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<p>While standard methods in fixed-point theory, such as the contraction mapping, Krasnosel’skii, and Mönch fixed-point results, are regularly adopted to demonstrate the existence of solutions, this paper introduces a novel approach using <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1815_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">\((\varvec{\varphi }, \digamma )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mrow> <mi mathvariant="bold-italic">φ</mi> </mrow> <mo>,</mo> <mi>ϝ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation><b>-contraction (or nonlinear</b> <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1815_Article_IEq1.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\( \digamma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϝ</mi> </math></EquationSource> </InlineEquation><b>-contraction)</b>. Nonlinear <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1815_Article_IEq1.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\( \digamma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϝ</mi> </math></EquationSource> </InlineEquation>-contraction provides a more versatile structure than contractive mappings, which depend on a constant and uniform contraction rate. We apply these nonlinear contractions to explore a broad class of neural networks with fractional-order derivatives. New sufficient conditions are derived to guarantee the existence and uniqueness of solutions for nonlinear mixed Volterra-Fredholm integral equations and neural network system with variable coefficients and multiple time delays. Moreover, we examined it’s uniform stability with time delay.</p>

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On the Uniform Stability of Fractional-Order Neural Networks and Solutions to Nonlinear Mixed Integral Equations via Nonlinear \( \digamma \)-contractions

  • 期刊论文

  • 发表日期：

    2025年9月20日

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  • Sumati Kumari Panda,
  • Velusamy Vijayakumar,
  • Ravi P Agarwal

摘要

While standard methods in fixed-point theory, such as the contraction mapping, Krasnosel’skii, and Mönch fixed-point results, are regularly adopted to demonstrate the existence of solutions, this paper introduces a novel approach using \((\varvec{\varphi }, \digamma )\) ( φ , ϝ ) -contraction (or nonlinear \( \digamma \) ϝ -contraction). Nonlinear \( \digamma \) ϝ -contraction provides a more versatile structure than contractive mappings, which depend on a constant and uniform contraction rate. We apply these nonlinear contractions to explore a broad class of neural networks with fractional-order derivatives. New sufficient conditions are derived to guarantee the existence and uniqueness of solutions for nonlinear mixed Volterra-Fredholm integral equations and neural network system with variable coefficients and multiple time delays. Moreover, we examined it’s uniform stability with time delay.