It is well known that the Sturm–Liouville equation has many applications in different areas of science. Thus, it is important to review different versions of the well-known equation. The technique of α-admissible α-ψ-contractions was introduced by Samet et al. in (Nonlinear Anal. 75:2154–2165, 2012). Our aim in this work is to study a fractional hybrid version of the Sturm–Liouville equation by mixing the technique of Samet. In fact, by using the technique of α-admissible α-ψ-contractions, we investigate the existence of solutions for the fractional hybrid Sturm–Liouville equation by using the multi-point boundary value conditions. Also, we review the existence of solutions for a fractional hybrid version of the problem under the integral boundary value conditions. Finally, we provide two examples to illustrate our main results.

众所周知，Sturm-Liouville方程在科学的不同领域都有许多应用。因此，重要的是要回顾众所周知的方程式的不同版本。的技术中α -admissible α - ψ -contractions通过沙美等人提出。于（非线性分析75：2154-2165，2012年）。我们在这项工作中的目的是通过混合Samet技术研究Sturm-Liouville方程的分数混合形式。实际上，通过使用α-容许α - ψ技术-收缩，我们使用多点边界值条件研究分数混合Sturm-Liouville方程解的存在性。另外，我们回顾了积分边值条件下问题的分数混合版本的解的存在性。最后，我们提供两个示例来说明我们的主要结果。