We study the unique solvability of linear equations in Banach spaces with discretely distributed Gerasimov–Caputo fractional derivative in terms of analytic resolving families of operators. Necessary and sufficient conditions for the existence of such a family of operators are obtained in terms of the resolvent of a closed operator from the right-hand side of the equation, and the properties of this family are studied. These results are used to prove the existence of a unique solution to the Cauchy problem for a linear equation of the corresponding class with inhomogeneity which is either continuous in the norm of the graph of the operator from the right-hand side of the equation or HЈolderian. Based on the abstract results obtained, we investigate the unique solvability of initial–boundary value problems for a class of equations with discretely distributed fractional time derivative and with polynomials in an elliptic self-adjoint differential operator with respect to spatial variables.
Keywords: Gerasimov–Caputo fractional derivative, discretely distributed fractional derivative, Cauchy problem, resolving family of operators, initial–boundary value problem
Received February 1, 2021
Revised March 6, 2021
Accepted March 15, 2021
Funding Agency: This work was supported by the Russian Foundation for Basic Research (project 21-51-54003), is a part of the research carried out at the Ural Mathematical Center and supported by the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-02-2021-1383).
Vladimir Evgenyevich Fedorov, Dr. Phys.-Math. Sci., Prof., Chelyabinsk State University, Chelyabinsk, 454001 Russia; Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: kar@csu.ru
Nikolay Vladimirovich Filin, Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Yugra State University, Khanty-Mansiysk, 628012 Russia; Chelyabinsk State University, Chelyabinsk, 454001 Russia; e-mail: nikolay_filin@inbox.ru
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Cite this article as: V.E. Fedorov, N.V. Filin. Linear equations with discretely distributed fractional derivative in Banach spaces, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2021, vol. 27, no. 2, pp. 264–280.
