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高阶分数阶微分方程边值问题解的唯一性
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Abstract:
分数阶微积分是整数阶微积分的推广，主要研究任意阶微分和积分的理论和应用问题，由于其更加适合描述具有遗传和记忆特质的材料与过程，因此广泛用于解决混沌与湍流、随机游走、统计与随机过程、粘弹性力学、电化学等诸多领域所面临的问题。目前，高阶分数阶微分方程边值问题解的存在唯一性是研究的重点课题之一。本文将借助泛函分析等相关工具，对非线性项含未知函数分数阶导数的边值问题进行深入探讨。首先通过降阶将原边值问题转变成非线性项不含导数的等价边值问题，接着分析Green函数的性质，然后利用混合单调算子不动点定理得到边值问题解的唯一性结果，最后举例说明结果的正确性。
Fractional calculus is a generalization of integer calculus, mainly studies the theory and application of differential and integral of arbitrary order, because it is more suitable to describe materials and processes with genetic and memory characteristics, so it is widely used to solve problems in many fields such as chaos and turbulence, random walk, statistics and random process, viscoelastic mechanics, electrochemistry and so on. At present, the existence and uniqueness of solutions to boundary value problem of higher order fractional differential equation is one of the key research topics. In this paper, with the help of functional analysis and other relevant tools, the boundary value problem of nonlinear terms including fractional derivatives of unknown functions is deeply discussed. First, the original boundary value problem is transformed into an equivalent boundary value problem without derivative by reducing the order, then the properties of Green function are analyzed, and then the uniqueness of the solution of the boundary value problem is obtained by using the fixed point theorem of mixed monotone operator. Finally, an example is given to illustrate the correctness of the results.
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