$\ell_{p}$ -maximal regularity for a class of fractional difference equations on UMD spaces: The case $1\lt \alpha\leq2$

Abstract

By using Blunck’s operator-valued Fourier multiplier theorem, we completely characterize the existence and uniqueness of solutions in Lebesgue sequence spaces for a discrete version of the Cauchy problem with fractional order 1<α≤2. This characterization is given solely in spectral terms on the data of the problem, whenever the underlying Banach space belongs to the UMD-class.