Nonlinear Langevin-type Erdélyi-Kober and quantum-Caputo fractional operators with separated boundary conditions

Fractional Langevin equations have emerged as a powerful tool for modeling systems subject to memory effects, anomalous damping, and non-uniform temporal dynamics. Motivated by the need to capture simultaneously weighted-memory effects - naturally described by the Erdélyi–Kober (EK) fractional operator - and discrete-scale quantum dynamics - described by the quantum-Caputo ( $q$-Caputo) operator - we studied a nonlinear Langevin-type boundary value problem that couples these two structurally distinct fractional operators subject to separated boundary conditions. Our analysis proceeded in three stages. First, we derived an equivalent integral formulation of the boundary value problem in the quantum-EK setting, requiring new mixed EK– $q$–integral identities of independent interest. Second, we established existence and uniqueness in $C([0,T],\mathbb{R})$ via three fixed-point theorems: the Banach contraction principle (uniqueness under a global Lipschitz condition on $f$), the Krasnosel'skiĭ theorem (existence for broader nonlinearities via a sum-of-operators decomposition), and the Leray–Schauder nonlinear alternative (existence under polynomial growth without contraction). These approaches are complementary, progressively relaxing hypotheses on $f$ while trading sharpness for generality. Third, we presented numerical experiments based on the spectral collocation method, together with analytical examples verifying that the theoretical conditions can be realized with explicit data.