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

Sandra Pinelas; Cemil Tunç; Osman Tunç; Merve Şengün Oğuz; Sandra Pinelas; Cemil Tunç; Osman Tunç; Merve Şengün Oğuz

doi:10.3934/math.2026052

AIMS Mathematics

2026, Volume 11, Issue 1: 1219-1238. doi: 10.3934/math.2026052

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

1.
Departamento de Ciências Exatas e Engenharia, Academia Militar, Av. Conde Castro Guimarães, 2720-113 Amadora, Portugal
2.
Center for Research and Development in Mathematics and Applications (CIDMA), Departamento de Matemática, Universidade de Aveiro, 3810-193 Aveiro, Portugal
3.
School of Engineering and Natural Sciences, Istanbul Medipol University, 34810, Istanbul, Turkey
4.
Department of Computer Programming, Baskale Vocational School, Van Yuzuncu Yil University, 65080, Turkey
5.
Maltepe, Istanbul

Received: 12 November 2025 Revised: 26 December 2025 Accepted: 08 January 2026 Published: 16 January 2026
MSC : 34K05, 45D05, 45J05, 47H10

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.
- Ulam-type stability,
- iterative methods,
- integro-delay differential equation,
- higher order,
- Banach fixed point theorem,
- Bielecki metric
Citation: Sandra Pinelas, Cemil Tunç, Osman Tunç, Merve Şengün Oğuz. On the Ulam-type stability of higher-order iterative Volterra integro-delay differential equations[J]. AIMS Mathematics, 2026, 11(1): 1219-1238. doi: 10.3934/math.2026052

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Abstract

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.

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