SECOND ORDER FORWARD-BACKWARD DYNAMICAL SYSTEMS FOR MONOTONE INCLUSION PROBLEMS

We begin by considering second order dynamical systems of the from x (t)+ gamma(t) x (t)+ lambda(t) B (x (t)) = 0, where B : H -> H is a cocoercive operator de fined on a real Hilbert space H, lambda : [0, + infinity) -> [0, + infinity) is a relaxation function, and gamma : [0; + infinity) -> [0; + infinity) is a damping function, both depending on time. For the generated trajectories, we show existence and uniqueness of the generated trajectories as well as their weak asymptotic convergence to a zero of the operator B. The framework allows us to address from similar perspectives second order dynamical systems associated with the problem of finding zeros of the sum of a maximally monotone operator and a cocoercive one. This captures as a particular case the minimization of the sum of a nonsmooth convex function with a smooth convex one. Furthermore, we prove that when B is the gradient of a smooth convex function the value of the latter converges along the ergodic trajectory to its minimal value with a rate of O (1/t).