Maximal Regularity for Fractional Difference Equations with Finite Delay on UMD Spaces

In this paper, we study the \(\ell ^p\) ${\ell }^p$ -maximal regularity for the fractional difference equation with finite delay: \(\begin{aligned} \ \ \ \ \ \ \ \ \left\{ \begin{array}{ll} \Delta ^{\alpha }u(n)=Au(n)+B u(n-\lambda )+f(n), \ n\in {\mathbb {N}}_0, \lambda \in {\mathbb {N}}; \\ u(i)=0,\ \ i=-\lambda , -\lambda +1,\cdots , 1, 2, \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mathrm{\Delta }}^{\alpha }u(n)=Au(n)+Bu(n-\lambda )+f(n),n\in {\mathbb{N}}_0,\lambda \in \mathbb{N};\hfill \\ u(i)=0,i=-\lambda ,-\lambda +1,\cdots ,1,2,\hfill \end{array}\right)\end{array}$$ where A and B are bounded linear operators defined on a Banach space X, \(f:{\mathbb {N}}_0\rightarrow X\) $f:{\mathbb{N}}_0\to X$ is an X-valued sequence and \(2<\alpha <3\) $2<\alpha <3$ . We introduce an operator theoretical method based on the notion of \(\alpha \) $\alpha$ -resolvent sequence of bounded linear operators, which gives an explicit representation of solution. Further, using Blunck’s operator-valued Fourier multipliers theorems on \(\ell ^p(\mathbb {Z}; X)\) ${\ell }^p(\mathbb{Z};X)$ , we completely characterize the \(\ell ^p\) ${\ell }^p$ -maximal regularity of solution when \(1< p < \infty \) $1<p<\infty$ and X is a UMD space.