In this paper, we consider fractional Langevin equation and derive a formula of solutions for fractional Langevin equation involving two Caputo fractional derivatives. Secondly, we implement the concept of Ulam–Hyers as well as Ulam–Hyers–Rassias stability. Then, we choose Generalized Diaz–Margolis’s fixed point approach to derive Ulam–Hyers as well as Ulam–Hyers–Rassias stability results for our proposed model, over generalized complete metric space. We give several examples which support our main results.

中文翻译：

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分数阶Langevin方程的存在论与乌拉姆稳定性

在本文中，我们考虑分数Langevin方程并推导出包含两个Caputo分数阶导数的分数Langevin方程的解公式。其次，我们实现了 Ulam-Hyers 和 Ulam-Hyers-Rassias 稳定性的概念。然后，我们选择广义 Diaz-Margolis 不动点方法在广义完全度量空间上为我们提出的模型推导出 Ulam-Hyers 和 Ulam-Hyers-Rassias 稳定性结果。我们给出了几个支持我们主要结果的例子。