Comparative symmetry analysis and qualitative properties of impulsive multi-delay systems across a family of power Caputo fractional kernels

Yasir A. Madani; Mohammed Almalahi; Bakri Younis; Alawia Adam; Nidal Eljaneid; Khaled Aldwoah; Amer Alsulami; Yasir A. Madani; Mohammed Almalahi; Bakri Younis; Alawia Adam; Nidal Eljaneid; Khaled Aldwoah; Amer Alsulami

doi:10.3934/math.2026428

AIMS Mathematics

2026, Volume 11, Issue 4: 10372-10399. doi: 10.3934/math.2026428

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Research article

Comparative symmetry analysis and qualitative properties of impulsive multi-delay systems across a family of power Caputo fractional kernels

1.
Department of Mathematics, College of Science, University of Ha'il, 55473 Ha'il, Saudi Arabia
2.
Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Chennai 602105, Tamil Nadu, India
3.
Department of Mathematics, College of Computer and Information Technology, Al-Razi University, Sana'a 12544, Yemen
4.
Department of Mathematics, College of Science, King Khalid University, Abha 61421, Saudi Arabia
5.
Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia
6.
Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia
7.
Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah 42351, Saudi Arabia
8.
Department of Mathematics, Turabah University College, Taif University, Taif 21944, Saudi Arabia

Received: 17 January 2026 Revised: 02 April 2026 Accepted: 08 April 2026 Published: 15 April 2026
MSC : 34A08, 34K20, 34K45, 47H10

In this study, we perform a comparative symmetry analysis and investigate the qualitative properties of a nonlinear impulsive fractional differential system with multiple delays and nonlocal boundary conditions. By utilizing the generalized power Caputo fractional derivative, we present a unified theoretical framework that encompasses several operators—including the Atangana-Baleanu, Caputo-Fabrizio, and weighted Hattaf derivatives—as special cases. This generality guarantees that our findings are relevant to a range of fractional kernels, emphasizing the inherent symmetry characteristics of these operators. We establish adequate criteria for the existence and uniqueness of solutions through fixed-point theory. We also show that the system is Ulam-Hyers stable, an important property for maintaining its strength in the face of change. A convergent numerical scheme confirms the theoretical results, and a sensitivity analysis demonstrates the effect of kernel symmetry on stability margins. The networked control system application shows how useful the framework is in real life. The results show that the framework can include complex genetic phenomena and spontaneous interactions that are often ignored in traditional models.
- fractional derivatives,
- impulsive delay,
- stability,
- fixed-point theory,
- symmetry analysis
Citation: Yasir A. Madani, Mohammed Almalahi, Bakri Younis, Alawia Adam, Nidal Eljaneid, Khaled Aldwoah, Amer Alsulami. Comparative symmetry analysis and qualitative properties of impulsive multi-delay systems across a family of power Caputo fractional kernels[J]. AIMS Mathematics, 2026, 11(4): 10372-10399. doi: 10.3934/math.2026428

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Abstract

In this study, we perform a comparative symmetry analysis and investigate the qualitative properties of a nonlinear impulsive fractional differential system with multiple delays and nonlocal boundary conditions. By utilizing the generalized power Caputo fractional derivative, we present a unified theoretical framework that encompasses several operators—including the Atangana-Baleanu, Caputo-Fabrizio, and weighted Hattaf derivatives—as special cases. This generality guarantees that our findings are relevant to a range of fractional kernels, emphasizing the inherent symmetry characteristics of these operators. We establish adequate criteria for the existence and uniqueness of solutions through fixed-point theory. We also show that the system is Ulam-Hyers stable, an important property for maintaining its strength in the face of change. A convergent numerical scheme confirms the theoretical results, and a sensitivity analysis demonstrates the effect of kernel symmetry on stability margins. The networked control system application shows how useful the framework is in real life. The results show that the framework can include complex genetic phenomena and spontaneous interactions that are often ignored in traditional models.

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