A generalized Z-number approach to solvability, best approximation, and Ulam–Hyers–Rassias stability of Ψ-Hilfer fractional differential equations

This paper investigates the solvability, best approximation, and Ulam–Hyers–Rassias stability of a class of Ψ-Hilfer fractional Hammerstein integro-differential equations in a generalized Z-number setting. The generalized Z-number framework is used to describe uncertainty together with the reliability of the available information. By applying a fixed-point approach in a generalized metric space, sufficient conditions for the existence and uniqueness of solutions are established, together with best-approximation estimates and Ulam–Hyers–Rassias stability results. An illustrative example is provided to verify the main assumptions and the corresponding contraction condition, demonstrating the applicability of the theoretical results.