Determination of the order in Abstract fractional differential equations

In this paper we identify, for small $t$ and a fixed $T>0,$ the order $\alpha >0$ in the abstract fractional differential equation $${\mathrm{\partial }}^{\alpha }u(t)=Au(t),$$ where the time-fractional derivative ${\mathrm{\partial }}^{\alpha }$ is understood in the sense of Caputo and Riemann-Liouville, $A$ is a closed (possibly unbounded) linear operator in a Banach space $X,$ and $0<\alpha <1$ or $1<\alpha <2.$