Langevin differential equations with fractional orders play a significant role due to their applications in vibration theory, viscoelasticity and electrical circuits. In this paper, we mainly study the explicit analytical representation of solutions to a class of Langevin time-delay differential equations with general fractional orders, for both homogeneous and inhomogeneous cases. First, we propose a new representation of the solution via a recently defined delayed Mittag-Leffler type function with double infinite series to homogeneous Langevin differential equation with a constant delay using the Laplace transform technique. Second, we obtain exact formulas of the solutions of the inhomogeneous Langevin type delay differential equation via the fractional analogue of the variation constants formula and apply them to vibration theory. Moreover, we prove the existence and uniqueness problem of solutions of nonlinear fractional Langevin equations with constant delay using Banach’s fixed point theorem in terms of a weighted norm with respect to exponential functions. Furthermore, the concept of stability analysis in the mean of solutions to Langevin time-delay differential equations based on the fixed point approach is proposed. Finally, an example is given to demonstrate the effectiveness of the proposed results.

具有分数阶的朗之万微分方程由于其在振动理论、粘弹性和电路中的应用而发挥着重要作用。在本文中，我们主要研究齐次和非齐次情况下一类具有一般分数阶的Langevin时滞微分方程的解的显式解析表示。首先，我们通过最近定义的具有双无穷级数的延迟 Mittag-Leffler 型函数对具有恒定延迟的齐次 Langevin 微分方程使用拉普拉斯变换技术提出了解决方案的新表示。其次，我们通过变分常数公式的分数模拟得到非齐次Langevin型时滞微分方程解的精确公式，并将其应用到振动理论中。而且，我们根据指数函数的加权范数，使用 Banach 不动点定理证明了具有恒定延迟的非线性分数阶 Langevin 方程解的存在性和唯一性问题。此外，提出了基于不动点法的Langevin时滞微分方程解均值稳定性分析的概念。最后，给出一个例子来证明所提出结果的有效性。提出了基于不动点法的Langevin时滞微分方程解均值稳定性分析的概念。最后，给出一个例子来证明所提出结果的有效性。提出了基于不动点法的Langevin时滞微分方程解均值稳定性分析的概念。最后，给出一个例子来证明所提出结果的有效性。