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Russia Voronezh
Year
2022
Volume
32
Issue
3
Pages
415-432
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Section Mathematics
Title On a boundary value problem for a class of fractional Langevin type differential equations in a Banach space
Author(-s) Petrosyan G.G.ab
Affiliations Voronezh State Pedagogical Universitya, Voronezh State University of Engineering Technologiesb
Abstract In this paper, we consider a boundary value problem for differential equations of Langevin type with the Caputo fractional derivative in a Banach space. It is assumed that the nonlinear part of the equation is a Caratheodory type map. Equations of this type generalize equations of motion in various kinds of media, for example, viscoelastic media or in media where a drag force is expressed using a fractional derivative. We will use the theory of fractional mathematical analysis, the properties of the Mittag-Leffler function, as well as the theory of measures of non-compactness and condensing operators to solve the problem. The initial problem is reduced to the problem of the existence of fixed points of the corresponding resolving integral operator in the space of continuous functions. We will use Sadovskii type fixed point theorem to prove the existence of fixed points of the resolving operator. We will show that the resolving integral operator is condensing with respect to the vector measure of non-compactness in the space of continuous functions and transforms a closed ball in this space into itself.
Keywords Caputo fractional derivative, Langevin type differential equation, boundary value problem, fixed point, condensing map, measure of noncompactness, Mittag-Leffler function
UDC 517.927.4
MSC 34B15, 34B30, 34G20, 47H08, 47H10
DOI 10.35634/vm220305
Received 23 June 2022
Language Russian
Citation Petrosyan G.G. On a boundary value problem for a class of fractional Langevin type differential equations in a Banach space, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2022, vol. 32, issue 3, pp. 415-432.
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