Sectorial Tuples of Operators and Quasilinear Fractional Equations with Multi-Term Linear Part

The incomplete Cauchy problem to an equation with several Riemann–Liouville derivatives in a Banach space is studied. It is proved that the tuple of linear operators from the linear multi-term equation is sectorial, if and only if the incomplete Cauchy problem to this equation has an unique solution. A local existence of an unique solution to a quasilinear equation with a sectorial tuple of operators in the multi-term linear part is proved under the condition of local Lipschitz continuity of the nonlinear operator and its continuity in the sum of the graph norms of the linear operators from the equation. Obtained results are applied to the research of a class of initial boundary value problems for equations with Riemann–Liouville derivatives in time and with polynomials of a self-adjoint elliptic differential operator with respect to spatial variables.