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) = A u(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