On fractional derivatives

We introduce some types of fractional derivatives in one real variable, and try to solve some analogs of ordinary differential equations, in the form \[ D^\alpha u(t) = Au(t) + f(t), \quad t \in [0, T], \] where $A$ is a square $N \times N-$matrix, $f : [0, T] \to \C^N$, and prescribed proper initial conditions are given. Finally, we try to explain how to extend these elementary results to the important case that $A$ is a linear, not necessarily continuous, operator in a Banach space $X$.