【作者】 许晓婕；

【导师】 蒋达清；

【作者基本信息】 东北师范大学 ， 应用数学， 2012， 博士

【摘要】 本文首先在第一章给出分数阶微分和分数阶积分的相关概念及分数阶微分方程的研究历史.第二章,我们给出一些已知的结论并给出了两个积分方程新的正解的存在性结果.第三章,我们考虑非线性分数阶微分方程D0+α(t)=f(t,u(t)),0<t<1,在不同的两点边值条件下相应的Green函数的性质.作为Green函数性质的应用,我们应用Leray-Schauder非线性抉择,锥不动点定理等非线性分析方法给出奇异和非奇异,正的和半正边值问题多重正解的存在性,同时我们也给出奇异问题正解的存在唯一性.本章我们讨论以下几种情况：1<α≤2,2<α≤3,3<α≤4.第四章,我们考虑非线性分数阶微分方程D0+αu(t)=f(t,u(t)),0<t<1,在不同的三点边值条件下相应的Green函数的性质.以Green函数性质研究为基础,我们仍然应用非线性分析的方法和技巧给出奇异正的和半正边值问题多重正解的存在性,同时我们也给出奇异问题正解的存在唯一性.本章我们讨论以下几种情况：1<α≤2,2<α≤3,3<α≤4.更多还原

【Abstract】 In this paper, wo first present the necessary definitions from fractional calculustheory and some introduction about fractional diferential equation in Section1.In Section2, we state some known results and give some new results on the existenceof positive solutions of two integral equations.In Section3, we consider the properties of Green’s function for the nonlinear frac-tional diferential equationD₀₊^αu（t）=f （t, u（t））,0<t <1,with suitable two point boundary value problem. As an application of Green’s func-tion, we give some multiple positive solutions for singular and nonsingular, positone andsemipositone boundary value problems, and also we give uniqueness of solution for sin-gular problem by means of Leray-Schauder nonlinear alternative, a fixed-point theoremon cones etc.. Here we consider the case:1<α≤2,2<α≤3,3<α≤4.In Section4, we consider the properties of Green’s function for the nonlinear frac-tional diferential equationD₀₊^αu（t）=f （t, u（t））,0<t <1,with suitable three point boundary value problem. As an application of Green’s function,we also give some multiple positive solutions for singular positone and semipositoneboundary value problems, and also we give uniqueness of solution for singular problemby means of Leray-Schauder nonlinear alternative, a fixed-point theorem on cones etc..Here we consider the case:1<α≤2,2<α≤3,3<α≤4.更多还原