Ergodic convergence to a zero of the extended sum of two maximal monotone operators

Abstract

In this note we show that the splitting scheme of Passty [13] as well as the barycentric-proximal method of Lehdili & Lemaire [8] can be used to approximate a zero of the extended sum of maximal monotone operators. When the extended sum is maximal monotone, we generalize a convergence result obtained by Lehdili & Lemaire for convex functions to the case of maximal monotone operators. Moreover, we recover the main convergence results of Passty and Lehdili & Lemaire when the pointwise sum of the involved operators in maximal monotone.