In the current manuscript, we examine the existence and uniqueness of solution for generalized Langevin equation involving two distinct fractional orders with a three-point boundary value problem. We implement the notions of fractional calculus simultaneously with immutable point types to create the existence and uniqueness results. To explore our problem, we apply the Banach contraction principle, Krasnoselskii's fixed point theorem, nonlinear alternative Leray-Schauder theorem and Leray-Schauder degree theorem. Expository examples are rended to clarify the applicability of our main results.

中文翻译：

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涉及具有三点边界条件的两个分数阶的Langevin方程

在当前的手稿中，我们研究了涉及两个不同分数阶和三点边值问题的广义Langevin方程解的存在性和唯一性。我们用不可变点类型同时实现分数演算的概念，以创建存在性和唯一性结果。为了探讨我们的问题，我们应用了Banach压缩原理，Krasnoselskii不动点定理，非线性替代Leray-Schauder定理和Leray-Schauder度定理。借用说明性的例子来阐明我们主要结果的适用性。