Qualitative Analysis and Numerical Approximation of Nonlinear Caputo–Hadamard Fractional Boundary Value Problems

This paper investigates a class of nonlinear fractional differential equations with boundary value problems involving Caputo–Hadamard-type derivatives. By relaxing the monotonicity constraints on the nonlinear terms and considering more general nonlinear structures, the paper extends the theoretical framework and application scope of the relevant fractional models. Using the upper-lower solution method in conjunction with Schauder’s fixed-point theorem, we establish the existence of exact solutions; furthermore, by applying the Banach contraction mapping principle, we prove the uniqueness of the solutions. Concurrently, we construct a convergent iterative approximation scheme and provide a priori and a posteriori error estimates for numerical solution. Furthermore, the robustness of the solutions to perturbations is characterised via Ulam–Hyers stability analysis, ensuring the reliability of the approximate solutions. Finally, numerical examples are employed to validate all theoretical results. The qualitative theory and numerical analysis methods for Caputo–Hadamard-type fractional boundary value problems have been improved.