Well-posedness of a triply coupled system of fractional Langevin equations with closed boundary conditions

Wei Zhang; Zhongyuan Wang; Yu Zhang; Jinbo Ni; Wei Zhang; Zhongyuan Wang; Yu Zhang; Jinbo Ni

doi:10.3934/era.2025309

Electronic Research Archive

2025, Volume 33, Issue 11: 7017-7033. doi: 10.3934/era.2025309

Previous Article Next Article

Research article Special Issues

Well-posedness of a triply coupled system of fractional Langevin equations with closed boundary conditions

School of Mathematics and Big Data, Anhui University of Science and Technology, Huainan 232001, Anhui, China

Received: 28 September 2025 Revised: 06 November 2025 Accepted: 14 November 2025 Published: 20 November 2025

In this paper, we investigated a class of nonlinear triply coupled systems of fractional Langevin equations subject to closed boundary conditions. The existence of solutions to the proposed boundary value problem was first established by applying Krasnoselskii's fixed point theorem. Furthermore, the uniqueness of the solution was obtained via the Banach contraction mapping principle. To demonstrate the effectiveness of the theoretical results, illustrative examples are provided.
- triply coupled system,
- fractional Langevin equation,
- closed boundary condition,
- existence and uniqueness,
- fixed point theorem
Citation: Wei Zhang, Zhongyuan Wang, Yu Zhang, Jinbo Ni. Well-posedness of a triply coupled system of fractional Langevin equations with closed boundary conditions[J]. Electronic Research Archive, 2025, 33(11): 7017-7033. doi: 10.3934/era.2025309

Related Papers:

Abstract

In this paper, we investigated a class of nonlinear triply coupled systems of fractional Langevin equations subject to closed boundary conditions. The existence of solutions to the proposed boundary value problem was first established by applying Krasnoselskii's fixed point theorem. Furthermore, the uniqueness of the solution was obtained via the Banach contraction mapping principle. To demonstrate the effectiveness of the theoretical results, illustrative examples are provided.

References

[1]	H. G. Sun, Y. Zhang, D. Baleanu, W. Chen, Y. Q. Chen, A new collection of real world applications of fractional calculus in science and engineering, Commun. Nonlinear Sci. Numer. Simul., 64 (2018), 213–231. https://doi.org/10.1016/j.cnsns.2018.04.019 doi: 10.1016/j.cnsns.2018.04.019
[2]	R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. https://doi.org/10.1142/3779
[3]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Research, Elsevier Science, 204 (2006).
[4]	E. Lutz, Fractional Langevin equation, Phys. Rev. E, 64 (2001), 051106. https://doi.org/10.1103/PhysRevE.64.051106
[5]	S. Burov, E. Barkai, Critical exponent of the fractional Langevin equation, Phys. Rev. Lett., 100 (2008), 070601. https://doi.org/10.1103/PhysRevLett.100.070601 doi: 10.1103/PhysRevLett.100.070601
[6]	W. Chen, H. G. Sun, X. Li, Fractional Derivative Modeling in Mechanics and Engineering, Springer, Singapore, 2022. https://doi.org/10.1007/978-981-16-8802-7
[7]	A. Hamiaz, Lyapunov-type inequality and existence of solution for a nonlinear fractional differential equation with anti-periodic boundary conditions, Math. Sci., 18 (2024), 79–90. https://doi.org/10.1007/s40096-022-00486-w doi: 10.1007/s40096-022-00486-w
[8]	M. Alghanmi, R. P. Agarwal, B. Ahmad, Existence of solutions for a coupled system of nonlinear implicit differential equations involving $\varrho$-fractional derivative with anti periodic boundary conditions, Qual. Theory Dyn. Syst., 23 (2024), 6. https://doi.org/10.1007/s12346-023-00861-5 doi: 10.1007/s12346-023-00861-5
[9]	J. L. Edward, A. Chanda, H. K. Nashine, Solutions of higher order fractional differential equations in Riesz space with anti-periodic boundary conditions, J. Nonlinear Convex Anal., 24 (2023), 1929–1938.
[10]	B. Zoubida, M. S. Souid, H. Günerhan, H. Rezazadeh, Fractional differential equations of Riemann-Liouville of variable order with anti-periodic boundary conditions, Eng. Comput., 42 (2025), 595–610. https://doi.org/10.1108/EC-01-2024-0029 doi: 10.1108/EC-01-2024-0029
[11]	H. Baghani, J. Alzabut, J. J. Nieto, A coupled system of Langevin differential equations of fractional order and associated to antiperiodic boundary conditions, Math. Methods Appl. Sci., 47 (2024), 10900–10910. https://doi.org/10.1002/mma.6639 doi: 10.1002/mma.6639
[12]	W. Zhang, J. Ni, Qualitative analysis of tripled system of fractional Langevin equations with cyclic anti-periodic boundary conditions, Fract. Calc. Appl. Anal., 26 (2023), 2392–2420. https://doi.org/10.1007/s13540-023-00201-z doi: 10.1007/s13540-023-00201-z
[13]	S. O. Shah, R. Rizwan, Y. Xia, A. Zada, Existence, uniqueness, and stability analysis of fractional Langevin equations with anti-periodic boundary conditions, Math. Methods Appl. Sci., 46 (2023), 17941–17961. https://doi.org/10.1002/mma.9539 doi: 10.1002/mma.9539
[14]	N. M. Dien, T. Quoc Viet, On mild solutions of the $p$-Laplacian fractional Langevin equations with anti-periodic type boundary conditions, Int. J. Comput. Math., 99 (2022), 1823–1848. https://doi.org/10.1080/00207160.2021.2012167 doi: 10.1080/00207160.2021.2012167
[15]	H. Fazli, J. J. Nieto, Fractional Langevin equation with anti-periodic boundary conditions, Chaos Solitons Fractals, 114 (2018), 332–337. https://doi.org/10.1016/j.chaos.2018.07.009 doi: 10.1016/j.chaos.2018.07.009
[16]	M. M. Matar, J. Alzabut, J. M. Jonnalagadda, A coupled system of nonlinear Caputo-Hadamard Langevin equations associated with nonperiodic boundary conditions, Math. Methods Appl. Sci., 44 (2021), 2650–2670. https://doi.org/10.1002/mma.6711 doi: 10.1002/mma.6711
[17]	H. Baghani, J. Alzabut, J. J. Nieto, A. Salim, Existence and uniqueness of solution of a tripled system of fractional Langevin differential equations with cyclic boundary conditions, Carpathian J. Math., 41 (2025), 273–298. https://doi.org/10.37193/CJM.2025.02.02 doi: 10.37193/CJM.2025.02.02
[18]	J. Liu, T. Shen, X. Shen, The study on the cyclic generalized anti-periodic boundary value problems of the tripled fractional Langevin differential systems, J. Appl. Anal. Comput., 15 (2025), 2301–2326. https://doi.org/10.11948/20240446 doi: 10.11948/20240446
[19]	Z. Zhao, G. Jiang, T. Shen, A class of the cyclic anti-periodic and nonlocal boundary value problem to the self-adjoint tripled fractional Langevin differential systems, Filomat, 38 (2024), 10983–11005. https://doi.org/10.2298/FIL2431983Z doi: 10.2298/FIL2431983Z
[20]	A. Alsaedi, H. A. Saeed, H. Alsulami, Existence and stability of solutions for a nonlocal multi-point and multi-strip coupled boundary value problem of nonlinear fractional Langevin equations, Bull. Math. Sci., 15 (2025), 2450014. https://doi.org/10.1142/S1664360724500140 doi: 10.1142/S1664360724500140
[21]	B. Ahmad, H. A. Saeed, S. K. Ntouyas, A study of a nonlocal coupled integral boundary value problem for nonlinear Hilfer-Hadamard-type fractional Langevin equations, Fractal Fract., 9 (2025), 229. https://doi.org/10.3390/fractalfract9040229 doi: 10.3390/fractalfract9040229
[22]	A. Alsaedi, M. Alnahdi, B. Ahmad, S. K. Ntouyas, On a nonlinear coupled Caputo-type fractional differential system with coupled closed boundary conditions, AIMS Math., 8 (2023), 17981–17995. https://doi.org/10.3934/math.2023914 doi: 10.3934/math.2023914
[23]	B. Ahmad, J. Henderson, R. Luca, Boundary Value Problems for Fractional Differential Equations and Systems, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 9 (2021). https://doi.org/10.1142/11942
[24]	Y. Zhou, Basic Theory of Fractional Differential Equations, 3nd edition, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2024. https://doi.org/10.1142/13289

Reader Comments

Your name:*

Email:*
© 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)