Chicone C., Latushkin Y.
On an Integral Equation for Center Manifolds: a Direct Proof for
Nonautonomous Differential Equations on Banach Spaces
(108K, AMS-LaTeX)
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.