Maximal regularity for fractional difference equations with finite delay on UMD space

In this paper, we study the ${\mathrm{\ell }}^p$-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 $X$, $f:{\mathbb{N}}_0\to X$ is an $X$-valued sequence and $2<\alpha <3$. We introduce an operator theoretical method based on the notion of $\alpha$-resolvent sequence of bounded linear operators, which gives an explicit representation of solution. Further, using Blunck’s operator-valued Fourier multipliers theorems on ${\mathrm{\ell }}^p(\mathbb{Z};X)$, we completely characterize the ${\mathrm{\ell }}^p$-maximal regularity of solution when $1<p<\mathrm{\infty }$ and $X$ is a UMD space.