Center Manifolds of Differential Equations in Banach Spaces

TL;DR

This paper establishes the existence, uniqueness, and regularity of center manifolds for autonomous differential equations in Banach spaces with exponential trichotomy, using the Lyapunov-Perron method.

Contribution

It extends the theory of center manifolds to Banach spaces with exponential trichotomy, providing new proofs and generalizations to non-autonomous systems.

Findings

Proved existence and uniqueness of center manifolds under spectral gap conditions.

Established regularity of the center manifold with additional nonlinear assumptions.

Methodology based on the Lyapunov-Perron fixed-point approach.

Abstract

The center manifold is useful for describing the long-term behavior of a system of differential equations. In this work, we consider an autonomous differential equation in a Banach space that has the exponential trichotomy property in the linear terms and Lipschitz continuity in the nonlinear terms. Using the spectral gap condition we prove the existence and uniqueness of the center manifold. Moreover, we prove the regularity of the manifold with a few additional assumptions on the nonlinear term. We approach the problem using the well-known Lyapunov-Perron method, which relies on the Banach fixed-point theorem. The proofs can be generalized to a non-autonomous system.