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. Potapov; Vetlugin D. Beybalaev; Abutrab A. Aliverdiev; Alexander A. Potapov; Vetlugin D. Beybalaev; Abutrab A. Aliverdiev

doi:10.3934/era.2025259

Electronic Research Archive

2025, Volume 33, Issue 9: 5829-5844. doi: 10.3934/era.2025259

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

1.
Kotel'nikov Institute of Radioengineering and Electronics, Russian Academy of Sciences, Moscow 125009, Russia
2.
JNU-IRE RAS Joint Laboratory of Information Technology and Fractal Processing of Signals, Jinan University, Guangzhou 510632, China
3.
Dagestan State University, Gadzhieva Str., 43A, Makhachkala 367000, Russia
4.
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
5.
Dagestan State University of National Economy, J. Atayev Str., 5, Makhachkala 367000, Russia
6.
Institute of Physics DFRC, Russian Academy of Sciences, M. Yaragskogo Str., 94, Makhachkala 367003, Russia

Received: 09 April 2025 Revised: 13 August 2025 Accepted: 15 August 2025 Published: 26 September 2025

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.
- fractional derivative,
- boundary conditions,
- Green function,
- non-stationary processes
Citation: Alexander A. Potapov, Vetlugin D. Beybalaev, Abutrab A. Aliverdiev. On a boundary value problem for a nonlinear differential equation with a Riemann-Liouville fractional derivative of variable order and nonlocal boundary conditions[J]. Electronic Research Archive, 2025, 33(9): 5829-5844. doi: 10.3934/era.2025259

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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.

References

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