Coupled Langevin system with generalized proportional fractional derivatives relative to a function and Riemann-Stieltjes integral boundary conditions

We investigate a coupled Langevin system driven by the generalized proportional fractional derivative with respect to a function, closed by Riemann-Stieltjes integral boundary data that intertwine the two unknowns at the right endpoint. Under the assumption that the sum of the two fractional orders within each component exceeds one, the problem is recast as an equivalent system of Volterra-type integral equations. Existence is then established via Krasnoselskii's fixed point theorem and, alternatively, the Leray-Schauder nonlinear alternative; uniqueness follows from the Banach contraction principle. Three numerical examples illustrate the results, with explicit verification of every hypothesis.