Subordination Principle for Equations with Riemann–Liouville Derivative

The subordination principle is studied for linear equations in Banach spaces resolved with regards to the Riemann–Liouville fractional derivative. It is assumed that a linear operator at the unknown function is closed. We have proved that the existence of a strongly continuous resolving family of operators for such equation of an order ₁ implies the existence of an analytic resolving family of operators for every such equation of an order ₂<₁. Two presentations of the analytic family through the initial strongly continuous family are obtained. The Wright type function with two parameters plays a key role in these considerations. Further it is shown that from the existence of a strongly continuous resolving family of operators for equation with the Riemann–Liouville derivative it follows that a strongly continuous family exists for equation of the same order, but with the Gerasimov–Caputo derivative. The resolving family of operators for the last equation is presented as the Riemann– Liouville fractional integral of the resolving family for the equation with the Riemann–Liouville derivative.