The Sturm–Liouville differential equation is one of interesting problems which has been studied by researchers during recent decades. We study the existence of a solution for partial fractional Sturm–Liouville equation by using the α-ψ-contractive mappings. Also, we give an illustrative example. By using the α-ψ-multifunctions, we prove the existence of solutions for inclusion version of the partial fractional Sturm–Liouville problem. Finally by providing another example and some figures, we try to illustrate the related inclusion result.

中文翻译：

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关于部分分数 Sturm-Liouville 方程和包含

Sturm-Liouville 微分方程是近几十年来研究人员研究的有趣问题之一。我们通过使用α - ψ -收缩映射研究了部分分数阶 Sturm-Liouville 方程解的存在性。另外，我们给出了一个说明性的例子。通过使用α - ψ -多重函数，我们证明了部分分数 Sturm-Liouville 问题的包含版本的解的存在性。最后通过提供另一个例子和一些数字，我们试图说明相关的包含结果。