Resumen
A first order method is considered for the discretization in time of an integro-differential equation, which can be written as $D^{\alpha} u(t) = A u(t) + f(t)$, $1 < \alpha < 2$, where $A : D(A) \subset X \to X$ is a sectorial operator in a Banach space X. Qualitative properties of the numerical solution, such as contractivity and positivity, are studied. A numerical illustration is provided.

Resumen
In this paper we provide an analysis of monotonicity properties for linear multistep methods. These monotonicity properties include positivity and the diminishing of total variation. We also pay particular attention to related boundedness properties such as the total variation bounded (TVB) property. In the analysis the multistep methods are considered in combination with suitable starting procedures. This allows for monotonicity statements for classes of methods which are important and often used in practice but which were thus far not covered by theoretical results.