This paper is intended as an attempt to prove the existence of asymptotic periodic solutions to a class of abstract differential equations with infinite delay of the form <Equation ID="Equ21"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1970_Article_Equ21.gif" Format="GIF" Height="38" Rendition="HTML" Resolution="72" Type="Linedraw" Width="209" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} \frac{d u(t)}{d t}=A u(t)+L(u_t)+f(t) \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mfrac> <mrow> <mi>d</mi> <mi>u</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="italic">dt</mi> </mrow> </mfrac> <mo>=</mo> <mi>A</mi> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>L</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mi>t</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where A is the generator of a strongly continuous semigroup of linear operators, L is a bounded linear operator from an axiomatically definite phase space <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1970_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathscr {B}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">B</mi> </math></EquationSource> </InlineEquation> to a general Banach space <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1970_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">${\textbf{X}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">X</mi> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1970_Article_IEq3.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$u_t$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>u</mi> <mi>t</mi> </msub> </math></EquationSource> </InlineEquation> is an element of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1970_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathscr {B}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">B</mi> </math></EquationSource> </InlineEquation> and f is assumed to be asymptotic periodic.

Asymptotic Periodic Solutions of Differential Equations with Infinite Delay

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发表日期：

2025年9月11日

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Nguyen Duc Huy,
Anh Minh Le,
Vu Trong Luong,
Nguyen Ngoc Vien

摘要

This paper is intended as an attempt to prove the existence of asymptotic periodic solutions to a class of abstract differential equations with infinite delay of the form $\begin{aligned} \frac{d u(t)}{d t}=A u(t)+L(u_t)+f(t) \end{aligned}$ $\begin{matrix} \frac{d u (t)}{dt} = A u (t) + L (u_{t}) + f (t) \end{matrix}$ where A is the generator of a strongly continuous semigroup of linear operators, L is a bounded linear operator from an axiomatically definite phase space $\mathscr {B}$ $B$ to a general Banach space ${\textbf{X}}$ $X$ , $u_t$ $u_{t}$ is an element of $\mathscr {B}$ $B$ and f is assumed to be asymptotic periodic.