Convergence of damped inertial dynamics governed by regularized maximally monotone operators

In a Hilbert space setting, we study the asymptotic behavior, as time t goes to infinity, of the trajectories of a second-order differential equation governed by the Yosida regularization of a maximally monotone operator with time-varying positive index λ(t). The dissipative and convergence properties are attached to the presence of a viscous damping term with positive coefficient γ(t). A suitable tuning of the parameters γ(t) and λ(t) makes it possible to prove the weak convergence of the trajectories towards zeros of the operator. When the operator is the subdifferential of a closed convex proper function, we estimate the rate of convergence of the values. These results are in line with the recent articles by Attouch-Cabot [3], and Attouch-Peypouquet [8]. In this last paper, the authors considered the case γ(t)=αt, which is naturally linked to Nesterov's accelerated method. We unify, and often improve the results already present in the literature.