Maximal ℓp-regularity for discrete time volterra equations with delay

Abstract

In this paper we investigate the existence and uniqueness of solutions belonging to the vector-valued space $ \ell_p(\Z,X)$ by using Blunck's theorem on the equivalence between operator-valued $\ell_p$-multipliers and the notion of $R$-boundedness for the discrete time volterra equation with delay given by $$ u(n) = \sum_{j=-\infty}^{n}b(n-j) Au(j)+\sum_{j=1}^k \beta_j u(n-\tau_j) + f(n), \qquad n \in \mathbb{Z}, $$ where $A$ is a closed linear operator with domain $D(A)$ defined on a Banach space $X$ and $b\in\ell_1(\Z)$ verifies suitable conditions such as $1$-regularity. We characterize maximal $\ell_p$-regularity of solutions of such problems in terms of the data and an spectral condition and we provide optimal estimates. Moreover, we illustrate our results providing different models that label into our general scheme such as the discrete time wave and Kuznetsov equations.