Quasilinear Equations with a Sectorial Set of Operators at Gerasimov–Caputo Derivatives

The issues of unique solvability of the Cauchy problem are studied for a quasilinear equation solved with respect to the highest fractional Gerasimov–Caputo derivative in a Banach space with closed operators from the class $A_{\alpha,G}^{n}$ in the linear part and with a nonlinear operator continuous in the graph norm. A theorem on the local existence and uniqueness of a solution to the Cauchy problem is proved in the case of a locally Lipschitz nonlinear operator. Under the nonlocal Lipschitz condition for the nonlinear operator, the existence of a unique solution on a predetermined interval is shown. Abstract results are illustrated by examples of initial–boundary value problems for partial differential equations with Gerasimov–Caputo time derivatives.