Center Manifolds for Infinite Dimensional Nonautonomous Differential Equations

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 operatorΛ. The operatorΛis formed by the Greenʹs 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 ofΛon corresponding spaces of differentiable functions. This gives a direct proof of the existence of a smooth global center manifold.