【作者】 李成福；

【导师】 周勇；

【作者基本信息】 湘潭大学 ， 应用数学， 2010， 博士

【摘要】 分数微分方程(Fractional Differential Equations)在许多学科领域有广泛的应用,这些学科领域的许多数学模型都是用分数微分方程来描述的.近十年来,分数微分方程边值问题得到了迅速的发展.随着这一方向研究的进一步深入,研究内容和研究方法不断得到丰富,非线性分数微分方程边值问题获得了许多研究成果.本文主要研究非线性分数微分方程边值问题解的存在性.本文由四章组成,主要内容如下：第一章主要介绍了分数微分方程的研究背景和发展状况,并简要介绍本文的主要工作.这一章也包括一些预备知识,如分数微积分的基本概念、分数导数与分数积分的基本性质、非线性泛函分析的一些基本知识和一些重要的不动点定理.第二章讨论了非线性分数微分方程具有分数边界条件的两点边值问题解的存在性.我们较早研究了积分号下求分数阶导数,推广了积分号下求整数阶导数的相应结果,在此基础上,研究了具有分数边界条件的两点边值问题解的存在性,我们的结果改进和推广了文献中的已有结果.第三章讨论了非线性分数微分方程具有分数边界条件的三点边值问题解的存在性。随着点数的增加,格林函数随之会变得复杂。我们求出对应的格林函数,并讨论其性质,把分数微分方程转化为等价的积分方程,在此基础上,利用Banach压缩映像原理、Leray-Schauder非线性备择性原理、锥拉伸-锥压缩等不动点定理,给出非线性分数微分方程三点边值问题正解的存在性结果.第四章研究了两类非线性分数泛函微分方程边值问题解的存在性,利用Schauder不动点定理,获得了解的存在性结果.更多还原

【Abstract】 Fractional differential equations have gained considerable importance due to their application in various sciences, many mathematical modellings in vari-ous science are discribed on basis of fractional differential equation. In the past decade, boundary value problem of fractional differential equation has developed very rapidly. With the further study of this direction, the studing contents and methods continuously get abundant. A lot of results have been obtained for nonlinear fractional equations. This dissertation focuses on the research of the existence of solution of the boundary value problem for nonlinear fractional dif-ferential equations and fractional functional differential equations. The paper consists of four chapters. Main contents are as follows:In Chapter 1, we give a survey to the development and situation of the boundary value problem for nonlinear fractional differential equations and frac-tional functional differential equations and introduces main results in this paper. We also introduces some preliminary material, including some basic concepts from the fractional differential equation, some results from fractional calculus and some importment fixed point theorems.In Chapter 2, we first extends the well-known rule for the differentiation of an integral depending on a parameter with the upper limit depending on the same parameter. We discuss the two points boundary value problem with fractional boundary conditions. We obtain the existence and multiplicity results of positive solutions by using some fixed theorems. Our results improve and extend results in the literature.In Chapter 3, we discuss the three points boundary value problem with frac-tional boundary conditions. We derive first the Green function and its propety. consequently, boundary value problem is deduced to a equivalent integral equa-tion. Next, by using some fixed-point theorems, we obtain the the existence and multiplicity results of positive solutions.In Chapter 4, we begin with the study of boundary value problem for two classes of fractional functional differential equations and give a series of sufficient conditions for the existence of solutions of the boundary value problem. Our results improve and extend a number of results in the literature.更多还原