非线性泛函分析在分数阶脉冲微分系统中的应用

Recently, the research of fractional differential equations has been paid attention widely. This is because a large number of systems in the nature are associated with the fractional derivatives, such as the elastic materials with memory and genetic characteristics, mass transfer, heat conduction and so on. Fractional systems are power systems described by fractional differential equations (sometimes in the form of plural). It plays a very important role both in theory and practice. In the existing literature, the research of fractional differential systems are focused on the uniqueness of Initial value problems, the existence of two point and three point boundary value problems. However, the research of nonlocal BVPs has not been seen as far as we know. Nonlinear functional analysis is an important role in the research of integer differential equations. This project plans to research the behavior of solutions of fractional differential systems with impulsive, to create the judgment of existence and multiplicities, and then research the image characteristics of the solutions using associated software. To solve the above problems can not only broaden the development field of fractional order system, but also provide new ideas and methods for the research of complex dynamic systems, and so it has a very important theoretical and practical significance.

近年来，分数阶微分方程的研究受到了学者的广泛关注，这是因为自然界的许多系统都是与分数阶导数相关的，例如在具有记忆与遗传特性的弹性材料、质量转移和热传导等动态过程等。分数阶系统就是由分数阶次（有时为复数形式的分数阶次）的微分方程所描述的动力系统，无论在理论还是在实践中它都有着非常重要的作用。在已有的文献中，对分数阶微分系统的研究主要集中在分数阶微分方程初值问题解的唯一性，有限区间上两点及三点边值问题解的存在性, 但对更多非局部边值问题，尤其是带脉冲的非局部边值问题的研究还不多见。非线性泛函分析是研究整数阶微分方程动力系统的重要工具，本项目拟利用该工具研究带脉冲的分数阶微分系统解的性态，建立这类边值问题解及多解存在的判断依据，进而利用相关软件研究解的图像特点。解决上述问题不仅可以拓展分数阶微分系统的应用领域，还能为复杂动力学系统的研究提供新的思想和方法，具有十分重要的理论意义。

本项目使用非线性泛函分析的方法研究了分数阶脉冲微分方程边值问题，主要就以下两个方面开展了研究：(1) 非线性项含有分数阶导数的无穷区间上带脉冲的分数阶微分方程边值问题。非线性项中是否含有分数阶导数对空间的构造和全连续性的讨论影响很大。本项目针对几种不同的边界条件,构造了不同的空间研究了解的存在性、多解性以及解的性态，并利用相关软件分析某些具体问题解的性态。(2) 脉冲点处函数和导数值均有跳跃且脉冲量彼此影响的无穷区间上带脉冲的分数阶微分方程边值问题。本项目讨论了脉冲点处函数和导数值均有跳跃且脉冲量彼此影响的情形，针对Sturm-Liouville 两点、多点以及带有积分边界条件的分数阶微分方程边值问题，得到了解的存在性、多解性以及解的性态。