分数阶时滞微分方程边值问题解的存在性与唯一性  

Existence and Uniqueness of Solutions for Boundary Value Problems of Fractional Differential Equations with Delay

作　　者：李帅 张志信[1] 蒋威[1] LI Shuai;ZHANG Zhixin;JIANG Wei(School of Mathematical Sciences,Anhui University,Hefei 230601)

机构地区：[1]安徽大学数学科学学院,合肥230601

出　　处：《工程数学学报》2021年第5期700-708,共9页Chinese Journal of Engineering Mathematics

基　　金：国家自然科学基金(11371027,11471015,11601003);安徽省自然科学基金(1608085MA12);高等学校博士点专项科研基金(20123401120001).

摘　　要：时滞因素对分数阶微分系统的解有重要影响,系统解的变化不仅取决于现在状态,而且受到过去状态的约束,因此在分数阶微分系统中考虑时滞效应具有重要的意义.本文主要研究一类分数阶时滞微分方程边值问题解的存在性与唯一性问题.首先通过构建Green函数并利用分数阶微积分的相关性质给出该类分数阶时滞微分方程的等价方程.然后将此等价方程的求解问题转换为Banach空间中的不动点问题.再利用Banach压缩映像原理和Schauder不动点定理分别给出了保证分数阶时滞微分方程边值问题解的唯一性与存在性的充分性条件.最后,通过两个例子验证了定理结论的有效性.在考虑该类边值问题解的存在唯一性时,本文利用Banach空间中一个特殊的范数,得到系统解的存在唯一性充分性条件较以往的研究结果更为简单.这种方法是新颖的,在后续的研究过程中将尝试利用这种方法讨论带有时滞的分数阶Langevin方程边值问题的正解存在唯一性问题.The time delay factor has an important influence on the solution of the fractional differential system.The change of the system solution not only depends on the current state,but also is constrained by the past state.Therefore,it is of great significance to consider the delay effect in the fractional differential system.In this paper,the existence and uniqueness of solutions for a class of boundary value problem of fractional order differential equation with delay are disscussed.Firstly,the equivalent equation of fractional order differential equation with delay is given by constructing the Green function and the relevant properties of the fractional calculus.Then,the problem of solving this equivalent equation is transformed into a fixed point problem in Banach space.By applying Banach contraction mapping principle and Schauder fixed point theorems,some sufficient conditions ensure that the uniqueness and the existence of solutions of boundary value problem for fractional order differential equation with delay are obtained respectively.Finally,two examples are presented to verify the theoretical results.In this paper,we use a special norm in Banach space to obtain sufficient conditions for the existence and uniqueness of the solution of the system,which is simpler than the previous results.This method is novel and can be used to discuss the existence and uniqueness of positive solutions for boundary value problems of fractional Langevin equations with time delay.

关 键 词：分数阶 时滞 边值问题 CAPUTO导数 不动点定理 

分 类 号：O175.8[理学—数学]