Existence, Uniqueness, and Stability of Solutions to the Langevin Equation via the $(k,\psi)$-Hilfer Proportional Fractional Operator with Multi-Point Fractional Boundary Conditions
DOI:
https://doi.org/10.5890/DNC.2026.12.002Abstract
In this work, we consider a nonlinear fractional Langevin equation involving the $(k,\psi)$-Hilfer proportional fractional operator, which unifies several well-known fractional derivatives as particular cases, together with nonlocal multi-point fractional boundary conditions. This setting combines the operator and the boundary conditions within a coherent mathematical framework that allows the treatment of a broad class of Langevin-type problems in a unified manner. By means of Banach’s contraction principle and Krasnoselskii’s fixed-point theorem, sufficient conditions ensuring the existence and uniqueness of solutions are derived. Moreover, Ulam-Hyers and Ulam-Hyers-Rassias stability of the solutions are investigated. Illustrative examples are included to highlight the applicability of the obtained results.References
[1] Samko, S.G., Kilbas, A.A., and Marichev, O.I. (1993), Fractional integrals and derivatives, Gordon and Breach Science Publishers, Yverdon Yverdon-les-Bains, Switzerland.
[2] Daftardar-Gejji, V. (2019), Fractional calculus and fractional differential equations, Springer Singapore.
[3] Coffey, W. and Kalmykov, Y.P. (2012), The Langevin equation: with applications to stochastic problems in physics, chemistry and electrical engineering, (Vol. 27), World Scientific.
[4] Hilal, K., Kajouni, A., and Lmou, H. (2023), Existence and stability results for a coupled system of Hilfer fractional Langevin equation with non local integral boundary value conditions, Filomat, 37(4), 1241-1259.
[5] Zhou, Y. (2023), Basic theory of fractional differential equations, World Scientific.
[6] Sudsutad, W., Ntouyas, S.K., and Thaiprayoon, C. (2021), Nonlocal coupled system for $psi$-Hilfer fractional order Langevin equations, AIMS Mathematics, 6(9), 9731-9756.
[7] Hilfer, R. (2000), Applications of fractional calculus in physics, World Scientific.
[8] Da Vanterler, J., Sousa, C.C., and de Oliveira, E. (2018), On the $psi$-Hilfer fractional derivative, Communications in Nonlinear Science and Numerical Simulation, 60, 72-91.
[9] Oliveira, D.S. and Capelas de Oliveira, E. (2018), Hilfer-Katugampola fractional derivatives, Journal of Computational and Applied Mathematics, 37(3), 3672-3690.
[10] Kucche, K.D. and Mali, A.D. (2021), On the nonlinear (k,$psi$)-Hilfer fractional differential equations, Chaos, Solitons & Fractals, 152, 111335.
[11] Sudsutad, W., Kongson, J., and Thaiprayoon, C. (2024), On generalized (k, $psi$)-Hilfer proportional fractional operator and its applications to the higher-order Cauchy problem, Boundary Value Problems, 2024(1), 83.
[12] Granas, A. and Dugundji, J. (2003), Fixed point theory, Springer.
[13] Benchohra, M. and Lazreg, J.E. (2014), Existence and uniqueness results for nonlinear implicit fractional differential equations with boundary conditions, Romanian Journal of Mathematics and Computer Science, 4(1), 79-92.
[14] Zhou, Y. (2009), Existence and uniqueness of solutions for a system of fractional differential equations, Fractional Calculus and Applied Analysis, 12(2), 195-204.
[15] Wongcharoen, A., Ahmad, B., Ntouyas, S.K., and Tariboon, J. (2020), Three-point boundary value problems for the Langevin equation with the Hilfer fractional derivative, Advances in Mathematical Physics, 2020(1), 9606428.
[16] Abbas, M.I. (2021), Investigation of Langevin equation in terms of generalized proportional fractional derivatives with respect to another function, Filomat, 35(12), 4073-4085.
[17] Abbas, S., Ahmad, B., Benchohra, M., and Salim, A. (2024), Fractional difference, differential equations, and inclusions: analysis and stability, Elsevier.
[18] Lmou, H., Hilal, K., and Kajouni, A. (2022), A new result for $psi$-Hilfer fractional pantograph-type Langevin equation and inclusions, Mathematics, 2022(1), 2441628.
[19] Fazli, H., Sun, H., and Nieto, J.J. (2020), Fractional Langevin equation involving two fractional orders: existence and uniqueness revisited, Mathematics, 8(5), 743.
[20] Cheng, H., Naila, A., Zada, I., Popa, L., and Kallekh, A. (2024), (k, $varphi$)-Hilfer fractional Langevin differential equation having multipoint boundary conditions, Boundary Value Problems, 2024(1), 113.
[21] Li, B., Sun, S., Li, Y., and Zhao, P. (2014), Multi-point boundary value problems for a class of Riemann-Liouville fractional differential equations, Advances in Difference Equations, 2014(1), 151.
[22] D{i}az, R. and Pariguan, E. (2007), On hypergeometric functions and Pochhammer k-symbol, Divulgaciones Matemáticas, 15(2), 179-192.
[23] Abbas, S., Benchohra, M., Lagreg, J.E., Alsaedi, A., and Zhou, Y. (2017), Existence and Ulam stability for fractional differential equations of Hilfer-Hadamard type, Advances in Difference Equations, 2017(1), 180.
[24] Nuchpong, C., Ntouyas, S.K., Vivek, D., and Tariboon, J. (2021), Nonlocal boundary value problems for $psi$-Hilfer fractional-order Langevin equations, Boundary Value Problems, 2021(1), 34.
[25] Dhaniya, S., Kumar, A., Khan, A., Abdeljawad, T., and Alqudah, M.A. (2023), Existence results of Langevin equations with Caputo–Hadamard fractional operator, Journal of Mathematics, 2023(1), 2288477.
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