Analysis of Fractional Pantograph Equations with Riemann–Liouville Derivatives and Nonlocal Conditions

This work develops a rigorous analytical framework for the study of nonlinear fractional pantograph equations involving sequential Riemann–Liouville derivatives of different fractional orders together with nonlocal integral boundary conditions. These models arise naturally in systems that exhibit memory, scaling effects, and nonlocal interactions, making their qualitative analysis both challenging and essential. The methodological approach relies on a combination of Schauder’s fixed-point theorem and the Banach contraction principle to obtain fundamental results on the solvability of the problem. Within the Banach space $L^{1}(J)$, we establish the existence of at least one solution by demonstrating the compactness of the associated integral operator through the Kolmogorov compactness criterion. Under suitable Lipschitz-type assumptions, uniqueness is further guaranteed by showing that the corresponding operator is a contraction.
In addition, we investigate the continuous dependence of solutions on initial data and nonlinear terms, confirming the stability and robustness of the proposed model under small perturbations. An illustrative example is included to verify the theoretical framework and to demonstrate how the analytical results can be applied in practice. Overall, the findings of this study contribute to the qualitative theory of fractional pantograph systems and provide a unified basis for modeling complex nonlocal phenomena with memory effects in applied mathematics, physics, and engineering.