Asymptotic properties of solutions to Caputo-Hadamard fractional differential equations

Haonan Zhang; Zidi Zhao; Qixiang Dong; Haonan Zhang; Zidi Zhao; Qixiang Dong

doi:10.3934/math.2026695

AIMS Mathematics

2026, Volume 11, Issue 6: 16983-17008. doi: 10.3934/math.2026695

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Research article

Asymptotic properties of solutions to Caputo-Hadamard fractional differential equations

School of Mathematics, Yangzhou University, No. 180 Siwangting Road, Hanjiang District, Yangzhou, Jiangsu 225002, China

Received: 28 March 2026 Revised: 19 May 2026 Accepted: 22 May 2026 Published: 12 June 2026
MSC : 34A08, 34D05, 34D20

This paper investigates the stability properties of Caputo-Hadamard fractional differential equations. We first analyze the asymptotic behavior and rigorously prove a specific convergence rate for these equations. Then, a novel stability criterion called logarithmic Mittag-Leffler stability is proposed. By employing the fixed-point theorem in an innovative Banach space equipped with a designed weighted norm, we demonstrate that when the linearization spectrum of the Caputo-Hadamard fractional differential equation lies within a prescribed sector, the equilibrium point of the equation exhibits logarithmic Mittag-Leffler stability. This result leads to a version of Lyapunov's first method for Caputo-Hadamard fractional differential equations, demonstrating its stability in the presence of logarithmic memory.
Citation: Haonan Zhang, Zidi Zhao, Qixiang Dong. Asymptotic properties of solutions to Caputo-Hadamard fractional differential equations[J]. AIMS Mathematics, 2026, 11(6): 16983-17008. doi: 10.3934/math.2026695

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Abstract

This paper investigates the stability properties of Caputo-Hadamard fractional differential equations. We first analyze the asymptotic behavior and rigorously prove a specific convergence rate for these equations. Then, a novel stability criterion called logarithmic Mittag-Leffler stability is proposed. By employing the fixed-point theorem in an innovative Banach space equipped with a designed weighted norm, we demonstrate that when the linearization spectrum of the Caputo-Hadamard fractional differential equation lies within a prescribed sector, the equilibrium point of the equation exhibits logarithmic Mittag-Leffler stability. This result leads to a version of Lyapunov's first method for Caputo-Hadamard fractional differential equations, demonstrating its stability in the presence of logarithmic memory.

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