Discrete Asymptotic Behaviors for Skew-Evolution Semiflows on Banach Spaces

TL;DR

This paper investigates the asymptotic behaviors such as stability, instability, dichotomy, and trichotomy for skew-evolution semiflows on Banach spaces, providing a unified discrete-time framework for these properties.

Contribution

It introduces a comprehensive discrete-time characterization of asymptotic behaviors for skew-evolution semiflows, extending previous continuous and uniform results to nonuniform cases.

Findings

Unified discrete-time treatment of asymptotic properties

Extension of trichotomy concept to nonuniform skew-evolution semiflows

Generalization of stability and instability criteria

Abstract

The paper emphasizes asymptotic behaviors, as stability, instability, dichotomy and trichotomy for skew-evolution semiflows, defined by means of evolution semiflows and evolution cocycles and which can be considered generalizations for evolution operators and skew-product semiflows. The definition are given in continuous time, but the unified treatment for the characterization of the studied properties in the nonuniform case is given in discrete time. The property of trichotomy, introduced in finite dimension by S. Elaydi and O. Hajek in 1988 as a natural generalization for the dichotomy of linear time-varying differential systems, was studied by us in continuous time and from uniform point of view and in discrete time and from nonuniform point of view but for a particular case of one-parameter semiflows.