两类复杂分数阶微分方程边值问题的正解

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 contributions exist,as far as we know, concerning the content of this project. The results are new and are interesting to the theory of fractional differential equations.

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。

本项目研究具复杂非线性项的微分方程、分数阶微分方程边值问题解与正解的存在性，利用非线性分析方法，克服复杂非线性项带来的困难，建立了几类问题解与正解的存在性结果：.1、对无穷区间上分数阶微分方程边值问题，通过巧妙构造函数空间及其上的范数，克服无穷区间紧性不足带来的困难，结合解的先验估计和不动点定理，给出了一类无穷区间上分数阶微分方程边值问题解的存在性结论。.2、对高阶、分数阶微分方程共振边值问题，利用分析方法克服函数的导数符号与函数极值情况无必然蕴含关系的限制，利用范数形式的Leggett-Williams不动点定理研究其正解的存在性，建立正解的存在性结果; 利用Mawhin重合度理论研究了一类时间测度上微分方程四点边值问题在不同共振条件下解的存在性结论，得到了解存在的充分性条件。.3、对非线性项显含导数项、尤其是分数阶导数项的分数阶微分方程边值问题，利用解的先验估计、巧妙构造相应的函数空间，结合不动点定理给出了正解存在的充分条件。. 项目研究期间在SCI期刊发表研究论文10篇，培养研究生硕士研究生1人，参加国内学术会议5次。取得结果具有创新性，对分数阶微分方程理论有重要意义。