Discrete maximal regularity for Volterra equations and nonlocal time-stepping schemes

Abstract

In this paper we investigate conditions for maximal regularity of Volterra equations de ned on the Lebesgue space of sequences $l_p(Z)$ by using Bl unck's theorem on the equivalence between operator-valued $l_p$-multipliers and the notion of R-boundedness. We show sufficient conditions for maximal $l_p-l_q$ regularity of solutions of such problems solely in terms of the data. We also explain the signi cance of kernel sequences in the theory of viscoelasticity, establishing a new and surprising connection with schemes of approximation of fractional models.