C. Chicone et Y. Latushkin, CENTER MANIFOLDS FOR INFINITE-DIMENSIONAL NONAUTONOMOUS DIFFERENTIAL-EQUATIONS, Journal of differential equations, 141(2), 1997, pp. 356-399
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.