Center Manifolds of Differential Equations in Banach Spaces

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.