A Class of Quasilinear Equations with Distributed Gerasimov–Caputo Derivatives

Vladimir E. Fedorov () and Nikolay V. Filin
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Vladimir E. Fedorov: N.N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, 16, S.Kovalevskaya St., 620108 Yekaterinburg, Russia
Nikolay V. Filin: N.N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, 16, S.Kovalevskaya St., 620108 Yekaterinburg, Russia

Mathematics, 2023, vol. 11, issue 11, 1-17

Abstract: Quasilinear equations in Banach spaces with distributed Gerasimov–Caputo fractional derivatives, which are defined by the Riemann–Stieltjes integrals, and with a linear closed operator A , are studied. The issues of unique solvability of the Cauchy problem to such equations are considered. Under the Lipschitz continuity condition in phase variables and two types of continuity over all variables of a nonlinear operator in the equation, we obtain two versions on a theorem on the nonlocal existence of a unique solution. Two similar versions of local unique solvability of the Cauchy problem are proved under the local Lipschitz continuity condition for the nonlinear operator. The general results are used for the study of an initial boundary value problem for a generalization of the nonlinear phase field system of equations with distributed derivatives with respect to time.

Keywords: distributed fractional derivative; fractional differential equation; Cauchy problem; quasiliner equation; fixed point theorem; initial boundary value problem (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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