Quasilinear equations with fractional Gerasimov–Caputo derivative. Sectorial case

We study initial-value problems for quasilinear equations with Gerasimov–Caputo fractional derivatives in Banach spaces whose linear part has an analytic resolving family of operators in the sector. The nonlinear operator is assumed to be a locally Lipschitz operator. We consider equations that are solved with respect to the highest derivative and equations containing a degenerate linear operator acting on the highest derivative. The theorem on the unique solvability of the Cauchy problem proved in the paper is used for the study of the unique solvability of the Showalter–Sidorov problem for degenerate equations. Abstract results are applied to the initial-boundary-value problem for partial differential equations that are not solvable with respect to the highest fractional derivative in time.