Existence, Uniqueness and Ulam-Hyers Stability for a Coupled System of Sequential Hilfer Fractional Differential Equations with Nonlocal Coupled Boundary Conditions

This paper investigates the existence, uniqueness, and stability of solutions for a new class of coupled systems of sequential fractional differential equations involving the Hilfer fractional derivative. Generalizing previous works based on Caputo derivatives, we employ the Hilfer operator, which interpolates between Riemann–Liouville and Caputo derivatives. The nonlinear terms are fully coupled, and the boundary conditions are nonlocal and coupled. The main results are established using the Banach Contraction Principle and Schaefer’s Fixed Point Theorem, with rigorous, detailed proofs for each step, addressing specific methodological requirements regarding operator invariance and space completeness. Furthermore, we provide a comprehensive analysis of the Ulam–Hyers stability of the proposed system, with explicitly tracked stability constants. An illustrative example with numerical verification is provided to validate the theoretical findings.