  While it is known that one can consider the Cauchy problem for evolution
equations with Caputo derivatives, the situation for the initial value problems
for the Riemann-Liouville derivatives is less understood. In this paper we
propose new type initial, inner and inner-boundary value problems for
fractional differential equations with the Riemann-Liouville derivatives. The
results on the existence and uniqueness are proved, and conditions on the
solvability are found. The well-posedness of the new type initial, inner and
inner-boundary conditions are also discussed. Moreover, we give explicit
formulas for the solutions. As an application fractional partial differential
equations for general positive operators are studied.
