Resolvents of operators and partial functional differential equations with nonautonomous past

To a backward evolution family U = (U(t, s))(t) less than or equal to s less than or equal to 0 on a Banach space X we associate an abstract differential operator G through the integral equation u(t) = U(t, s) u (s) + integral(t)(s) U (t, xi) f (xi) dxi on a Banach space of X-valued functions on R- We compute the resolvent of the restriction of this operator to a smaller domain to obtain a generator. We then apply the results to prove existence, exponential stability and exponential dichotomy of solutions to partial functional equations with nonautonomous past as discussed in [S. Brendle, R. Nagel, Dist. Comm. Dynam. Systems 8 (2002) 953-966]. Our main tools are spectral mapping theorems for evolution semigroups and hyperbolicity criteria. (C) 2003 Elsevier Inc. All rights reserved.