两类复杂分数阶微分方程边值问题的正解

项目名称： 两类复杂分数阶微分方程边值问题的正解

项目编号： No.11201109

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 杨刘

作者单位： 合肥师范学院

项目金额： 20万元

中文摘要： 具有共振边值条件的分数阶微分方程边值问题和无穷区间上分数阶微分方程边值问题是分数阶微分方程研究的重要内容。由于无法直接将这两类问题转化为等价的u=Tu型算子方程，目前对这两类问题正解存在性的的研究未取得深入结果。 本项目拟以具有共振边值条件的分数阶微分方程边值问题和无穷区间上分数阶微分方程边值问题为研究对象，利用非线性泛函分析方法和拓扑度理论， （1）建立研究Lu=Tu型算子方程解存在性的新的锥上不动点定理，克服具共振边值条件的分数阶微分方程边值问题转化为u=Tu型算子方程的困难，建立问题正解的存在性结果； （2）通过推广分数阶微积分的相关概念和性质，克服无法将无穷区间上分数阶微分方程边值问题转化为u=Tu型算子方程的困难，建立问题正解的存在性结果。 本项目研究内容均未见系统的文献，因此预期结果具有明确创新性，对分数阶微分方程理论有重要意义。

中文关键词： 边值问题；非线性泛函分析；不动点；拓扑度；

英文摘要： Recently much attention has been paid to the study of certain boundary value problems of fractional differential equations at resonance and boundary value problems of fractional differential equations on an infinite interval. However there are few literatures available on the existence of positive solutions for these boundary value problems because of the difficulties in transforming these problems to operator equation of u=Tu type. In this project, we will consider the positive solutions of boundary value problems of fractional differential equations at resonance and boundary value problems of fractional differential equations on an infinite interval. By means of the nonlinear functional analysis method and the topological degree method, (1) we will establish a new fixed point theorem on cones to overcome the difficulty in transforming the problem to operator equation of u=Tu type and establish the existence of positive solutions for the boundary value problems of fractional differential equations at resonance. (2) we will give some new properties of fractional calculus to overcome the difficulty in transforming the problem to operator equation of u=Tu type and establish the existence of positive solutions for the boundary value problems of fractional differential equations on an infinite interval. Few

英文关键词： Boundary value problem；nonlinear functional analysis；fixed point theorem；topological degree；