Near equality, near convexity, sums of maximally monotone operators, and averages of firmly nonexpansive mappings

TL;DR

This paper investigates the properties of nearly convex sets, maximally monotone operators, and firmly nonexpansive mappings, revealing that averages of such mappings preserve key range and regularity properties with applications to resolvent averages.

Contribution

It establishes that the range of an average of firmly nonexpansive mappings approximates the average of their ranges, and shows that asymptotic regularity is preserved under averaging.

Findings

Range of average mappings is nearly equal to average of ranges.

Averages of asymptotically regular mappings remain asymptotically regular.

Detailed properties of resolvent averages are characterized.

Abstract

We study nearly equal and nearly convex sets, ranges of maximally monotone operators, and ranges and fixed points of convex combinations of firmly nonexpansive mappings. The main result states that the range of an average of firmly nonexpansive mappings is nearly equal to the average of the ranges of the mappings. A striking application of this result yields that the average of asymptotically regular firmly nonexpansive mappings is also asymptotically regular. Throughout, examples are provided to illustrate the theory. We also obtain detailed information on the domain and range of the resolvent average.