The non-homogeneous problem

${u}^{\text{'}}+Au=f$ with periodic boundary conditions and a closed linear operator on a UMD-space is considered. The authors characterize the maximal

${L}^{p}$ regularity of the problem in terms of R-boundedness of the resolvent. This approach was applied by

*L. Weis* [Lect. Notes Pure Appl. Math. 215, 195-214 (2001;

Zbl 0981.35030)] in the case of Dirichlet boundary conditions when

$A$ is an infinitesimal generator of a bounded analytic semigroup on a UMD-space. The present paper is a generalization of these results. The main tool (with respect to the periodic boundary conditions) is a discrete analog of the Marcinkiewicz operator-valued multiplier theorem. The authors present a direct and easy proof of this theorem. One of the main results of the paper is that the considered problem is strongly

${L}^{p}$-well-posed for

$1<p<\infty $ if and only if the set

$\{k{(ik-A)}^{-1}:k\in \mathbb{Z}\}$ is R-bounded. The maximal regularity of the second order problem for periodic, Dirichlet or Neumann boundary conditions is also characterized.