Fractional relaxation equations on Banach spaces

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