  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.
