<p>Hadamard fractional calculus has emerged as a prominent mathematical alternative for accurately capturing the creep behavior of ultra-slow varying physical evolution. In practical modeling scenarios, Hadamard fractional systems frequently demonstrate high-dimensionality and robust nonlinearity, which profoundly influence further detection and analysis. Consequently, this paper establishes the Lyapunov-Schmidt reduction for the Caputo-Hadamard fractional differential system and conducts both theoretical and computational analyses of the reduced equations. As a by-product and auxiliary of the reduction, an integration by parts formula and a modified inner product definition are presented to align with the characteristics of the Caputo-Hadamard fractional operator. Then, the linearized operators associated with the Caputo-Hadamard fractional differential system with fractional orders <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11071_2025_11651_Article_IEq1.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">$0<\alpha <1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>α</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11071_2025_11651_Article_IEq2.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">$1<\alpha <2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>α</mi> <mo><</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> are demonstrated to be Fredholm operators with index zero separately, thereby establishing the standard formulations of the Lyapunov-Schmidt reduction for the Caputo-Hadamard fractional differential system. Finally, by employing higher-order Gâteaux derivatives of multivariable functions and leveraging the fundamental properties of reduced equations, two illustrative examples are provided to substantiate the efficacy of the Lyapunov-Schmidt reduction for the Caputo-Hadamard fractional differential system.</p>

Lyapunov-Schmidt reduction for Caputo-Hadamard fractional differential system

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发表日期：

2025年7月31日

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Li Ma,
Ya Shu

摘要

Hadamard fractional calculus has emerged as a prominent mathematical alternative for accurately capturing the creep behavior of ultra-slow varying physical evolution. In practical modeling scenarios, Hadamard fractional systems frequently demonstrate high-dimensionality and robust nonlinearity, which profoundly influence further detection and analysis. Consequently, this paper establishes the Lyapunov-Schmidt reduction for the Caputo-Hadamard fractional differential system and conducts both theoretical and computational analyses of the reduced equations. As a by-product and auxiliary of the reduction, an integration by parts formula and a modified inner product definition are presented to align with the characteristics of the Caputo-Hadamard fractional operator. Then, the linearized operators associated with the Caputo-Hadamard fractional differential system with fractional orders $0<\alpha <1$ $0 < α < 1$ and $1<\alpha <2$ $1 < α < 2$ are demonstrated to be Fredholm operators with index zero separately, thereby establishing the standard formulations of the Lyapunov-Schmidt reduction for the Caputo-Hadamard fractional differential system. Finally, by employing higher-order Gâteaux derivatives of multivariable functions and leveraging the fundamental properties of reduced equations, two illustrative examples are provided to substantiate the efficacy of the Lyapunov-Schmidt reduction for the Caputo-Hadamard fractional differential system.