Discontinuity, Nonlinearity, and Complexity 15(4) (2026) 491--507 | DOI:10.5890/DNC.2026.12.002

Mehdi Selmani$^{1}$, Chahrazed Harrat$^{2}$, Youcef Bouizem$^{3}$

$^{1}$ Department of Mathematics, University of Science and Technology of Oran - Mohamed Boudiaf (USTO-MB), Laboratory of geometry and Analyse "GEANLAB" Oran, 31000, Algeria

$^{2}$ University of Science and Technology of Oran - Mohamed Boudiaf (USTO-MB), Laboratory for Research in Pure and Applied Mathematics (LRMPA), USTO-MB Oran, 31000, Algeria

$^{3}$ Institute of Maintenance and Industrial Safety, University of Oran 2 Mohamed Ben Ahmed, Laboratory of geometry and Analyse "GEANLAB" Oran, 31000, Algeria

Abstract

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.

Acknowledgments

This research work is supported by the General Direction of Scientific Research and Technological Development (DGRSDT), Algeria.

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