Sum theorems for maximally monotone operators of type (FPV)

TL;DR

This paper proves the maximal monotonicity of the sum of two maximally monotone operators under certain conditions, extending and unifying recent sum theorems in Monotone Operator Theory.

Contribution

It establishes new sum theorems for maximally monotone operators of type (FPV) under relaxed conditions, generalizing previous results.

Findings

Maximal monotonicity of A+B under specific domain conditions.

Sum operator A+B is of type (FPV) when dom A is convex.

Unifies several recent sum theorems in the field.

Abstract

The most important open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that the classical Rockafellar's constraint qualification holds. In this paper, we establish the maximal monotonicity of $A+B$ provided that $A$ and $B$ are maximally monotone operators such that $\sta(\dom A)\cap\inte\dom B\neq\varnothing$, and $A$ is of type (FPV). We show that when also $\dom A$ is convex, the sum operator: $A+B$ is also of type (FPV). Our result generalizes and unifies several recent sum theorems.