Linear and quasilinear equations with several Gerasimov — Caputo derivatives

A representation of a solution of the Cauchy problem for a linear inhomogeneous equation solved with respect to the oldest derivative with several fractional Gerasimov — Caputo derivatives and with a sectorial pencil of linear closed operators at them in the case of the Hölder function in the right-hand side of the equation is obtained; the uniqueness of the solution is proved. This result is used to reduce the Cauchy problem for the corresponding quasilinear equation to an integro-differential equation. The existence of a unique local solution is proved by the method of contraction operators in the case of local Lipschitz nonlinear operator depending on several Gerasimov — Caputo derivatives in the equation and a single global solution under the Lipschitz condition for this operator.