  We define fractional derivatives $\pppa$ in Sobolev spaces based on
$L^p(0,T)$ by an operator theory, and characterize the domain of $\pppa$ in
subspaces of the Sobolev-Slobodecki spaces $W^{\alpha,p}(0,T)$. Moreover we
define $\pppa u$ for $u\in L^p(0,T)$ in a sense of distribution. Then we
discuss initial value problems for linear fractional ordinary differential
equations by means of such $\pppa$ and establish several results on the unique
existence of solutions within specified classes according to the regularity of
the coefficients and the non-homogeneous terms in the equations.
