In this paper, we examine a coupled system of fractional integrodifferential equations of Liouville-Caputo form with nonlinearities depending on the unknown functions, as well as their lower-order fractional derivatives and integrals supplemented with coupled nonlocal and Erdélyi-Kober fractional integral boundary conditions. We explain that the topic discussed in this context is new and gives more analysis into the research of coupled boundary value problems. We have two results: the first is the existence result of the given problem by using the Leray-Schauder alternative, whereas the second referring to the uniqueness result is derived by Banach’s fixed-point theorem. Sufficient examples were also supplemented to substantiate the proof, and some variations of the problem were discussed.

中文翻译：

---

具有Erdélyi-Kober积分条件的Liouville-Caputo型分数积分微分方程耦合系统解的存在性和唯一性

在本文中，我们研究了基于未知函数的具有非线性的Liouville-Caputo形式的分数阶积分微分方程的耦合系统，以及它们的低阶分数阶导数和积分以及耦合的非局部和Erdélyi-Kober分数阶积分边界条件。我们解释说，在这种情况下讨论的主题是新的，并为耦合边值问题的研究提供了更多的分析。我们有两个结果：第一个是使用Leray-Schauder备选方案的给定问题的存在性结果，而第二个是指唯一性结果是通过Banach不动点定理得出的。还补充了足够的示例以证实证据，并讨论了该问题的一些变体。