Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 20, pp. 1-17.

Title: Maximal regularity for fractional difference  equations of order 2&lts;alpha<3 on UMD spaces

Authors: Jichao Zhang (Hubei Univ. of Technology, Wuhan, China)
         Shangquan Bu (Tsinghua Univ., Beijing, China)

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 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.