Limits of maximal monotone operators driven by their representative functions

TL;DR

This paper investigates the stability and continuity properties of maximal monotone operators in Banach spaces through their representative functions, establishing convergence results that enhance understanding of their limiting behavior.

Contribution

It demonstrates the continuity of the representation of maximal monotone operators with respect to epi-convergence and the stability under Mosco-convergence of their representative functions.

Findings

Continuity of the representation with respect to epi-convergence.

Stability of maximal monotone operators under Mosco-convergence.

Enhanced understanding of operator limits in Banach spaces.

Abstract

In a previous paper, the authors showed that in a reflexive Banach space the lower limit of a sequence of maximal monotone operators is always representable by a convex function. The present paper gives precisions to the latter result by demonstrating the continuity of the representation with respect to the epi-convergence of the representative functions, and the stability of the class of maximal monotone operators with respect to the Mosco-convergence of their representative functions.