Fractional relaxation equations on Banach spaces

We study existence and qualitative properties of solutions for the abstract fractional relaxation equation (0.1) u ′ ( t ) − A D t α 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, D t α 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 L p integrability. We characterize these properties in the Hilbert space case.