【作者】 吕志伟；

【导师】 梁进； 肖体俊；

【作者基本信息】 中国科学技术大学 ， 应用数学， 2011， 博士

【摘要】 本文主要使用非线性泛函分析中的不动点定理与不动点指数理论以及迭代方法和锥理论研究了几类整数阶及分数阶非线性微分方程初边值问题解与多解的存在性。全文共分六章。第一章简要地介绍了本文的研究背景和研究工作。第二章是预备知识,主要介绍了分数阶积分与导数的基本概念和一些性质及有关锥的一些基础知识和相应的定理。第三章研究了三阶三点边值问题(?)借助于u与u之间的关系,运用不动点指数定理和不动点定理,我们得到了多个正解的存在性。第四章研究了Banach空间中两类脉冲微分方程边值问题。4.1节研究了Banach空间中二阶奇异脉冲微分方程(?)通过构造一个凸闭集来克服奇异性,利用连续算子的不动点定理我们得到了解的存在性。4.2节研究了Banach空间中带积分边值条件的二阶脉冲微分方程本节利用Green函数的性质结合锥理论与不动点指数定理得到了多个正解的存在性。第五章考虑了分数阶微分方程m点边值问题(?)我们用迭代的方法结合Green函数的性质和分析技巧得到最大最小正解,与上下解方法不同。第六章在Banach空间中研究了两类分数阶微分方程的初值问题。6.1节在Banach空间中研究了分数阶微分方程(?)通过一个估计,利用Mo¨nch不动点定理我们得到了解的存在性。6.2节在Banach空间中利用锥理论研究了分数阶微分方程(?)利用比较定理,迭代技术,结合锥理论得到了最大最小解及最小正解的存在性。更多还原

【Abstract】 This thesis mainly investigates the existence of solutions to initial and boundaryvalue problems for several nonlinear differential equations of integer order and frac-tional order by utilizing the fixed point theorems, fixed point index theory of nonlinearfunctional analysis, iterative technique and cone theory. There are six chapters in thisthesis.In Chapter 1, the research background and results of this thesis is brie?y pre-sented.In Chapter 2, the contents are about preliminaries, mainly including the defini-tions, some properties of fractional integral and derivative and the basic definitions andtheorems of cone.In Chapter 3, the following third-order three-point boundary value problems(?)is studied. By utilizing the connection between u and u , the fixed point indextheory and the fixed point theorem, we get the existence of multiple positive solutions.In Chapter 4, two classes of boundary value problems of impulsive differentialequations in Banach spaces are discussed.In Section 4.1, the existence of solutions for the following second order singulardifferential equations in Banach spaces(?)is considered. By constructing a convex closed set to overcome the singularity, andusing the fixed point theorem for continuous operators, we obtain the existence ofsolution. In Section 4.2, the following second order differential equations with integralboundary condition in Banach spaces(?)is investigated. Combining the properties of Green function with cone theory and thefixed point index theory, we show the existence of multiple positive solutions.In Chapter 5, we deal with m-point boundary value problems for nonlinear frac-tional differential equations(?)Using the properties of the Green function, analysis technique and iterative technique,we get the existence of minimal and maximal positive solutions, and the method usedis different from the method of upper and lower solutions.In Chapter 6, we consider two classes of initial value problems of fractional dif-ferential equations.In Section 6.1, the following fractional differential equations with integral bound-ary condition (?)is studied. By an estimation and the Mo¨nch fixed point theorem, we obtain the exis-tence of solution.In Section 6.2, the existence of solutions for fractional differential equations(?)is investigated. By using a comparability theorem, iterative technique and cone theory,we get the existence of minimal and maximal solutions and minimal positive solution.更多还原