On Regularized Forward-Backward Dynamical Systems Associated with Structured Monotone Inclusions

Recently, a regularized forward-backward dynamical system associated with additively structured monotone inclusions involving a multi-valued maximally monotone operator A and a single-valued co-coercive operator B has been studied in Bot et al. (Adv. Nonlinear Anal. 10, 450-476, 2021). In this work, we establish strong convergence of the generated trajectories to a solution of the original monotone inclusion under a weaker assumption on the operator B, namely B is Lipschitz continuous and such that the sum S := A+ B is maximally monotone. It is well known that the co-coerciveness of B implies its monotonicity and Lipschitz continuity, which in turn infers the maximal monotonicity of S. If the operator A+ B is maximally monotone and strongly pseudomonotone, we obtain a convergence estimate. A time discretization of the dynamical system provides an iterative regularization forward-backward method with relaxation parameters. The performance of the regularized dynamical system approach is illustrated by numerical experiments.