Real-world processes that display non-local behaviours or interactions, and that are subject to external impulses over non-zero periods, can potentially be modelled using non-instantaneous impulsive fractional differential equations or systems. These have been the subject of many recent papers, which rely on re-formulating fractional differential equations in terms of integral equations, in order to prove results such as existence, uniqueness, and stability. However, specifically in the non-instantaneous impulsive case, some of the existing papers contain invalid re-formulations of the problem, based on a misunderstanding of how fractional operators behave. In this work, we highlight the correct ways of writing non-instantaneous impulsive fractional differential equations as equivalent integral equations, considering several different cases according to the lower limits of the integro-differential operators involved.

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关于非瞬时脉冲分数阶微分方程及其等效积分方程

显示非局部行为或交互以及在非零周期内受到外部脉冲影响的现实世界过程，可以潜在地使用非瞬时脉冲分数阶微分方程或系统进行建模。这些是最近许多论文的主题，这些论文依赖于根据积分方程重新制定分数阶微分方程，以证明诸如存在性、唯一性和稳定性等结果。然而，特别是在非瞬时脉冲情况下，基于对分数运算符行为方式的误解，一些现有论文包含对问题的无效重新表述。在这项工作中，我们强调了将非瞬时脉冲分数阶微分方程写成等效积分方程的正确方法，