A Novel Discrete Approximation for Nonlinear Variable-Order Hadamard Fractional BVP With Finite Delay via Measures of Noncompactness

This paper investigates a class of nonlinear boundary value problems (BVPs) for Hadamard-type variable-order (VO) fractional differential equations (FDEs) with delay. The VO framework poses substantial analytical challenges because several structural properties that are available for constant-order (CO) fractional models are no longer directly applicable, thereby limiting the use of classical nonlinear analytical techniques. To address this difficulty, we develop an approximation approach in which the variable fractional order is replaced by a sequence of piecewise constant functions. This construction transforms the original VO problem into a family of auxiliary CO problems, providing a rigorous bridge between the VO and CO settings while preserving the essential characteristics of the original model. Within this framework, the existence of solutions is established by combining fixed-point theory with the Kuratowski measure of noncompactness (KMNC), whereas uniqueness is obtained through a contraction argument. Furthermore, the Ulam–Hyers stability of the proposed problem is investigated, yielding quantitative estimates that characterize the robustness of solutions under perturbations. The convergence of the piecewise approximation to the solution of the original VO problem is rigorously justified, providing a solid theoretical foundation for the proposed methodology. Finally, a numerical example is presented to illustrate the applicability and effectiveness of the theoretical results. The proposed approach enriches the analytical theory of Hadamard-type VO fractional models and provides an effective framework for studying delay fractional systems with evolving memory effects.