Existence, Uniqueness, and Stability of Solutions to the Langevin Equation via the $(k,ψ)$-Hilfer Proportional Fractional Operator with Multi-Point Fractional Boundary Conditions - Selmani, Mehdi; Harrat, Chahrazed; Bouizem, Youcef (Discontinuity, Nonlinearity, and Complexity, 2026)
In this work, we consider a nonlinear fractional Langevin equation involving the $(k,ψ)$-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.