Every maximally monotone operator of Fitzpatrick-Phelps type is actually of dense type

TL;DR

This paper proves that all maximally monotone operators of Fitzpatrick-Phelps type are of dense type in real Banach spaces, confirming a long-standing question and unifying key concepts in monotonicity theory.

Contribution

It establishes that Fitzpatrick-Phelps type operators are necessarily of dense type, resolving a question posed by Stephen Simons in 2001.

Findings

All Fitzpatrick-Phelps type operators are of dense type

Confirms the equivalence of important monotonicity notions

Provides an affirmative answer to a longstanding open problem

Abstract

We show that every maximally monotone operator of Fitzpatrick-Phelps type defined on a real Banach space must be of dense type. This provides an affirmative answer to a question posed by Stephen Simons in 2001 and implies that various important notions of monotonicity coincide.