分数阶微分方程的一些新结果

第一部分，介绍分数阶导数的定义和著名的Mittag—Leffler函数的性质．第二部分，利用单调迭代方法给出了具有2序列Riemann—Liouville分数阶导数微分方程初值问题解的存在性和唯一性．第三部分，利用上下解方法和Schauder不动点定理给出了具有2序列Riemann—Liouville分数阶导数微分方程周期边值问题解的存在性．第四部分，利用Leray—Schauder不动点定理和Banach压缩映像原理建立了具有n序列Riemann—Liouville分数阶导数微分方程初值问题解的存在性、唯一性和解对初值的连续依赖性．第五部分，利用锥上的不动点定理给出了具有Caputo分数阶导数微分方程边值问题，在超线性（次线性）条件下C310，11正解存在的充分必要条件．最后一部分，通过建立比较定理和利用单调迭代方法给出了具有Caputo分数阶导数脉冲微分方程周期边值问题最大解和最小解的存在性．

Some New Results on Fractional Differential Equations

In this report,there are six parts.In the first part,we introduce some definitions of fractional derivative and discuss the properties of the well-known Mittag-Leffier function.In the second part,we consider the existence and uniqueness of solution of the initial value problem for fractional differential equation involving 2 sequential Riemann-Liouville fractional derivative by using monotone iterative method.In the third part,we consider the existence of solution of the periodic boundary value problem for fractional differential equation involving 2 sequential RiemannLiouville fractional derivative by means of the method of upper and lower solutions and Schauder fixed point theorem.In the fourth part,we have established the existence,uniqueness and continuous dependence results of solutions for the initial value problem of fractional differential equation involving n sequential Riemann-Liouville fractional derivative by means of the Leray-Schauder type fixed point theorem and Banach contraction principle.In the fifth part,we investigate the existence of positive solutions of singular superlinear (or sublinear) boundary value problems for fractional differential equation involving Caputo fractional derivative,a necessary and sufficient condition for the existence of C30,1] positive solutions is given by means of the fixed point theorems on cones.In the last part,we consider the existence of minimal and maximal solutions for the periodic boundary value problem of impulsive fractional differential equation involving Caputo fractional derivative by using a comparison result and the monotone iterative method.