On uniqueness of solution to functional-integral equation of fractional order with involution

L.M. Eneeva

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Abstract: The paper studies a functional integral equation with a fractional integration operator and an involution operator, which arise when solving boundary value problems for differential equations that contain a composition of left- and right-sided fractional derivatives. These equations underlie mathematical models of various physical and geophysical processes, such as describing dissipative oscillatory systems.

Aim. The study aims to investigate a functional integral equation with an operator of fractional integration involving an involution operator in the critical case. Research methods. To solve the problem, we employ methods of the theory of integral equations of the first kind, operator theory and properties of completely monotone functions.
Results. It has been shown that the equation under study can be reduced to the problem of solving an integral equation of the first kind with a positive kernel, in a class of functions that change sign under the action of an operator, and for this class of functions, a theorem on the uniqueness of the solution has been proven.

Keywords: functional integral equation, Riemann–Liouville fractional integral, involution, Mittag-Leffler function, positive operator, completely monotone function

For citation. Eneeva L.M. On uniqueness of solution to functional-integral equation of fractional order with involution. News of the Kabardino-Balkarian Scientific Center of RAS. 2025. Vol. 27. No. 6. Pp. 39–46. DOI:
10.35330/1991-6639-2025-27-6-39-46

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Information about the authors

Liana M. Eneeva, Candidate of Physics and Mathematics, Senior Researcher, Department of Mathematical Modeling of Geophysical Processes, Institute of Applied Mathematics and Automation – branch of the Kabardino-Balkarian Scientific Center of the Russian Academy of Sciences;
89 A, Shortanov street, Nalchik, 360000, Russia;
eneeva72@list.ru, ORCID: https://orcid.org/0000-0003-2530-5022, SPIN-code: 3403-8412