In this work, we define Sturm–Liouville problems through Hilfer fractional derivative operator using its different types. First, we obtain an integral representation of solutions in eight different forms including Riemann–Liouville, Caputo, and ordinary form. These different forms are stemmed from the type of Hilfer fractional derivative. Then, we analyze the existence and uniqueness of solutions for the problem by the Banach fixed point theorem. Moreover, we investigate the behavior of solutions and analyze eigenvalues and eigenfunctions for three-point Hilfer fractional boundary value problem under different values of \(\alpha \) and types of \(\beta \).

在这项工作中，我们使用不同类型的Hilfer分数导数算子定义Sturm-Liouville问题。首先，我们获得了八种不同形式的解决方案的完整表示，包括黎曼-利维尔，卡普托和普通形式。这些不同形式源自希尔弗分数导数的类型。然后，我们通过Banach不动点定理来分析问题解的存在性和唯一性。此外，我们研究了在不同的\（\ alpha \）值和\（\ beta \）类型的情况下三点Hilfer分数边值问题的解的行为，并分析了特征值和特征函数。