【作者】 李晓艳；

【导师】 蒋威；

【作者基本信息】 安徽大学 ， 基础数学， 2012， 博士

【摘要】 最近几十年,由于分数阶微分方程在自然科学、物理学、工程学等很多领域的广泛应用,引起了人们较多的关注.分数阶差分方程由于其数学模型的不断出现及对微分方程近似计算的需要,也在近两年逐渐成为了学者关注的研究课题.本文主要讨论了分数阶微分方程与分数阶泛函微分方程解的延拓理论、初值问题的解与正解存在性与唯一性、边值问题正解的存在性；分数阶差分方程解的存在性与唯一性、边值问题正解的存在性.本文主要研究结果如下：1.探讨了分数阶微分方程解的延拓问题.分数阶微分方程与泛函微分方程解的存在性结果有很多(文献[10-12],[21-24]等),但是我们发现在这些结果中解的存在区间大多有限制,这给研究结果的实际使用带来了不便,因为在实际使用中我们总是希望解的存在区间越大越好.我们首先研究了一般分数阶微分方程解的延拓问题,得出解在什么情况下可以延拓,可以延拓至什么程度?其次我们讨论了分数阶泛函方程解的延拓,分别对含有无限和有限时滞的分数阶微分方程解的延拓予以分析,得出了有关结论.2.研究了分数阶微分方程正解的存在性与唯一性问题.首先,我们考虑了一类等式左边含有未知函数分数阶导数常系数多项式且右边非线性项含有未知函数导数的微分方程正解的存在性与唯一性,主要利用了正规锥上的不动点定理及压缩映像原理,得出方程正解存在与唯一的充分条件.其次,我们讨论了分数阶泛函微分方程的边值问题正解的存在性,通过探寻Green函数的表达式及性质,利用不动点定理,得出相关定理.3.讨论了分数阶差分方程解的存在性问题.我们主要关注了一类分数阶差分方程解的表达式,并使用压缩映像原理证明了解的存在唯一性.还研究了一类分数阶差分方程三点边值问题解的存在唯一性及正解的存在性问题.更多还原

【Abstract】 In recent years, the fractional differential equation have caused people’wides-pread concern,because of extensive applications in the natural science, physics, engineering, and many other areas. Meanwhile, Fractional discrete equations ap-peared ceaselessly in mathematical model and the need of approximation of Frac-tional differential equation approximation. The study of fractional discrete equa-tion have become a new research topic in recent years. In this paper we mainly discuss the fractional differential equation theory, for example the continuation of solutions for initial value problems, the existence and uniqueness of the solution and boundary value problem positive solution. Also the existence and uniqueness of solution of initial value problem and positive solution of boundary value problem to fractional discrete equations. Our main results are as follows:1. We study the continuation problem of fractional differential equations, we find that there are many existence results of fractional differential equations and functional differential equations([10-12],[21-24] and so on), but we also find that in these results solutions are restricted to the given interval. But in practical using, these conclusions bring inconvenience, we always hope the existence interval of solutions the bigger the better.2.We discuss the positive solution existence and uniqueness problems of frac-tional differential equation. Firstly, we consider a class of equations with the poly-nomial with constant coefficient of fractional derivative of unknown function on the left, and on the right side there contain fractional derivative of unknown function in the nonlinear term, the conclusion about the existence and uniqueness of positive solution to these equations are given, we will use the regular fixed-point theorem in cones and contraction mapping principle to get the conclusion. And we discuss the existence of positive solution for boundary value problem to fractional functional differential equation, through exploring the expression of Green function and the property, we get the related theorem by using the fixed point theorem.3. We search for the conditions of the existence of solutions and positive solutions for the fractional discrete equation. We will mainly focus on a class of fractional order differential equation solution, and to prove the existence and uniqueness of the solutions of these equations using the compression of solutions and mapping principle. At last, we study existence the positive solution for boundary value problem of fractional discrete equations.更多还原