A Grobman-Hartman theorem for a differential equation with piecewise constant generalized argument

TL;DR

This paper establishes a Grobman-Hartman type theorem for differential equations with piecewise constant generalized arguments, providing conditions for a homeomorphism between linear and nonlinear solutions.

Contribution

It generalizes previous results by introducing new conditions for the existence of a homeomorphism in systems with piecewise constant arguments, using a novel exponential dichotomy concept.

Findings

Established sufficient conditions for solution homeomorphism

Extended Grobman-Hartman theorem to piecewise constant argument systems

Unified linear and nonlinear solution analysis

Abstract

We obtain sufficient conditions ensuring the existence of a uniformly continuous and H\"older continuous homeomorphism between the solutions of a linear system of differential equations with piecewise constant argument of generalized type and the solutions of the quasilinear corresponding system. We use a definition (recently introduced by M. Akhmet) of exponential dichotomy for those systems combined with technical assumptions on the nonlinear part. Our result generalizes a previous work of G. Papaschinopoulos.