In this paper, we investigate new types of nonlocal implicit problems involving piecewise Caputo fractional operators. The existence and uniqueness results are proved by using some fixed point theorems. Furthermore, we present analogous results involving piecewise Caputo-Fabrizio and Atangana–Baleanu fractional operators. The ensuring of the existence of solutions is shown by Ulam-Hyer’s stability. At last, two examples are given to show and approve our outcomes.

中文翻译：

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分段非局部隐式分数阶微分方程解的定性研究

在本文中，我们研究了涉及分段 Caputo 分数运算的新型非局部隐式问题。利用一些不动点定理证明了结果的存在性和唯一性。此外，我们提出了涉及分段 Caputo-Fabrizio 和 Atangana–Baleanu 分数运算符的类似结果。Ulam-Hyer 的稳定性表明解决方案存在的保证。最后，给出了两个例子来展示和验证我们的结果。