Cauchy type problems for fractional differential equations

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