Fractional Langevin equation describes the evolution of physical phenomena in fluctuating environments for the complex media systems. It is a sequential fractional differential equation with two fractional orders involving a memory kernel, which leads to non-Markovian dynamics and subdiffusion. Here by establishing a general solution of the linear fractional Langevin equations involving initial conditions with the help of well-known Mittag–Leffler functions and using the special properties of these functions, we construct a new comparison result related to linear fractional Langevin equation. Meanwhile, we investigate the existence of extremal solutions for nonlinear boundary value problems with advanced arguments. The method is a constructive method that yields monotone sequences that converge to the extremal solutions. At last an example is presented to illustrate the main results.

中文翻译：

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涉及非线性边界条件的分数阶Langevin方程极值解的存在性

分数朗之万方程描述了复杂介质系统在波动环境中物理现象的演变。它是一个具有两个分数阶的序列分数阶微分方程，涉及一个记忆核，导致非马尔可夫动力学和子扩散。在此，我们借助著名的 Mittag-Leffler 函数建立了包含初始条件的线性分数阶 Langevin 方程的通解，并利用这些函数的特殊性质，构造了与线性分数阶 Langevin 方程相关的新比较结果。同时，我们研究了具有高级参数的非线性边值问题的极值解的存在性。该方法是一种构造方法，它产生收敛到极值解的单调序列。