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 Aα,Gn 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.
Keywords: Gerasimov–Caputo fractional derivative, Cauchy problem, sectorial set of operators, resolving family of operators, quasilinear equation, local solution, nonlocal solution, initial–boundary value problem
Received February 28, 2023
Revised March 15, 2023
Accepted March 20, 2023
Funding Agency: This work was supported by the RF President’s Grant for State Support of Leading Scientific Schools (project no. 2708.2022.1.1).
Vladimir Evgenyevich Fedorov, Dr. Phys.-Math. Sci., Prof., Chelyabinsk State University, Chelya-binsk, 454001 Russia, e-mail: kar@csu.ru
Kseniya Vladimirovna Boyko, doctoral student, Chelyabinsk State University, Chelyabinsk, 454001 Russia, e-mail: kvboyko@mail.ru
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Cite this article as: V.E. Fedorov, K.V. Boyko. Quasilinear equations with a sectorial set of operators at Gerasimov–Caputo derivatives. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2023, vol. 29, no. 2, pp. 248–259; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2023, Vol. 321, Suppl. 1, pp. S78–S89.
