New Results on Linearization of Differential Equations with Piecewise Constant Argument

Abstract

In this work, we give a version of Grobman–Hartman theorem for the nonautonomous differential equations with piecewise constant argument of generalized type when the nonlinear term is unbounded and its linear system partially satisfies $$\alpha $$-exponential dichotomy. More specifically, we divide the linear system into three subsystems, one of the subsystems is not necessary to admit $$\alpha $$-exponential dichotomy. It was assumed that the whole linear system admits exponential dichotomy in Pinto and Robledo (Z Anal Anwend 37:101–126, 2018), Zou and Shi (J Appl Anal Comput 7(1):309–333, 2017) and Zou et al. (Qual Theory Dyn Syst 18:495–531, 2019). Thus, we extend and improve the previous results in the literature. Moreover, we prove that the conjugated function H(t, x) is also $$\omega $$-periodic when the systems are $$\omega $$-periodic.