On the Nature of Impulsive Differential Equations and the Existence of its Solutions. The Emerging facts. – Z. Lipcsey, J. A. Ugboh and I. M. Esuabana
ABSTRACT
Examples of delayed impulsive differential equations show that the delay converts a differential equation with smooth right side to one with measurable right side. A traditional impulsive differential equation describing the impacts of impulses can be accompanied by another one that describes the impulse process. In order to support handling impulsive differential equations with delay, we formulated and proved existence theorems for impulsive differential equations with measurable right sides following Caratheodory’s techniques. The new setup had 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 (‘(0) = 0) which selects one of the impulsive differential equations by the position of 0 in [a, b]. 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 ( , ) × 0 =: by the conditions in the existence theorem. However, the jump at a discontinuity point may land outside. Thus, there is no continuation of the solution from such points of. If we restrict the jumps into, then all trajectories reaching a discontinuity point will be continued. If range of jumps 6= then trajectories from outside the range of jumps cannot be continued backwards. (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 interactions of generations.