分数阶微分方程的概周期型解基础数学专业论文.docxVIP

哈尔滨工业大学理 哈尔滨工业大学理学硕士学位论文 -I -II- Abstract This paper consists of three parts: the first part introduces the space of almost periodic type functions and the related nature of the gradual expansion; the second part describes the interger-order differential equations of the almost periodic solutions of relevant theories; the last part takes on the existence and uniqueness of almost periodic type solutions of abstract fractional differential equations. Theory of almost periodic function was first proposed by Danish mathematician H. Bohr in 1925-1926, and later was developed by S. Bochner, H. Weyl, A. Besicovitch, and V.V.Stepanov. Fractional calculus includes fractional derivatives and fractional integrals. It means to generalize the differentiation and integration into fractional and complex order. In a related article, the authors Daniela Araya introduced the concept of ? -resolvent families to prove the existence of almost automorphic mild solutions to the following fractional differential equation in the sense of Riemann-Liouville. Later, the author Hui- Sheng relaxed the conditions in literature and obtained the same results. In this paper, we utilize Banach fix-point theorem to get pseudo almost periodic mild solutions and pseudo almost automorphic mild solutions which extends the results in the paper introduced before to a certain extent under appropriate assumptions. Keywords: almost periodic type functions ； fractional differential equations； mild solution； solution operator -I -III- 目 录 摘 要 I Abstract II 第 1 章 绪论 1 1.1 课题来源及研究的目的和意义 1 1.2 国内外在该方向的研究现状及分析 1 1.3 本文的主要工作及结构 1 第 2 章 概周期型函数的定义和基本理论 3 2.1 使用的符号及定义 3 2.2 概周期型函数空间之间的包含关系 7 2.3 概周期函数的等价定义 8 2.4 概周期型函数的相关性质及定理 9 2.5 本章小结 15 第 3 章 整数阶微分方程的概周期型函数解 16 3.1 基本定义及性质 16 3.2 抽象微分方程概周期型解的基本理论 17 3.3 本章小结 18 第 4 章 分数阶微分方程的概周期型函数解 19 4.1 预备知识 19 4.2 分数阶微分方程的伪概周期解 20 4.3 本章小结 27 结 论 28 参考文献 29 哈尔滨工业大学硕士学位论文原创性声明 32 哈尔滨工业大学硕士学位论文使用授权书 32 致 谢 33 - - 第1章 绪论 1.1 课题来源及研究的目的和意义