Limits of sequences of maximal monotone operators

TL;DR

This paper investigates the behavior of sequences of maximal monotone operators in reflexive Banach spaces, showing their limits are representable and exploring implications for variational sums and compositions.

Contribution

It establishes that the lower limit of a sequence of maximal monotone operators is representable, extending understanding of their stability and composition properties.

Findings

Lower limit of maximal monotone operators is representable.

Variational sum of maximal monotone operators is representable.

Variational composition with linear operators is representable.

Abstract

We show that the lower limit of a sequence of maximal monotone operators on a reflexive Banach space is a representable monotone operator. As a consequence, we obtain that the variational sum of maximal monotone operators and the variational composition of a maximal monotone operator with a linear continuous operator are both representable monotone operators.