巴拿赫空间中微分方程的中心流形
Center Manifolds of Differential Equations in Banach Spaces
摘要：
中心流形对于描述微分方程系统的长期行为是有用的。本文考虑了巴拿赫空间中的一类自治微分方程，该方程的线性项具有指数可分性，非线性项具有Lipschitz连续性.利用谱间隙条件证明了中心流形的存在唯一性。此外，我们证明了流形的正则性与一些额外的非线性项的假设。我们使用著名的Lyapunov-Perron方法来处理这个问题，该方法依赖于Banach不动点定理。证明可以推广到非自治系统。
英文摘要：
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.
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