A forward-backward-forward differential equation and its asymptotic properties

In this paper, we approach the problem of finding the zeros of the sum of a maximally monotone operator and a monotone and Lipschitz continuous one in a real Hilbert space via an implicit forward-backward-forward dynamical system with nonconstant relaxation parameters and stepsizes of the resolvents. Besides proving existence and uniqueness of strong global solutions … Read more

Second order forward-backward dynamical systems for monotone inclusion problems

We begin by considering second order dynamical systems of the from $\ddot x(t) + \Gamma (\dot x(t)) + \lambda(t)B(x(t))=0$, where $\Gamma: {\cal H}\rightarrow{\cal H}$ is an elliptic bounded self-adjoint linear operator defined on a real Hilbert space ${\cal H}$, $B: {\cal H}\rightarrow{\cal H}$ is a cocoercive operator and $\lambda:[0,+\infty)\rightarrow [0,+\infty)$ is a relaxation function depending … Read more

A Continuous Dynamical Newton-Like Approach to Solving Monotone Inclusions

We introduce non-autonomous continuous dynamical systems which are linked to Newton and Levenberg-Marquardt methods. They aim at solving inclusions governed by maximal monotone operators in Hilbert spaces. Relying on Minty representation of maximal monotone operators as lipschitzian manifolds, we show that these dynamics can be formulated as first-order in time differential systems, which are relevant … Read more