Existence and Stability of Nonlinear Langevin FDEs with Mixed ψ-Hilfer and ψ-Caputo Derivatives Under Nonlocal Boundary Conditions

Abstract

In this paper, we investigate the existence, uniqueness, and Hyers-Ulam stability of a nonlinear fractional Langevin equation involving a generalized fractional derivative that unifies the ψ-Hilfer and ψ-Caputo types. The problem incorporates mixed and nonlocal boundary conditions. Existence results are derived using Krasnoselskii's fixed-point theorem, while uniqueness is established through Banach's contraction principle. Moreover, Ulam-type stability is analyzed under both Ulam-Hyers and Ulam-Hyers-Rassias criteria. Finally, a numerical example is presented, and the computed results are summarized in a table, showing the influence of different ψ-functions on the stability constants and confirming the consistency between theoretical and numerical findings.