On the Ulam-type stability of higher-order iterative Volterra integro-delay differential equations

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.