Electron. J. Differential Equations, Vol. 2024 (2024), No. 20, pp. 117.
Maximal regularity for fractional difference equations of order 2<alpha<3 on UMD spaces
Jichao Zhang, Shangquan Bu
Abstract:
In this article, we study the \(\ell^p\)maximal regularity for the fractional
difference equation
$$
\Delta^{\alpha}u(n)=Tu(n)+f(n), \quad (n\in \mathbb{N}_0).
$$
We introduce the notion of \(\alpha\)resolvent sequence of bounded linear operators
defined by the parameters \(T\) and \(\alpha\), which gives an explicit representation
of the solution. Using Blunck's operatorvalued Fourier multipliers theorems on
\(\ell^p(\mathbb{Z}; X)\), we give a characterization of the \(\ell^p\)maximal regularity
for \(1 < p < \infty\) and \(X\) is a UMD space.
Submitted March 12, 2023. Published February 26, 2024.
Math Subject Classifications: 47A10, 35R11, 35R20, 43A22.
Key Words: Fractional difference equation; maximal regularity; UMD space; Rbounded.
DOI: 10.58997/ejde.2024.20
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Jichao Zhang
School of Science
Hubei University of Technology
Wuhan 430068, China
email: 156880717@qq.com


Shangquan Bu
Department of Mathematical Science
Tsinghua University
Beijing 100084, China
email: bushangquan@tsinghua.edu.cn

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