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Article

Analysis of Delay-Type Integro-Differential Systems Described by the Φ-Hilfer Fractional Derivative

by
Ravichandran Vivek
1,
Waleed Mohammed Abdelfattah
2,3,* and
Elsayed Mohamed Elsayed
4,5
1
Department of Mathematics, Sri Krishna Arts and Science College, Coimbatore 641008, India
2
College of Engineering, University of Business and Technology, Jeddah 23435, Saudi Arabia
3
Department of Engineering Mathematics and Physics, Faculty of Engineering, Zagazig University, Zagazig P.O. Box 44519, Egypt
4
Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
5
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
*
Author to whom correspondence should be addressed.
Axioms 2025, 14(8), 629; https://doi.org/10.3390/axioms14080629
Submission received: 9 July 2025 / Revised: 4 August 2025 / Accepted: 7 August 2025 / Published: 11 August 2025
(This article belongs to the Special Issue Dynamics, Stability, Chaos, Control and Applications of Dynamical Systems)
Versions Notes

Abstract

In this article, a novel type of equation, namely the Φ-Hilfer fractional-order integro-differential delay system (Φ-HFOIDDS), is proposed. Here, we study the existence and Hyers–Ulam–Mittag–Leffler (H-U-M-L) stability of the aforementioned equation which are obtained by using the multivariate Mittag–Leffler function, Banach contraction principle, and Picard operator method as well as generalized Gronwall inequality. Finally, we conclude this paper by constructing a suitable example to illustrate the applicability of the principal outcomes.
Keywords: Φ-Hilfer integro-differential equations; delay differential equations; multivariate Mittag–Leffler function; Picard operator method; generalized Gronwall inequality Φ-Hilfer integro-differential equations; delay differential equations; multivariate Mittag–Leffler function; Picard operator method; generalized Gronwall inequality

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MDPI and ACS Style

Vivek, R.; Abdelfattah, W.M.; Elsayed, E.M. Analysis of Delay-Type Integro-Differential Systems Described by the Φ-Hilfer Fractional Derivative. Axioms 2025, 14, 629. https://doi.org/10.3390/axioms14080629

AMA Style

Vivek R, Abdelfattah WM, Elsayed EM. Analysis of Delay-Type Integro-Differential Systems Described by the Φ-Hilfer Fractional Derivative. Axioms. 2025; 14(8):629. https://doi.org/10.3390/axioms14080629

Chicago/Turabian Style

Vivek, Ravichandran, Waleed Mohammed Abdelfattah, and Elsayed Mohamed Elsayed. 2025. "Analysis of Delay-Type Integro-Differential Systems Described by the Φ-Hilfer Fractional Derivative" Axioms 14, no. 8: 629. https://doi.org/10.3390/axioms14080629

APA Style

Vivek, R., Abdelfattah, W. M., & Elsayed, E. M. (2025). Analysis of Delay-Type Integro-Differential Systems Described by the Φ-Hilfer Fractional Derivative. Axioms, 14(8), 629. https://doi.org/10.3390/axioms14080629

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