Evolution equations for maximal monotone operators: asymptotic analysis in continuous and discrete time

TL;DR

This survey reviews the asymptotic behavior of solutions to evolution equations driven by maximal monotone operators in Hilbert spaces, comparing continuous and discrete time approaches and unifying various convergence results.

Contribution

It provides a comprehensive comparison of continuous and discrete time asymptotic analyses for maximal monotone operator evolution equations, unifying proofs and concepts.

Findings

Weak convergence for the average process established

Strong convergence results discussed

Connections with almost orbit analysis made

Abstract

This survey is devoted to the asymptotic behavior of solutions of evolution equations generated by maximal monotone operators in Hilbert spaces. The emphasis is in the comparison of the continuous time trajectories to sequences generated by implicit or explicit discrete time schemes. The analysis covers weak convergence for the average process, for the process itself and strong convergence and aims at highlighting the main ideas and unifying the proofs. We further make the connection with the analysis in terms of almost orbits that allows for a broader scope.