Complex Powers of Fractional Sectorial Operators and Quasilinear Equations with Riemann-Liouville Derivatives

For a continuously invertible operator -A fractional powers A(?) and spaces Z(?) as domains of A(? )with the graph norms are defined from the class of generators of analytic resolving families of fractional order differential equations. Properties of these powers and of resolving families in these spaces are studied. This allows to prove the existence of a unique solution for the incomplete Cauchy type problem for a differential equation in a Banach space, resolved with respect to the oldest Riemann-Liouville derivative, with the operator A in the linear part of the equation and with a nonlinear operator depending on Riemann-Liouville derivatives and integrals. For the nonlinear operator is fulfilled Holder condition with respect to the norm in Z(?). This result applied to research of unique solvability of an initial boundary value problem for a quasilinear partial differential equation.