Forward-backward approximation of nonlinear semigroups in finite and infinite horizon

This work is concerned with evolution equations and their forwardbackward discretizations, and aims at building bridges between differential equations and variational analysis. 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 numerical optimization and variational inequalities. 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 constructed by interpolating 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 +∞, of sequences generated by forward-backward algorithms, based solely on the knowledge of the behavior, as time goes to +∞, of the solutions of differential inclusions, and viceversa.