  This research is concerned with evolution equations and their
forward-backward discretizations. Our first contribution is an estimation for
the distance between iterates of sequences generated by forward-backward
schemes, useful in the convergence and robustness analysis of iterative
algorithms of widespread use in variational analysis and optimization. Our
second contribution is the approximation, on a bounded time frame, of the
solutions of evolution equations governed by accretive (monotone) operators
with an additive structure, by trajectories defined using forward-backward
sequences. This provides a short, simple and self-contained proof of existence
and regularity for such solutions; unifies and extends a number of classical
results; and offers a guide for the development of numerical methods. Finally,
our third contribution is a mathematical methodology that allows us to deduce
the behavior, as the number of iterations tends to $+\infty$, of sequences
generated by forward-backward algorithms, based solely on the knowledge of the
behavior, as time goes to $+\infty$, of the solutions of differential
inclusions, and viceversa.
