In this paper, we introduce a new concept of dual anti-periodic boundary conditions. One of these conditions relates to the end points of an interval of arbitrary length, while the second one involves two nonlocal positions within the interval. Equipped with these conditions, we present the criteria for the existence of solutions for a fractional integro-differential equation involving two Caputo fractional derivatives of different orders and a Riemann-Liouville integral. Our study relies on the modern methods of functional analysis. Examples are constructed for illustrating the obtained results.

中文翻译：

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具有双重反周期边界条件的分数阶积分微分方程

在本文中，我们引入了双重反周期边界条件的新概念。这些条件之一涉及任意长度间隔的端点，而第二个条件涉及该间隔内的两个非局部位置。配备了这些条件，我们给出了分数积分微分方程的解的存在性准则，该分数微分方程包含两个不同阶的Caputo分数阶导数和Riemann-Liouville积分。我们的研究依赖于功能分析的现代方法。举例说明所获得的结果。