具有积分边界条件的分数阶微分方程边值问题正解的存在性
The Existence of Positive Solutions for Boundary Value Problem of Fractional Differential Equation with Integral Boundary Conditions
摘要: 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。
Abstract: 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.
文章引用：尚海蕊. 具有积分边界条件的分数阶微分方程边值问题正解的存在性[J]. 应用数学进展, 2025, 14(6): 222-235. https://doi.org/10.12677/aam.2025.146314

参考文献

[1] Apostol, T.M. (1991) Calculus, Volume 1. John Wiley and Sons.
[2] Kiryakova, V.S. (2012). Some Operational Tools for Solving Fractional and Higher Integer Order Differential Equations: A Survey on Their Mutual Relations. American Institute of Physics Conference Proceedings, 1497, 273-289.[CrossRef] 
[3] Lan, C.X., Sun, J.P. and Zhao, Y.H. (2020) Positive Solution of Fractional Differential Equation with Integral Boundary Conditions. Baltica, 33, 26-35.
[4] Guezane-Lakoud, A. and Khaldi, R. (2012) Solvability of a Fractional Boundary Value Problem with Fractional Integral Condition. Nonlinear Analysis: Theory, Methods & Applications, 75, 2692-2700. [Google Scholar] [CrossRef] 
[5] Xu, J., Wei, Z. and Dong, W. (2012) Uniqueness of Positive Solutions for a Class of Fractional Boundary Value Problems. Applied Mathematics Letters, 25, 590-593. [Google Scholar] [CrossRef] 
[6] Zhang, X., Liu, L., Wu, Y. and Lu, Y. (2013) The Iterative Solutions of Nonlinear Fractional Differential Equations. Applied Mathematics and Computation, 219, 4680-4691. [Google Scholar] [CrossRef] 
[7] Zhong, Q., Zhang, X., Lu, X. and Fu, Z. (2018) Uniqueness of Successive Positive Solution for Nonlocal Singular Higher-Order Fractional Differential Equations Involving Arbitrary Derivatives. Journal of Function Spaces, 2018, Article ID: 6207682. [Google Scholar] [CrossRef] 
[8] Zhai, C., Wang, W. and Li, H. (2018) A Uniqueness Method to a New Hadamard Fractional Differential System with Four-Point Boundary Conditions. Journal of Inequalities and Applications, 2018, Article No. 207. [Google Scholar] [CrossRef] [PubMed]
[9] Choi, H., Sin, Y. and Jong, K. (2020) Existence Results for Nonlinear Multiorder Fractional Differential Equations with Integral and Antiperiodic Boundary Conditions. Journal of Applied Mathematics, 2020, Article ID: 1212040. [Google Scholar] [CrossRef] 
[10] Su, C., Sun, J. and Zhao, Y. (2017) Existence and Uniqueness of Solutions for BVP of Nonlinear Fractional Differential Equation. International Journal of Differential Equations, 2017, Article ID: 4683581. [Google Scholar] [CrossRef] 
[11] Bai, Z. and Qiu, T. (2009) Existence of Positive Solution for Singular Fractional Differential Equation. Applied Mathematics and Computation, 215, 2761-2767. [Google Scholar] [CrossRef] 
[12] Liu, W. and Zhuang, H. (2017) Existence of Solutions for Caputo Fractional Boundary Value Problems with Integral Conditions. Carpathian Journal of Mathematics, 33, 207-217. [Google Scholar] [CrossRef] 
[13] Chandran, K., Gopalan, K., Zubair, S.T. and Abdeljawad, T. (2021) A Fixed Point Approach to the Solution of Singular Fractional Differential Equations with Integral Boundary Conditions. Advances in Difference Equations, 2021, Article No. 56. [Google Scholar] [CrossRef] 
[14] Agarwal, R.P., Hodis, S. and O’Regan, D. (2019) 500 Examples and Problems of Applied Differential Equations. Springer.
[15] Adigüzel, R.S., Aksoy, Ü., Karapinar, E. and Erhan İ.M., (2020) On the Solution of a Boundary Value Problem Associated with a Fractional Differential Equation. Mathematical Methods in the Applied Sciences, 47, 10928-10939.
[16] Debao, Y. (2021) Existence and Uniqueness of Positive Solutions for a Class of Nonlinear Fractional Differential Equations with Singular Boundary Value Conditions. Mathematical Problems in Engineering, 2021, Article ID: 6692620. [Google Scholar] [CrossRef] 
[17] Cabada, A. and Hamdi, Z. (2014) Nonlinear Fractional Differential Equations with Integral Boundary Value Conditions. Applied Mathematics and Computation, 228, 251-257. [Google Scholar] [CrossRef] 
[18] Li, M., Sun, J. and Zhao, Y. (2020) Positive Solutions for BVP of Fractional Differential Equation with Integral Boundary Conditions. Discrete Dynamics in Nature and Society, 2020, Article ID: 6738379. [Google Scholar] [CrossRef] 
[19] Samko, S.G., Kilbas, A.A. and Marichev, O.I. (1993) Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers.
[20] Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. (2006) Theory and Applications of Fractional Differential Equations. Elsevier.
[21] Guo, D.J. and Lakshmikantham, V. (1988) Nonlinear Problems in Abstract Cones. Academic Press.
[22] Brodsky, A.R., Krasnoselskii, M.A., Flaherty, R.E. and Boron, L.F. (1967) Positive Solutions of Operator Equations. The American Mathematical Monthly, 74, 343. [Google Scholar] [CrossRef] 
[23] Granas, A. and James, D. (2003) Fixed Point Theory. Springer.

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