Resolvents of operators and partial functional differential equations with nonautonomous past

AbstractTo 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)+∫tsU(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