Resolvents of operators and partial functional differential equations with nonautonomous past

To a backward evolution family U = (U(t , s))t s 0 on a Banach space X we associate an abstract differential operator G through the integral equation u(t ) = U(t, s)u(s) + s t U(t, ξ)f (ξ)dξ 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. Contin. Dynam. Systems 8 (2002) 953–966]. Our main tools are spectral mapping theorems for evolution semigroups and hyperbolicity criteria.  2003 Elsevier Inc. All rights reserved.