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Vol. 24, No. 4. P. 47–54

Adyghe Int. Sci. J. Vol. 24, No. 4. P. 47–54. 

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DOI: https://doi.org/10.47928/1726-9946-2024-24-4-47-54
EDN: BQQHJC

MATHEMATICS

MSC 34A08, 34B05 Original Article

Nonlocal boundary value problem

for a linear ordinary delay differential equation with Gerasimov–Caputo derivative

Mazhgikhova Madina Gumarovna
Candidate of Physical and Mathematical Sciences, Junior Researcher Department of Fractional Calculus Institute of Applied Mathematics and Automation KBSC RAS (360000, Kabardino-Balkarian Republic, Nalchik, 89A Shortanov St.), ORCID: https://orcid.org/0000-0001-7612-8850, madina.mazhgihova@yan-dex.ru

Abstract. In this paper, for a linear ordinary delay differential equation with constant coefficients and with the Gerasimov–Caputo derivative, a solution to the nonlocal boundary value problem with conditions, connecting the value of the unknown function at the end of the interval with the values at interior points, is constructed. The solution to the problem is obtained in explicit form. The condition for the unique solvability of the problem is written out.
A representation of Green’s function is obtained. Green’s function is defined in terms of the special function W\nu (t), which, in turn, is defined using the generalized Mittag-Leffler functions. The properties of the Green’s function are proven. The solution to the problem is formulated in terms of Green’s function. The existence and uniqueness theorem to the problem under study is formulated and proved. The proofs of the lemma and theorem are given using methods of the theory of fractional calculus, the theory of special functions, Green’s function method, and the theory of integral equations.

Keywords: fractional differential equation, Gerasimov–Caputo derivative, delay differential equation, nonlocal boundary value problem, nonlocal conditions, generalized Mittag-Leffler function.

Funding. The work was carried out within the framework of the state assignments of the Ministry of Education and Science of the Russian Federation under the projects: Nonlinear singular integro-differential equations and boundary value problems (FEGS-2020-0001); Investigation of boundary value problems for equations with generalised fractional differentiation operators, their application to mathematical modelling of physical and socioeconomic processes. (1021032424223-6).
Competing interests. There are no conflicts of interest regarding authorship and publication.
Contribution and Responsibility. The author participated in the writing of the article and is fully responsible for submitting the final version of the article to the press.

For citation. Mazhgikhova M. G. Nonlocal boundary value problem for a linear ordinary delay differential equation with Gerasimov–Caputo derivative. Adyghe Int. Sci. J. 2024. Vol. 24, No. 4. Pp. 47–54.
DOI: https://doi.org/10.47928/1726-9946-2024-24-4-47-54; EDN: BQQHJC

Submitted 29.11.2024; approved after reviewing 06.12.2024; accepted for publication 13.12.2024.

© Mazhgikhova M. G., 2024

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