Maximal regularity for fractional difference equations of order 2

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. For more information see https://ejde.math.txstate.edu/Volumes/2024/20/abstr.html