The Existence of Positive Solutions for Boundary Value Problem of Fractional Differential Equation with Integral Boundary Conditions

分数阶微积分是对函数进行非整数阶积分和微分的定量分析，它是整数阶微积分的推广，且在实际应用过程中一直彰显着其独特优势和不可替代性。分数阶微分方程已应用于生物传染病、金融市场、控制系统、异常扩散等多个领域。目前，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，对具有积分边界条件的非线性边值问题进行深入探讨。首先讨论对应线性边值问题的Green函数解，接着分析Green函数的性质，然后利用锥上的不动点定理得到边值问题正解的存在性结果，最后举例说明结果的正确性。Fractional calculus is a quantitative analysis of non-integer order integration and differentiation of functions. It is an extension of integer order calculus and has always demonstrated its unique advantages and irreplaceability in practical applications. Fractional order differential equations have been applied in various fields such as biological infectious diseases, financial markets, control systems, and anomalous diffusion. At present, the existence and uniqueness of solutions to boundary value problems of fractional differential equations is one of the key research topics. This article will use functional analysis and other related tools to explore in depth nonlinear boundary value problems with integral boundary conditions. Firstly, we will discuss the Green function solution for the corresponding linear boundary value problem. Then, we will analyze the properties of the Green function and use the fixed point theorem on cones to obtain the existence of positive solutions for the boundary value problem. Finally, we will give an example to demonstrate the correctness of the results.