................................................5

2.1 分数阶积分微分的基本概念及理论....................................................................5

2.2 算子与不动点定理................................................................................................6

第三章一类分数阶微分方程反周期边值问题解的存在性.........................................8

3.1 基本引理...............................................................................................................8

3.2 解的存在性..........................................................................................................10

第四章分数阶脉冲微分方程反周期边值问题解的存在性.......................................20

4.1 预备引理.............................................................................................................20

4.2 解的存在唯一性.................................................................................................27

4.3 多解性..................................................................................................................30

参考文献........................................................................................................................33

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摘要：

分数阶微分方程反周期边值问题解的存在性摘要本文研究两类分数阶微分方程反周期边值问题解的存在性.一类是在非线性项中含有未知函数分数导数的分数阶微分方程反周期边值问题,通过构造Banach空间,利用Schauder不动点定理及压缩映射原理进行研究.另一类是脉冲点与边界条件都涉及到分数阶导数的分数阶脉冲微分方程反周期边值问题,利用Schaefer定理,Krasnoselskii’s不动点定理及压缩映射原理进行研究.本文对两类问题的多解性及有唯一解分别进行阐述,使得已有的结果得到推广,并有所创新.本文内容分为四章:第一章绪论,首先是有关分数阶微分方程的一些简介,然后介绍了本课题研究的意义及国内外关于本...

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