Parabolic Volterra integrodifferential equations of convolution type (Q1899923)

The main object of this paper is the existence of differentiable solutions of the following problem \[ u' (t)= Au(t)+ \int_0^t k(t- s) Bu(s) ds+ f(t), \qquad t\in ]0,1 [, \quad u(0)=x, \] where \(f\) and \(x\) are given, \(X\) is a Banach space, \(A: D(A) \subset X\to X\) is the infinitesimal generator of an analytic semigroup \(S(t)\) on \(X\), and \(B: D(B) \subset X\to X\) is a linear operator with domain \(D(B) \supseteq D(A)\).