We discuss the existence and uniqueness of solutions for the Langevin fractional differential equation and its inclusion counterpart involving the Hilfer fractional derivatives, supplemented with three-point boundary conditions by means of standard tools of the fixed-point theorems for single and multivalued functions. We make use of Banach’s fixed-point theorem to obtain the uniqueness result, while the nonlinear alternative of the Leray-Schauder type and Krasnoselskii’s fixed-point theorem are applied to obtain the existence results for the single-valued problem. Existence results for the convex and nonconvex valued cases of the inclusion problem are derived via the nonlinear alternative for Kakutani’s maps and Covitz and Nadler’s fixed-point theorem respectively. Examples illustrating the obtained results are also constructed. (2010) Mathematics Subject Classifications. This study is classified under the following classification codes: 26A33; 34A08; 34A60; and 34B15.

中文翻译：

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带希尔弗分数阶导数的兰文方程的三点边值问题

我们讨论了Langevin分数阶微分方程及其包含Hilfer分数阶导数的包含对应项的解的存在性和唯一性，并通过单点和多值函数不动点定理的标准工具补充了三点边界条件。我们利用Banach不动点定理获得唯一性结果，而Leray-Schauder型非线性替代和Krasnoselskii不动点定理则被用于获得单值问题的存在性结果。包含问题的凸值和非凸值情况的存在性结果分别通过Kakutani映射的非线性替代以及Covitz和Nadler定点定理得出。还构造了说明获得的结果的实例。（2010）数学学科分类。这项研究分为以下分类代码：26A33; 34A08；34A60；和34B15。