Fractional relaxation equations on Banach spaces

We study existence and qualitative properties of solutions for the abstract fractional relaxation equation (0.1) u′(t)−ADα t u(t) + u(t) = f(t), 0 < α < 1, t ≥ 0, u(0) = 0, on a complex Banach space X, where A is a closed linear operator, Dα t is the Caputo derivative of fractional order α ∈ (0, 1), and f is an X-valued function. We also study conditions under which the solution operator has the properties of maximal regularity and Lp integrability. We characterize these properties in the Hilbert space case.