# American Institute of Mathematical Sciences

April  1999, 5(2): 233-268. doi: 10.3934/dcds.1999.5.233

## The optimal gap condition for invariant manifolds

 1 Department of Mathematics, University of Missouri, Columbia, MO 65211, United States, United States

Received  June 1998 Revised  June 1998 Published  January 1999

We give optimal gap conditions using Lipschitz constants of the nonlinear terms and growth bounds of the linear terms that imply the existence of infinite dimensional Lipschitz invariant manifolds for systems of semilinear equations on Banach spaces. This result improves and generalizes recent theorems by C. Foias and by N. Castañeda and R. Rosa. The result is also shown to imply the existence of invariant manifolds for nonautonomous equations and semilinear skew-product flows. Also, generalizations for smoothness of invariant manifolds are given.
Citation: Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233
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