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<p>In this paper, we consider the generalized proportional fractional stochastic delay differential equations of Caputo type with nonlinear terms satisfying the non-Lipschitz condition. Initially, the solution of the addressed system is obtained by employing the Laplace transform and its inverse, along with the Picard iteration technique. Thereafter, the uniqueness of the solution is demonstrated through the method of contradiction. In addition, under the averaging conditions, the averaging principle in the sense of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2645_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathrm{L}^p\)</EquationSource> </InlineEquation> is explored by utilizing the Hölder inequality, the Jensen inequality, the Burkholder-Davis-Gundy inequality, the generalized Grönwall inequality with singular integral kernel and the interval translation technique. Lastly, the correctness of the conclusions is attested through a numerical simulation.</p>

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Averaging principle for Caputo-type generalized proportional fractional stochastic delay differential equations with non-Lipschitz coefficients

  • 期刊论文

  • 发表日期：

    2025年9月9日

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  • Xue Yang,
  • Danfeng Luo

摘要

In this paper, we consider the generalized proportional fractional stochastic delay differential equations of Caputo type with nonlinear terms satisfying the non-Lipschitz condition. Initially, the solution of the addressed system is obtained by employing the Laplace transform and its inverse, along with the Picard iteration technique. Thereafter, the uniqueness of the solution is demonstrated through the method of contradiction. In addition, under the averaging conditions, the averaging principle in the sense of \(\mathrm{L}^p\) is explored by utilizing the Hölder inequality, the Jensen inequality, the Burkholder-Davis-Gundy inequality, the generalized Grönwall inequality with singular integral kernel and the interval translation technique. Lastly, the correctness of the conclusions is attested through a numerical simulation.