On the existence of solutions and Hyers–Ulam stability for Caputo $ \vartheta $-fractional iterative integro–differential equations

This work investigated a Caputo $ \vartheta $-fractional iterative integro-differential equation (Caputo $ \vartheta $-FIIDE). First, the existence of solutions was established by means of Schauder's fixed point theorem, and uniqueness was obtained under suitable sufficient conditions. The continuous dependence of solutions on the given data was then analyzed. Furthermore, the Ulam-Hyers stability (the U-H stability) and the Ulam-Hyers-Rassias stability (the U-H-R stability) of the proposed problem were examined. A new illustrative example was presented to demonstrate the applicability of the theoretical results and to validate the developed analysis. Thus, this paper presents novel theoretical results, supported by an illustrative example that highlights the applicability of the proposed approach and contributes to the existing literature. Since the stability analysis of fractional iterative integro-differential equations plays a significant role in both theory and applications, the results obtained here provide a useful framework for future research and stimulate further advances in this area.