Most Maximally Monotone Operators Have a Unique Zero and a Super-regular Resolvent

TL;DR

This paper demonstrates that in Hilbert spaces, most maximally monotone operators have a unique zero and super-regular resolvents, with the set of strongly monotone mappings being of first category, using Baire Category Theorem.

Contribution

It establishes that most maximally monotone operators have a unique zero and super-regular resolvents, revealing generic properties in the space of such operators.

Findings

Most resolvents are super-regular.

Most maximally monotone operators have a unique zero.

The set of strongly monotone mappings is of first category.

Abstract

Maximally monotone operators play important roles in optimization, variational analysis and differential equations. Finding zeros of maximally monotone operators has been a central topic. In a Hilbert space, we show that most resolvents are super-regular, that most maximally monotone operators have a unique zero and that the set of strongly monotone mapping is of first category although each strongly monotone operator has a unique zero. The results are established by applying the Baire Category Theorem to the space of nonexpansive mappings.