Fractional relaxation equations on Banach spaces

Abstract

We study existence and qualitative properties of solutions for the abstract fractional relaxation equation

(0.1)$$u^{\prime }\left(t\right)-A{D}_t^{\alpha }u\left(t\right)+u\left(t\right)=f\left(t\right)\text{,}0<\alpha <1,t\ge 0\text{,}u\left(0\right)=0\text{,}$$

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. 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.