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

TL;DR

This paper establishes $ ext{ell}^p$-maximal regularity for fractional difference equations with finite delay on UMD spaces, using operator-theoretic methods and Fourier multiplier theorems, providing explicit solution representations.

Contribution

It introduces an operator-theoretic approach based on $ ext{alpha}$-resolvent sequences to characterize maximal regularity for fractional difference equations with delay on UMD spaces.

Findings

Explicit solution representation via $ ext{alpha}$-resolvent sequences.

Complete characterization of $ ext{ell}^p$-maximal regularity for $1<p< $.

Application of Fourier multiplier theorems on $ ext{ell}^p( ext{Z};X)$.

Abstract

In this paper, we study the $\ell^p$-maximal regularity for the fractional difference equation with finite delay: \begin{equation*} \ \ \ \ \ \ \ \ \left\{\begin{array}{cc} \Delta^{\alpha}u(n)=Au(n)+\gamma u(n-\lambda)+f(n), \ n\in \mathbb N_0, \lambda \in \mathbb N, \gamma \in \mathbb R; u(i)=0,\ \ i=-\lambda, -\lambda+1,\cdots, 1, 2, \end{array} \right. \end{equation*} where $A$ is a bounded linear operator defined on a Banach space $X$, $f:\mathbb N_0\rightarrow X$ 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.