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h-Dichotomies via noncritical uniformity and expansiveness for evolution families

Davor Dragicevic

Abstract

In a recent paper (Math. Ann. 393 (2025), 1769--1795), Elorreaga et al. have obtained a complete characterization of the notion of a $h$-dichotomy for ordinary differential equations on a finite-dimensional space in terms of the notions of $h$-expansiveness and $h$-noncriticality. Their results extended the previous results of Coppel and Palmer, which dealt with exponential dichotomies. The main objective of this note is to extend the results of Elorreaga et al. to arbitrary invertible evolution families that act on Banach spaces. We emphasize that our approach is completely different and considerably simpler from the one developed by Elorreaga et al. It is based on the time-rescaling method introduced by Dragicevic and Silva.

h-Dichotomies via noncritical uniformity and expansiveness for evolution families

Abstract

In a recent paper (Math. Ann. 393 (2025), 1769--1795), Elorreaga et al. have obtained a complete characterization of the notion of a -dichotomy for ordinary differential equations on a finite-dimensional space in terms of the notions of -expansiveness and -noncriticality. Their results extended the previous results of Coppel and Palmer, which dealt with exponential dichotomies. The main objective of this note is to extend the results of Elorreaga et al. to arbitrary invertible evolution families that act on Banach spaces. We emphasize that our approach is completely different and considerably simpler from the one developed by Elorreaga et al. It is based on the time-rescaling method introduced by Dragicevic and Silva.

Paper Structure

This paper contains 7 sections, 6 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Time-rescaling
  4. Noncritical uniformity and expansiveness
  5. Main results
  6. Characterization of exponential dichotomies
  7. Characterization of $h$-dichotomies

Key Result

Proposition 1

$\mathcal{T}^h:=\{T_h(t, s): \ t\ge s>-\infty\}$ is an evolution family.

Theorems & Definitions (31)

  • Definition 1
  • Remark 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Remark 2
  • Remark 3
  • Definition 5
  • Remark 4
  • Remark 5
  • ...and 21 more