In this work, we give sufficient conditions to investigate the existence and uniqueness of solution to fractional-order Langevin equation involving two distinct fractional orders with unprecedented conditions (three-point boundary conditions including two nonlocal integrals). The problem is introduced to keep track of the progress made on exploring the existence and uniqueness of solution to the fractional-order Langevin equation. As a result of employing the so-called Krasnoselskii and Leray-Schauder alternative fixed point theorems and Banach contraction mapping principle, some novel results are presented in regarding to our main concern. These results are illustrated through providing three examples for completeness.

中文翻译：

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分数阶Langevin方程解的存在唯一性。

在这项工作中，我们提供了充分的条件来研究分数阶Langevin方程的解的存在性和唯一性，该方程涉及两个具有不同的分数阶且具有前所未有的条件（三点边界条件包括两个非局部积分）。引入问题是为了跟踪在探索分数阶Langevin方程解的存在性和唯一性方面取得的进展。由于采用了所谓的Krasnoselskii和Leray-Schauder替代不动点定理和Banach压缩映射原理，对于我们的主要关注点，提出了一些新颖的结果。通过提供三个完整示例说明了这些结果。