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贵 州 大 学 2012届硕士研究生学位论文 分数阶微分方程 初值问题与非局部问题 学科专业： 基础数学 研究方向： 偏微分方程与控制论 导 师： 韦维、王锦荣 研 究 生： 董希望 中国﹒贵州﹒贵阳 2012年5 月 目录 摘要 I Abstract II 第一章 绪论 1 1.1 研究背景 1 1.2 国内外研究现状 2 1.3 研究主要内容 3 第二章 准备知识 4 2.1 分数阶导数与积分的定义 4 2.2 重要引理和不动点定理 5 2.3 常用积分不等式 6 第三章：分数阶微分方程初值问题 8 3.1解的存在性 8 3.2 极值解 17 第四章 分数阶微分方程非局部问题 26 第五章：进一步研究工作 35 参考文献 36 攻读硕士学位期间科研、论文和获奖情况 39 致谢 40 noselskii不动点定理结合解的先验估计，获得半线性非自治分数微分方程解的存在性和唯一性新结果。其次，引进- Lipschitz，-Carathéodory和- Chandrabha定义，运用混合不动点定理获得一类分数微分方程极值解的存在性。最后，我们在较弱的条件下讨论一类分数微分方程非局部问题解的存在性。 关键词：分数阶微分方程，初值问题，非局部问题，不动点定理，存在性，唯一性、极值解。 Initial value problem and nonlocal problem of fractional differential equations Abstract Fractional differential is a useful tool in describing the essence of memory and genetic process，and Fractional order impulsive differential equation is suitable for describing some real problems which have impulsive phenomenon、the essence of memory and genetic process. Therefore,it is widely used in the field of viscoelastic、electrochemistry and electromagnetic .This thesis studies the initialvalue problem and nonlocal problem of fractional order differential equations in a space of infinite dimension. Canceling the continuity of nonlinear term in the right side, we studies the existence and uinqueness of solution and the existence of extremal solution.Firstly, applying Banach fixed point theorem 、Krasnoselskii fixed point theorem and solution prior estimate to obtain the existence and uniqueness of solution of semilinear non-autonomous fractional differential equation .Secondly, introducing the definition of - Lipschitz，-Carathéodory and- Chandrabha. Applying mixed fixed point theorem to obtain the existence of extremal solution of an class of fraction differential equation. Finally, we discuss the existence of solution of an class of nonlocal fraction differential equation in the weaker condition. Keywords: Fractional differential equation initial value nonlocal fixed theorem