In this paper, we present a new type of Gronwall inequality and discuss some particular cases. We apply these results to investigate the uniqueness and $\delta$-Ulam–Hyers–Rassias stability of mild solutions of a fractional differential equation with non-instantaneous impulses in a P$\delta$-normed Banach space. In this sense, we present an example, in order to elucidate one of the results discussed.

中文翻译：

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关于Hilfer脉冲分数阶微分方程的温和解的一个注记

在本文中，我们提出了一种新型的Gronwall不等式，并讨论了一些特殊情况。我们将这些结果用于调查唯一性和$\delta$P中具有非瞬时脉冲的分数阶微分方程的温和解的-Ulam–Hyers–Rassias稳定性$\delta$-规范的Banach空间。从这个意义上讲，我们提供一个示例，以阐明所讨论的结果之一。