Asymptotic periodic solutions of differential equations with infinite delay
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866918134513926144 |
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| author | Huy, Nguyen Duc Minh, Le Anh Luong, Vu trong Vien, Nguyen Ngoc |
| author_facet | Huy, Nguyen Duc Minh, Le Anh Luong, Vu trong Vien, Nguyen Ngoc |
| contents | In this paper, by using the spectral theory of functions and properties of evolution semigroups, we establish conditions on the existence, and uniqueness of asymptotic 1-periodic solutions to a class of abstract differential equations with infinite delay of the form \begin{equation*} \frac{d u(t)}{d t}=A u(t)+L(u_t)+f(t) \end{equation*} where $A$ is the generator of a strongly continuous semigroup of linear operators, $L$ is a bounded linear operator from a phase space $\mathscr{B}$ to a Banach space $X$, $u_t$ is an element of $\mathscr{B}$ which is defined as $u_t(θ)=u(t+θ)$ for $θ\leq 0$ and $f$ is asymptotic 1-periodic in the sense that $\lim\limits_{t \rightarrow \infty}(f(t+1)-$ $f(t))=0$. A Lotka-Volterra model with diffusion and infinite delay is considered to illustrate our results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_02679 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Asymptotic periodic solutions of differential equations with infinite delay Huy, Nguyen Duc Minh, Le Anh Luong, Vu trong Vien, Nguyen Ngoc Classical Analysis and ODEs Analysis of PDEs 34K30, 34k13, 34K14, 34D05, 35B15 In this paper, by using the spectral theory of functions and properties of evolution semigroups, we establish conditions on the existence, and uniqueness of asymptotic 1-periodic solutions to a class of abstract differential equations with infinite delay of the form \begin{equation*} \frac{d u(t)}{d t}=A u(t)+L(u_t)+f(t) \end{equation*} where $A$ is the generator of a strongly continuous semigroup of linear operators, $L$ is a bounded linear operator from a phase space $\mathscr{B}$ to a Banach space $X$, $u_t$ is an element of $\mathscr{B}$ which is defined as $u_t(θ)=u(t+θ)$ for $θ\leq 0$ and $f$ is asymptotic 1-periodic in the sense that $\lim\limits_{t \rightarrow \infty}(f(t+1)-$ $f(t))=0$. A Lotka-Volterra model with diffusion and infinite delay is considered to illustrate our results. |
| title | Asymptotic periodic solutions of differential equations with infinite delay |
| topic | Classical Analysis and ODEs Analysis of PDEs 34K30, 34k13, 34K14, 34D05, 35B15 |
| url | https://arxiv.org/abs/2309.02679 |