Boundness and Linearisation of a class of differential equations with piecewise constant argument

TL;DR

This paper investigates bounded solutions and topological linearisation of a broad class of differential equations with piecewise constant argument, providing new criteria and generalized theorems that improve upon previous results in hybrid dynamical systems.

Contribution

It introduces a new criterion for the existence of unique bounded solutions and establishes a generalized Grobman-Hartman theorem for topological conjugacy in DEPCAs.

Findings

New criterion for bounded solutions established

Generalized topological linearisation theorem proved

Improves and extends previous results in hybrid systems

Abstract

The differential equations with piecewise constant argument (DEPCAs, for short) is a class of hybrid dynamical systems (combining continuous and discrete). In this paper, under the assumption that the nonlinear term is partially unbounded, we study the bounded solution and global topological linearisation of a class of DEPCAs of general type. One of the purpose of this paper is to obtain a new criterion for the existence of a unique bounded solution, which improved the previous results. The other aim of this paper is to establish a generalized Grobman-Hartman-type theorem for the topological conjugacy between a nonlinear perturbation system and its linear system. The method is based on the new obtained criterion for bounded solution. The obtained results generalized and improved some previous papers. Some novel techniques are employed.