On the Solvability of Equations with a Distributed Fractional Derivative Given by the Stieltjes Integral

Sergey M. Sitnik (), Vladimir E. Fedorov, Nikolay V. Filin and Viktor A. Polunin
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Sergey M. Sitnik: Applied Mathematics and Computer Modelling, Belgorod State National Research University, Pobedy St. 85, Belgorod 308015, Russia
Vladimir E. Fedorov: Department of Differential Equations, N.N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, 16, S.Kovalevskaya St., Yekaterinburg 620108, Russia
Nikolay V. Filin: Department of Differential Equations, N.N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, 16, S.Kovalevskaya St., Yekaterinburg 620108, Russia
Viktor A. Polunin: Applied Mathematics and Computer Modelling, Belgorod State National Research University, Pobedy St. 85, Belgorod 308015, Russia

Mathematics, 2022, vol. 10, issue 16, 1-20

Abstract: Linear equations in Banach spaces with a distributed fractional derivative given by the Stieltjes integral and with a closed operator A in the right-hand side are considered. Unlike the previously studied classes of equations with distributed derivatives, such kinds of equations may contain a continuous and a discrete part of the integral, i.e., a standard integral of the fractional derivative with respect to its order and a linear combination of fractional derivatives with different orders. Resolving families of operators for such equations are introduced into consideration, and their properties are studied. In terms of the resolvent of the operator A , necessary and sufficient conditions are obtained for the existence of analytic resolving families of the equation under consideration. A perturbation theorem for such a class of operators is proved, and the Cauchy problem for the inhomogeneous equation with a distributed fractional derivative is studied. Abstract results are applied for the research of the unique solvability of initial boundary value problems for partial differential equations with a distributed derivative with respect to time.

Keywords: distributed fractional derivative; fractional differential equation; analytic k-resolving family; Cauchy problem; initial boundary value problem (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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