On existence, uniqueness and numerical approximation of impulsive differential equations with adaptive state-dependent delays using equations with piecewise-constant arguments

TL;DR

This paper investigates impulsive nonlinear differential equations with adaptive state-dependent delays, establishing existence, uniqueness, and uniform approximation of solutions using equations with piecewise-constant arguments.

Contribution

It introduces a novel approach using EPCAs to approximate solutions of impulsive DDEs with adaptive delays, ensuring their existence and uniqueness.

Findings

Solutions exist and are unique under monotonic delay functions.

EPCAs provide uniform approximation of the original solutions.

The approach applies to a broad class of impulsive DDEs with adaptive delays.

Abstract

In this paper we consider a class of impulsive nonlinear differential equations with adaptive state-dependent delays. We discuss the existence and uniqueness of solutions of the initial value problem using a Picard-Lindel\"of type argument where we define approximate solutions with the help of equations with piecewise-constant arguments (EPCAs). Moreover, we show that the solutions of the associated EPCAs approximate the solutions of the original impulsive DDE with adaptive state-dependent delay uniformly on compact time intervals. The key assumption underlying both results is that the delayed time function is monotone, or piecewise strictly monotone.