Existence and Ulam stability results of hybrid Langevin pantograph ψ-fractional coupled systems

This study presents results on the solutions of a coupled system of hybrid Langevin fractional pantograph differential equations involving ?-Caputo type fractional derivatives within Banach spaces. We establish the uniqueness of solutions using Banach?s fixed-point theorem and confirm their existence through Dhage?s hybrid fixed-point theorem for the sum of three operators. Additionally, we investigate the stability of these solutions in both the Ulam-Hyers sense and its generalized form. The theoretical findings are further supported by several illustrative examples.