Geodesic
Quasilinear Equations with a Sectorial Set of Operators at Gerasimov--Caputo Derivatives
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 29 (2023) no. 2, pp. 248-259

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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.
Keywords: Gerasimov–Caputo fractional derivative, Cauchy problem, sectorial set of operators, resolving family of operators, quasilinear equation, local solution, initial–boundary value problem.
Mots-clés : nonlocal solution 
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     author = {V. E. Fedorov and K. V. Boyko},
     title = {Quasilinear {Equations} with a {Sectorial} {Set} of {Operators} at {Gerasimov--Caputo} {Derivatives}},
     journal = {Trudy Instituta matematiki i mehaniki},
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     url = {http://geodesic.mathdoc.fr/item/TIMM_2023_29_2_a17/}
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V. E. Fedorov; K. V. Boyko. Quasilinear Equations with a Sectorial Set of Operators at Gerasimov--Caputo Derivatives. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 29 (2023) no. 2, pp. 248-259. http://geodesic.mathdoc.fr/item/TIMM_2023_29_2_a17/