h-Dichotomies via noncritical uniformity and expansiveness for evolution families

2025/11/05 by Dragicevic, Davor
Engineering · Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Fixed Point Theorems Analysis #Nonlinear Differential Equations Analysis #Stability and Controllability of Differential Equations

paper · doi:10.48550/arxiv.2511.03453

openalex publication_date 2025/11/05 · openalex created_date 2025/11/07 · openalex updated_date 2026/07/28

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.

Citations

• Dichotomies for triangular systems on Hilbert spaces
• Generalized dichotomies via time rescaling
• Admissibility and general dichotomies for evolution families

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• Robustness of nonuniform exponential dichotomies under a wider class of perturbations
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• Generalized Evolution Semigroups and h-Dichotomies for Evolution Families on Banach Spaces
• Admissibility and general dichotomies for evolution families