In this paper, we investigate a class of fractional hybrid differential equations with impulses, which can be seen as nonlinear differential equations with a quadratic perturbation of second type and a linear perturbation in partially ordered Banach algebras. We deduce the existence and approximation of a mild solution for the initial value problems of this system by applying Dhage iteration principles and related hybrid fixed point theorems. Compared with previous works, we generalize the results to fractional order and extend some existing conclusions for the first time. Meantime, we take into consideration the effect of impulses. Our results indicate the influence of fractional order for nonlinear hybrid differential equations and improve some known results, which have wider applications as well. A numerical example is included to illustrate the effectiveness of the proposed results.

中文翻译：

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偏序 Banach 代数中带冲量的分数阶混合微分方程分析

在本文中，我们研究了一类带冲量的分数阶混合微分方程，它可以看作是具有第二类二次扰动和偏序巴拿赫代数中的线性扰动的非线性微分方程。我们通过应用Dhage迭代原理和相关的混合不动点定理，推导出该系统的初值问题的温和解的存在性和逼近性。与以前的工作相比，我们将结果推广到分数阶，并首次扩展了一些现有的结论。同时，我们考虑了脉冲的影响。我们的结果表明分数阶对非线性混合微分方程的影响并改进了一些已知结果，这些结果也具有更广泛的应用。