On the generalized coupled Hadamard-Gronwall-Bellman-type inequalities with applications to fractional delay systems

Sina Etemad; Ali Akgül; J. Alberto Conejero; Sina Etemad; Ali Akgül; J. Alberto Conejero

doi:10.3934/math.20251228

AIMS Mathematics

2025, Volume 10, Issue 11: 27954-27984. doi: 10.3934/math.20251228

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On the generalized coupled Hadamard-Gronwall-Bellman-type inequalities with applications to fractional delay systems

1.
Department of Mathematics, Saveetha School of Engineering, SIMATS, Saveetha University, Chennai 602105, Tamil Nadu, India
2.
Institute of Graduate Studies and Research, Cyprus International University, Nicosia, Northern Cyprus, via Mersin 10, Türkiye; Email: sina.etemad@gmail.com (S.E.)
3.
Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, 46022 València, Spain; Email:aakgul1@upvnet.upv.es (A.A.), aconejero@upv.es (J.A.C.)
4.
Department of Mathematics, Art and Science Faculty, Siirt University, 56100, Siirt, Turkey

Received: 15 August 2025 Revised: 09 November 2025 Accepted: 18 November 2025 Published: 28 November 2025
MSC : 26A33, 34A08, 35A23, 39A11

The Gronwall-Bellman inequality is a primary tool for proving various types of stability. For this importance, the present paper focuses on the generalized forms of the well-known Gronwall-Bellman inequality in the context of the Hadamard fractional calculus. We prove and generalize the coupled version of the Hadamard-Gronwall-Bellman inequality and then, generalize its extended form with the sum of two non-decreasing functions. In the sequel, the applicability of these inequalities is established in proving the existence and Ulam-Hyers stability of a Caputo-Hadamard coupled delay system and a Caputo-Hadamard damped initial value problem, which are appeared in population dynamic problems.
- Hadamard fractional integral,
- Gronwall-Bellman inequality,
- coupled system,
- stability of solutions,
- existence results
Citation: Sina Etemad, Ali Akgül, J. Alberto Conejero. On the generalized coupled Hadamard-Gronwall-Bellman-type inequalities with applications to fractional delay systems[J]. AIMS Mathematics, 2025, 10(11): 27954-27984. doi: 10.3934/math.20251228

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Abstract

The Gronwall-Bellman inequality is a primary tool for proving various types of stability. For this importance, the present paper focuses on the generalized forms of the well-known Gronwall-Bellman inequality in the context of the Hadamard fractional calculus. We prove and generalize the coupled version of the Hadamard-Gronwall-Bellman inequality and then, generalize its extended form with the sum of two non-decreasing functions. In the sequel, the applicability of these inequalities is established in proving the existence and Ulam-Hyers stability of a Caputo-Hadamard coupled delay system and a Caputo-Hadamard damped initial value problem, which are appeared in population dynamic problems.

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