Quasilinear multi-term equations with Riemann — Liouville derivatives of arbitrary orders

The existence of a unique local solution of the incomplete Cauchy type problem is proved for a quasilinear equation with several fractional Riemann — Liouville derivatives and with a sectorial tuple of operators at lower derivatives in the linear part in the case of the local Lipschitz continuity of a nonlinear operator with respect to the sum of the norms of graphs of unbounded operators from the equation. In this case, the nonlinear operator depends on the fractional Riemann—Liouville derivatives of lower orders with arbitrary fractional parts. The obtained abstract result is used in the study of an initial boundary value problem for an equation with several fractional derivatives in time.