Differentiability and growth bounds of solutions of delay equations

We consider the question of eventual differentiability of the delay semigroups associated with the retarded equation u (t ) = Au(t ) + Φut (t 0), where ut is the history function, A generates an immediately norm-continuous semigroup and Φ is bounded. We show that this is determined by the rate of decay of the resolvent of A along vertical lines.  2004 Elsevier Inc. All rights reserved