Electron. J. Differential Equations, Vol. 2024 (2024), No. 20, pp. 1-17.

Maximal regularity for fractional difference equations of order 2<alpha<3 on UMD spaces

Jichao Zhang, Shangquan Bu

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 operator-valued 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; R-bounded.
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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