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<p>In this paper, we explore the existence of analytic solutions for an iterative functional equation of the form</p><p><Equation ID="Equa"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="10255_2024_1030_Article_Equa.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="158" /> </MediaObject> <EquationSource Format="TEX">\(g(g(x))+x=f(g(x))\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>+</mo> <mi>x</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </math></EquationSource> </Equation></p><p>that originates from Painlevé equations. By an invertible transformation, we study the analytic solutions of an auxiliary equation under three different cases, and obtain the invertible analytic solutions for the original equation.</p>

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A Note on an Iterative Equation Related to Painlevé Equations

  • 期刊论文

  • 发表日期：

    2025年10月6日

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  • Wei Cheng

摘要

In this paper, we explore the existence of analytic solutions for an iterative functional equation of the form

\(g(g(x))+x=f(g(x))\) g ( g ( x ) ) + x = f ( g ( x ) )

that originates from Painlevé equations. By an invertible transformation, we study the analytic solutions of an auxiliary equation under three different cases, and obtain the invertible analytic solutions for the original equation.