where A is a closed linear operator with domain D(A) defined on a Banach space X, and b. l(1)(Z) verifies suitable conditions such as 1-regularity. We characterize maximal l(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.