Existence Theorem for Impulsive Differential Equations with Measurable Right Side for Handling Delay Problems

Abstract

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 (φσ0=ξ0) which selects one of the impulsive differential equations by the position of σ0 in a,bν. 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 α,β×Ω0≕Ω 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.