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<p>In this paper, we consider the quasi-periodically forced equation with strong damping. We are devoted to studying the existence of response solutions, which are the quasi-periodic solutions with the same frequency as the forcing. Precisely, we obtain the corresponding solutions in the analytic and highly differentiable setting by contraction mapping principle. Note that, our method does not involve any arithmetic conditions on the forcing frequency <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1279_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(\omega \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ω</mi> </math></EquationSource> </InlineEquation>. Moreover, we extend our results from 1-dimensional case to n-dimension. In the process of proof, we reformulate the existence of response solutions for the original system as a fixed point problem in some suitable Banach space.</p>

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Existence of Response Solutions to Quasi-Periodically Forced Equation with Damping

  • 期刊论文

  • 发表日期：

    2025年5月16日

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  • Xingkui Shu,
  • Fenfen Wang,
  • Lian Yang

摘要

In this paper, we consider the quasi-periodically forced equation with strong damping. We are devoted to studying the existence of response solutions, which are the quasi-periodic solutions with the same frequency as the forcing. Precisely, we obtain the corresponding solutions in the analytic and highly differentiable setting by contraction mapping principle. Note that, our method does not involve any arithmetic conditions on the forcing frequency \(\omega \) ω . Moreover, we extend our results from 1-dimensional case to n-dimension. In the process of proof, we reformulate the existence of response solutions for the original system as a fixed point problem in some suitable Banach space.