Linear and quasilinear equations with several Gerasimov~--- Caputo derivatives

K. V. Boyko

Čelâbinskij fiziko-matematičeskij žurnal, Tome 9 (2024) no. 1, pp. 5-22.

Résumé

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.

Détail

Keywords: Gerasimov — Caputo fractional derivative, multi-term fractional equation, sectorial pencil of operators, Hölder function.

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