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

Cemil Tunç; Ivanka Stamova; Osman Tunç; Jen-Chih Yao; Cemil Tunç; Ivanka Stamova; Osman Tunç; Jen-Chih Yao

doi:10.3934/era.2026290

Electronic Research Archive

2026, Volume 34, Issue 9: 6636-6657. doi: 10.3934/era.2026290

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

1.
School of Engineering and Natural Sciences, Istanbul Medipol University, Istanbul 34810, Turkey
2.
Department of Mathematics, The University of Texas at San Antonio, San Antonio, TX 78249, USA
3.
Department of Computer Programing, Baskale Vocational School, Van Yuzuncu Yil University, Van 65080, Turkey
4.
Center for General Education, China Medical University, Taichung 40402, Taiwan
5.
Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan

Received: 21 April 2026 Revised: 12 July 2026 Accepted: 28 July 2026 Published: 04 August 2026

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.
- iterative integro-differential equation,
- fractional derivative,
- existence and uniqueness of solutions,
- Ulam-type stability
Citation: Cemil Tunç, Ivanka Stamova, Osman Tunç, Jen-Chih Yao. On the existence of solutions and Hyers–Ulam stability for Caputo $ \vartheta $-fractional iterative integro–differential equations[J]. Electronic Research Archive, 2026, 34(9): 6636-6657. doi: 10.3934/era.2026290

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Abstract

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.

References

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