We present a result on the existence and non-existence of 2r-periodic solutions for the following delay equation: <Equation ID="Equ31"> <EquationSource Format="TEX">$\begin{aligned} \theta ''(t)+\theta (t)(1- \theta (t-r))=0. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msup> <mi>θ</mi> <mrow> <mo>′</mo> <mo>′</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>θ</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <mi>θ</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>-</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>Specifically, our result states that if <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$r= (2k-1)\hspace{0.55542pt}\pi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="0.55542pt" /> <mi>π</mi> </mrow> </math></EquationSource> </InlineEquation> for some <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$k\in \mathbb {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation>, the set of positive non-trivial and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$2\pi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </math></EquationSource> </InlineEquation>-periodic solutions (or 2r-periodic) is topologically equivalent to <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">R</mi> </math></EquationSource> </InlineEquation>. However, if <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$r\ne (2k-1)\hspace{0.55542pt}\pi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>≠</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="0.55542pt" /> <mi>π</mi> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$k\in \mathbb {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation>, the equation does not admit 2r-periodic solutions. We also present information on the qualitative behavior of the solutions.

Infinitely many periodic solutions for a delayed Fisher-KPP equation

期刊论文
发表日期：

2026年1月7日

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Nolbert Morales,
Manuel Zamora

摘要

We present a result on the existence and non-existence of 2r-periodic solutions for the following delay equation: $\begin{aligned} \theta ''(t)+\theta (t)(1- \theta (t-r))=0. \end{aligned}$ $\begin{matrix} θ^{''} (t) + θ (t) (1 - θ (t - r)) = 0 . \end{matrix}$ Specifically, our result states that if $r= (2k-1)\hspace{0.55542pt}\pi $ $r = (2 k - 1) π$ for some $k\in \mathbb {N}$ $k \in N$ , the set of positive non-trivial and $2\pi $ $2 π$ -periodic solutions (or 2r-periodic) is topologically equivalent to $\mathbb {R}$ $R$ . However, if $r\ne (2k-1)\hspace{0.55542pt}\pi $ $r \neq (2 k - 1) π$ for all $k\in \mathbb {N}$ $k \in N$ , the equation does not admit 2r-periodic solutions. We also present information on the qualitative behavior of the solutions.