Maximality of the sum of a maximally monotone linear relation and a maximally monotone operator

TL;DR

This paper proves that the sum of a maximally monotone linear relation and another maximally monotone operator is maximally monotone under Rockafellar's constraint qualification, advancing understanding in Monotone Operator Theory.

Contribution

It establishes the maximal monotonicity of the sum when one operator is a linear relation, under a key qualification, and shows the sum is of type (FPV).

Findings

Maximal monotonicity of A+B under the given conditions.

A+B is of type (FPV).

Advances the open problem in Monotone Operator Theory.

Abstract

The most famous open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that Rockafellar's constraint qualification holds. In this paper, we prove the maximal monotonicity of $A+B$ provided that $A, B$ are maximally monotone and $A$ is a linear relation, as soon as Rockafellar's constraint qualification holds: $\dom A\cap\inte\dom B\neq\varnothing$. Moreover, $A+B$ is of type (FPV).