Existence of Positive Solutions for Boundary Value Problems of Fractional Differential Equations with Sign-Changing Nonlinearity

本文探讨了一类符号变化的非线性项对分数阶微分方程边值问题正解存在性的影响。首先，求解非线性分数阶微分方程边值问题对应线性问题的Green函数及其性质。这些性质为所研究问题正解的存在性提供了基础。接下来，将所研究问题解的存在性转化为积分方程的可解性。针对非线性项变号的特点，选择一个辅助积分方程，通过该积分方程定义算子。进一步探究算子的性质，我们最终在适当的空间中运用Guo-Krasnoselskii不动点定理，证明了所研究的非线性边值问题至少存在一个正解，并举例说明所得理论结果的正确性。This paper explores the influence of a class of nonlinear terms with varying symbols on the existence of positive solutions for boundary value problems of fractional differential equations. Firstly, construct Green’s function associated with the linear case of the given nonlinear fractional differential equation under boundary conditions and study its characteristics. These properties provide a basis for the existence of the positive solution of the studied problem. Next, transform the existence of the solution of the studied problem into the solvability of the integral equation. In view of the characteristic of the change sign of the nonlinear term, construct an auxiliary integral equation, thereby defining the operator through this integral equation. Further exploring the properties of the operator, finally, with the help of the Guo-Krasnoselskii fixed point theorem, the existence of positive solutions to the studied problem is obtained and the correctness of the obtained theoretical results is illustrated with examples.