97-147 Chicone C., Latushkin Y.

On an Integral Equation for Center Manifolds: a Direct Proof for Nonautonomous Differential Equations on Banach Spaces (108K, AMS-LaTeX) Mar 25, 97

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Abstract. We study a nonlinear integral equation for a center manifold of a semilinear nonautonomous differential equation having mild solutions. We assume that the linear part of the equation admits, in a very general sense, a decomposition into center and hyperbolic parts. The center manifold is obtained directly as the graph of a fixed point for a Lyapunov-Perron type integral operator. We prove that this integral operator can be factorized as a composition of a nonlinear substitution operator and a linear integral operator $\Lambda$. The operator $\Lambda$ is formed by the Green's function for the hyperbolic part and composition operators induced by a flow on the center part. We formulate the usual gap condition in spectral terms and show that this condition is, in fact, a condition of boundedness of $\Lambda$ on corresponding spaces of differentiable functions. This gives a direct proof of the existence of a smooth global center manifold.

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