Maximal regularity for fractional difference equations with finite delay on UMD space
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909229316571136 |
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| author | Zhang, Jichao Bu, Shangquan |
| author_facet | Zhang, Jichao Bu, Shangquan |
| contents | In this paper, we study the $\ell^p$-maximal regularity for the fractional difference equation with finite delay: \begin{equation*} \ \ \ \ \ \ \ \ \left\{\begin{array}{cc}
Δ^αu(n)=Au(n)+γu(n-λ)+f(n), \ n\in \mathbb N_0, λ\in \mathbb N, γ\in \mathbb R;
u(i)=0,\ \ i=-λ, -λ+1,\cdots, 1, 2,
\end{array} \right. \end{equation*} where $A$ is a bounded linear operator defined on a Banach space $X$, $f:\mathbb N_0\rightarrow X$ is an $X$-valued sequence and $2<α<3$. We introduce an operator theoretical method based on the notion of $α$-resolvent sequence of bounded linear operators, which gives an explicit representation of solution. Further, using Blunck's operator-valued Fourier multipliers theorems on $\ell^p(\mathbb{Z}; X)$, we completely characterize the $\ell^p$-maximal regularity of solution when $1 < p < \infty$ and $X$ is a UMD space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_15417 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Maximal regularity for fractional difference equations with finite delay on UMD space Zhang, Jichao Bu, Shangquan Functional Analysis Analysis of PDEs In this paper, we study the $\ell^p$-maximal regularity for the fractional difference equation with finite delay: \begin{equation*} \ \ \ \ \ \ \ \ \left\{\begin{array}{cc} Δ^αu(n)=Au(n)+γu(n-λ)+f(n), \ n\in \mathbb N_0, λ\in \mathbb N, γ\in \mathbb R; u(i)=0,\ \ i=-λ, -λ+1,\cdots, 1, 2, \end{array} \right. \end{equation*} where $A$ is a bounded linear operator defined on a Banach space $X$, $f:\mathbb N_0\rightarrow X$ is an $X$-valued sequence and $2<α<3$. We introduce an operator theoretical method based on the notion of $α$-resolvent sequence of bounded linear operators, which gives an explicit representation of solution. Further, using Blunck's operator-valued Fourier multipliers theorems on $\ell^p(\mathbb{Z}; X)$, we completely characterize the $\ell^p$-maximal regularity of solution when $1 < p < \infty$ and $X$ is a UMD space. |
| title | Maximal regularity for fractional difference equations with finite delay on UMD space |
| topic | Functional Analysis Analysis of PDEs |
| url | https://arxiv.org/abs/2406.15417 |