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

Abstract

In this paper, we study the \(\ell ^p\)-maximal regularity for the fractional difference equation with finite delay:

$$\begin{aligned} \ \ \ \ \ \ \ \ \left\{ \begin{array}{ll} \Delta ^{\alpha }u(n)=Au(n)+B u(n-\lambda )+f(n), \ n\in {\mathbb {N}}_0, \lambda \in {\mathbb {N}}; \\ u(i)=0,\ \ i=-\lambda , -\lambda +1,\cdots , 1, 2, \end{array} \right. \end{aligned}$$

where A and B are bounded linear operators defined on a Banach space X, \(f:{\mathbb {N}}_0\rightarrow X\) 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 \(\ell ^p(\mathbb {Z}; X)\), we completely characterize the \(\ell ^p\)-maximal regularity of solution when \(1< p < \infty \) and X is a UMD space.