CENTER MANIFOLDS FOR INFINITE-DIMENSIONAL NONAUTONOMOUS DIFFERENTIAL-EQUATIONS

We study a nonlinear integral equation for a center manifold of a semi linear nonautonomous differential equation having mild solutions. We a ssume that the linear part of the equation admits, in a very general s ense, a decomposition into center and hyperbolic parts. The center man ifold is obtained directly as the graph of a fixed point for a Lyapuno v-Perron type integral operator. We prove that this integral operator can be factorized as a composition of a nonlinear substitution operato r and a linear integral operator Lambda. The operator Lambda is formed by the Green's function for the hyperbolic part and composition opera tors induced by a now on the center part. We formulate the usual gap c ondition in spectral terms and show that this condition is, in fact, a condition of boundedness of ii on corresponding spaces of differentia ble functions. This gives a direct proof of the existence of a smooth global center manifold. (C) 1997 Academic Press.