In the current manuscript, we investigate existence of solutions to a coupled system of fractional hybrid differential equations (FHDEs). With the help of mixed type Lipschitz and Caratheodory conditions, some conditions for the existence of solutions to the considered problem are established. Considering the tools of nonlinear analysis and hybrid fixed points theory, we establish our results. Further some new type results about stability including Ulam-Hyers (UH), generalized Ulam-Hyers (GUH) stability, Ulam-Hyers-Rassias (UHR) and generalized Ulam-Hyers-Rassias (GUHR) stability are developed. A test problem is given to demonstrate the establish results.

中文翻译：

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非线性分数阶混合微分方程系统的稳定性理论

在当前的手稿中，我们研究分数混合微分方程（FHDE）耦合系统解的存在性。借助混合型Lipschitz条件和Caratheodory条件，为存在的问题的解决方案建立了一些条件。考虑到非线性分析和混合不动点理论的工具，我们确定了结果。还开发了一些有关稳定性的新型结果，包括Ulam-Hyers（UH），广义Ulam-Hyers（GUH）稳定性，Ulam-Hyers-Rassias（UHR）和广义Ulam-Hyers-Rassias（GUHR）稳定性。给出一个测试问题以证明确定的结果。