This paper studies a class of nonlinear variable-order fractional boundary value problems involving the Riemann–Liouville fractional derivative. An infinite-partition approximation is introduced by replacing the variable-order function with a sequence of piecewise constant functions, transforming the original problem into a family of constant-order fractional boundary value problems. For each approximating problem, an equivalent integral equation and the corresponding Green function are derived. Existence, uniqueness, and Ulam–Hyers stability of solutions are established using Schauder's fixed-point theorem and the Banach contraction principle. It is further proved that the approximating solutions converge uniformly to a solution of the original variable-order problem, providing a rigorous connection between constant-order and variable-order fractional models. A numerical example illustrates the theoretical results.

In this study, we addressed a higher-order iterative Volterra integro-delay differential equation (HOIVIDDE) involving two variable time delays. Our primary focus was on establishing the uniqueness of solutions and analyzing Ulam-type stability properties of the considered HOIVIDDE. We presented three novel results concerning Ulam–Hyers–Rassias (U-H-R), $\sigma$-semi-Ulam–Hyers ( $\sigma$-semi-U-H), and Ulam–Hyers (U-H) stability for HOIVIDDE, along with uniqueness results for the associated initial value problem (IVP). The analysis was conducted using the properties of iterative functions, the Banach fixed point theorem, and the Bielecki metric. Notably, this was the first study that extended and enhanced these qualitative properties to an $n$th-order HOIVIDDE. To illustrate the applicability of the results obtained here, we provided an example verifying the requirements of the new theorems.