Pointwise Trichotomy for Skew-Evolution Semiflows on Banach Spaces

TL;DR

This paper develops a new framework for analyzing skew-evolution semiflows on Banach spaces, introducing pointwise trichotomy and connecting it with classical concepts, with potential extensions to controlled systems.

Contribution

It introduces the concept of pointwise trichotomy for skew-evolution semiflows on Banach spaces, extending classical evolution operator results with a uniform approach.

Findings

Established the notion of pointwise trichotomy for skew-evolution semiflows.

Connected the new concept with classical trichotomy for evolution operators.

Provided characterizations and potential for extension to controlled systems.

Abstract

The paper introduces the notion of skew-evolution semiflows and presents the concept of pointwise trichotomy in the case of skew-evolution semiflows on a Banach space. The connection with the classic notion of trichotomy presented by us in a previous paper in 2006 for evolution operators, is also emphasized, as well as some characterizations. The approach of the theory is from uniform point of view. The study can also be extended to systems with control whose state evolution can be described by skew-evolution semiflows.