Hyers-Ulam stability for coupled fractional (p, q)-difference systems with mixed boundary conditions and Picard-assisted DA-PINN verification

Abstract This paper investigates a class of coupled fractional (p, q)-difference systems with mixed boundary conditions. By transforming the boundary value problem into an equivalent integral fixed-point equation, we establish the existence of solutions by Schaefer's fixed point theorem and prove uniqueness by the Banach contraction principle. Under suitable Lipschitz assumptions, a Hyers-Ulam stability criterion is further derived, showing that the deviation caused by bounded perturbations is controlled by a constant multiple of the perturbation amplitude. A numerical example is presented to verify the theoretical assumptions and illustrate the stability estimate. In the numerical part, Picard iteration is used to generate auxiliary fixed-point data, and a data-assisted physics-informed neural network is employed to directly learn the perturbation-induced response behavior. The numerical results show that the perturbation error remains below the theoretical Hyers-Ulam bound and exhibits an approximately linear dependence on the perturbation amplitude. These findings support the theoretical results on existence, uniqueness, and stability.