Dichotomies and asymptotic equivalence in alternately advanced and delayed differential systems

TL;DR

This paper explores the concepts of dichotomies and asymptotic equivalence in differential systems with piecewise constant arguments, establishing conditions under which solutions of perturbed systems behave similarly to unperturbed ones.

Contribution

It introduces new definitions of dichotomies for such systems and proves their asymptotic equivalence under certain perturbations, advancing the theoretical understanding of these systems.

Findings

Defined ordinary and exponential dichotomies for systems with piecewise constant arguments

Proved asymptotic equivalence between solutions of linear and perturbed systems

Established conditions for bounded solutions to behave similarly in perturbed systems

Abstract

In this paper, ordinary and exponential dichotomies are defined in differential equations with equations with piecewise constant argument of general type. We prove the asymptotic equivalence between the bounded solutions of a linear system and a perturbed system with integrable and bounded perturbations.