Global Lipschitz Invariant Center Manifolds for ODEs with Generalized Trichotomies

TL;DR

This paper establishes conditions for the existence of global Lipschitz invariant center manifolds in Banach spaces for perturbed differential equations with very general trichotomies, extending and improving previous results.

Contribution

It introduces new conditions for invariant center manifolds in Banach spaces under generalized trichotomies, broadening the scope of existing theories.

Findings

Provides sufficient conditions for existence of center manifolds.

Includes new cases not covered by previous literature.

Enhances understanding of invariant manifolds in nonautonomous systems.

Abstract

In a Banach space, assuming that a linear nonautonomous differential equation $v'=A(t)v$ admits a very general type of trichotomy, we establish conditions for the existence of global Lipschitz invariant center manifold of the perturbed equation $v'=A(t)v+f(t,v)$. Our results not only improve results already existing in the literature, as well include new cases.