Results reported in this paper study the existence and stability of a class of implicit generalized proportional fractional integro-differential Langevin equations with nonlocal fractional integral conditions. The main theorems are proved with the help of Banach’s, Krasnoselskii’s, and Schaefer’s fixed point theorems and Ulam’s approach. Finally, an example is given to demonstrate the applicability of our theoretical findings.

中文翻译：

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广义Caputo比例分数导数的积分微分Langevin方程的非局部边值问题

本文报道的结果研究了一类具有非局部分数积分条件的隐式广义比例分式积分-微分Langevin方程的存在性和稳定性。主要定理在Banach，Krasnoselskii和Schaefer的不动点定理以及Ulam的方法的帮助下得到了证明。最后，给出一个例子来证明我们的理论发现的适用性。