Eventual differentiability of functional differential equations in Banach spaces ✩

This paper concerns the regularity of a functional differential equation in the form: ˙u(t) = Au(t) + B1u(t −r)+ 0 −r a(s)B2u(t +s)ds, t > 0, where A is the generator of an analytic semigroup on a Banach space X, and B1,B2 are (γ −A)α-bounded linear operator for 0<α <1. By spectral analysis, it is shown that the associated solution semigroup of this equation is eventually differentiable. © 2006 Elsevier Inc. All rights reserved