Fractional relaxation equations on Banach spaces (Q710978)

The authors study the abstract fractional relaxation equation \[ u^{\prime}(t)-AD_t^{\alpha}u(t)+u(t)=f(t), \quad 0<\alpha < 1,\quad t \geq 0, \quad u(0)=0, \] on a complex Banach space \(X\), where \(A\) is a closed linear operator, \(D_t^{\alpha}\) is the Caputo derivative of fractional order \(\alpha \in (0,1)\), and \(f\) is an \(X\)-valued function. They obtain some results on existence and qualitative properties of solutions and also study conditions under which the solution operator has the properties of maximal regularity and \(L^p\) integrability. They characterize these properties in the Hilbert space case.