L-shadowing lemma for the Cauchy equation

TL;DR

This paper establishes a connection between hyperbolicity and the L-shadowing property for the Cauchy problem in Banach spaces, extending previous results and providing applications.

Contribution

It proves that hyperbolicity implies the L-shadowing property in Banach spaces and conversely in finite dimensions, generalizing earlier work on linear homeomorphisms.

Findings

Hyperbolic Cauchy problems have the L-shadowing property.

Finite-dimensional L-shadowing implies hyperbolicity.

Applications demonstrate the theoretical results.

Abstract

We prove that if the Cauchy problem $\dot{u}=Au$ in a Banach space is hyperbolic, then the problem has the L-shadowing property. Conversely, if the space is finite-dimensional and the L-shadowing property is satisfied, then the problem is hyperbolic. This generalizes a previous result by Ombach \cite{o, o1} for linear homeomorphisms. Some short applications are given.