<p>In this paper, we consider the equation <Equation ID="Equ1"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_109_Article_Equ1.gif" Format="GIF" Height="45" Rendition="HTML" Resolution="72" Type="Linedraw" Width="256" /> </MediaObject> <EquationSource Format="TEX">$\overline{y}\underline{y}+\alpha(x)\frac{y^{(d)}}{y^{2}}=R(x,y)=\frac{H(x, y)}{G(x, y)},$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mover> <mi>y</mi> <mo>¯</mo> </mover> <munder> <mi>y</mi> <mo>̲</mo> </munder> <mo>+</mo> <mi>α</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mfrac> <msup> <mi>y</mi> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </msup> <msup> <mi>y</mi> <mn>2</mn> </msup> </mfrac> <mo>=</mo> <mi>R</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mi>H</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>G</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> <mo>,</mo> </mrow> </math></EquationSource> </Equation> where <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_109_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="34" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is a nonzero rational, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_109_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">$H(x,y)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>H</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_109_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">$G(x,y)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> are co-prime polynomials of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_109_Article_IEq4.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$y$</EquationSource> <EquationSource Format="MATHML"><math> <mi>y</mi> </math></EquationSource> </InlineEquation> with rational coefficients. If there exists a non-rational meromorphic solution with <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_109_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="69" /> </InlineMediaObject> <EquationSource Format="TEX">$\rho_{2}(y)<1 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ρ</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, then <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_109_Article_IEq6.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="60" /> </InlineMediaObject> <EquationSource Format="TEX">$\deg_{y}(H)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>deg</mo> <mi>y</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_109_Article_IEq7.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">$\deg_{y}(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>deg</mo> <mi>y</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> must satisfy certain conditions.</p>

Higher-order delay differential equations with meromorphic solutions

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2025年9月3日

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Y. Liu,
W. L. Liu

摘要

In this paper, we consider the equation $\overline{y}\underline{y}+\alpha(x)\frac{y^{(d)}}{y^{2}}=R(x,y)=\frac{H(x, y)}{G(x, y)},$ $\bar{y} \underset{̲}{y} + α (x) \frac{y^{(d)}}{y^{2}} = R (x, y) = \frac{H (x, y)}{G (x, y)},$ where $\alpha(x)$ $α (x)$ is a nonzero rational, $H(x,y)$ $H (x, y)$ and $G(x,y)$ $G (x, y)$ are co-prime polynomials of $y$ $y$ with rational coefficients. If there exists a non-rational meromorphic solution with $\rho_{2}(y)<1 $ $ρ_{2} (y) < 1$ , then $\deg_{y}(H)$ ${deg}_{y} (H)$ and $\deg_{y}(G)$ ${deg}_{y} (G)$ must satisfy certain conditions.