Due to noncontinuous solution, impulsive differential equations with delay may have a measurable right side and not a continuous one. In order to support handling impulsive differential equations with delay like in other chapters of differential equations, we formulated and proved existence and uniqueness theorems for impulsive differential equations with measurable right sides following Caratheodory’s techniques. The new setup had an impact on the formulation of initial value problems (IVP), the continuation of solutions, and the structure of the system of trajectories. (a) We have two impulsive differential equations to solve with one IVP () which selects one of the impulsive differential equations by the position of in . Solving the selected IVP fully determines the solution on the other scale with a possible delay. (b) The solutions can be continued at each point of by the conditions in the existence theorem. (c) These changes alter the flow of solutions into a directed tree. This tree however is an in-tree which offers a modelling tool to study among other interactions of generations.

中文翻译：

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右侧可测的时滞脉冲微分方程的存在性定理

由于非连续解，带时滞的脉冲微分方程可能具有可测量的右侧而不是连续的。为了像在微分方程的其他各章中一样支持延迟处理脉冲微分方程，我们遵循Caratheodory的技术，制定并证明了具有可测量右侧的脉冲微分方程的存在性和唯一性定理。新的设置对初始值问题（IVP）的制定，解决方案的延续以及轨迹系统的结构产生了影响。（a）我们有两个脉冲微分方程要用一个IVP（）通过in的位置选择一个脉冲微分方程。解决选定的IVP可能会完全延迟其他规模的解决方案。（b）解决方案可以在以下各点继续进行存在定理中的条件。（c）这些更改将解决方案的流程更改为有向树。但是，该树是树中的树，它提供了建模工具来研究世代之间的其他交互。