Abstract

The 'heavy ball with friction' dynamical system x + ?x + ? f (x) = 0 is a nonlinear oscillator with damping (? > 0). It has been recently proved that when H is a real Hilbert space and f: H ? ? is a differentiable convex function whose minimal value is achieved, then each solution trajectory t ? x(t) of this system weakly converges towards a solution of ? f (x) = 0. We prove a similar result in the discrete setting for a general maximal monotone operator A by considering the following iterative method: xk+1 - xk - ?k(xk - xk-1) + ?kA(xk+1) ? 0, giving conditions on the parameters ?k and ?k in order to ensure weak convergence toward a solution of 0 ? A(x) and extending classical convergence results concerning the standard proximal method.