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Research Article | Open Access

On a boundary value problem for a nonlinear differential equation with a Riemann-Liouville fractional derivative of variable order and nonlocal boundary conditions

Alexander A. Potapov1,2( ), Vetlugin D. Beybalaev3,4,5, Abutrab A. Aliverdiev3,4,6
Kotel'nikov Institute of Radioengineering and Electronics, Russian Academy of Sciences, Moscow 125009, Russia
JNU-IRE RAS Joint Laboratory of Information Technology and Fractal Processing of Signals, Jinan University, Guangzhou 510632, China
Dagestan State University, Gadzhieva Str., 43A, Makhachkala 367000, Russia
Institute for Geothermal Researches and Renewable Energy — A Branch of the Joint Institute for High Temperatures, Russian Academy of Sciences Pr. I. Shamilya, 39A, Makhachkala 367030, Russia
Dagestan State University of National Economy, J. Atayev Str., 5, Makhachkala 367000, Russia
Institute of Physics DFRC, Russian Academy of Sciences, M. Yaragskogo Str., 94, Makhachkala 367003, Russia
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Abstract

The paper studies the existence of at least two positive solutions for a nonlinear differential equation with a fractional Riemann-Liouville derivative of variable order and nonlocal boundary conditions. To transform the original equation into an integral equation, the Green function is constructed and boundedness conditions for the Green function are obtained. A theorem on the existence of at least two positive solutions to the problem is proven. An example is given that shows the existence of two positive solutions, and approximation graphs are constructed for clarity.

Keywords

fractional derivativeboundary conditionsGreen functionnon-stationary processes

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Electronic Research Archive
Volume 33 Issue 9,
September 2025
Pages 5829-5844
DOI: 10.3934/era.2025259

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Cite this article:
Potapov AA, Beybalaev VD, Aliverdiev AA. On a boundary value problem for a nonlinear differential equation with a Riemann-Liouville fractional derivative of variable order and nonlocal boundary conditions. Electronic Research Archive, 2025, 33(9): 5829-5844. https://doi.org/10.3934/era.2025259

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Received: 09 April 2025
Revised: 13 August 2025
Accepted: 15 August 2025
Published: 26 September 2025
©2025 the Author(s), licensee AIMS Press.

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)