Firmly nonexpansive mappings and maximally monotone operators: correspondence and duality

TL;DR

This paper explores the deep relationship between firmly nonexpansive mappings and maximal monotone operators, highlighting duality properties and providing illustrative examples to enhance understanding in fixed point theory.

Contribution

It systematically analyzes the correspondence and duality between firmly nonexpansive mappings and maximal monotone operators, expanding theoretical understanding.

Findings

Identification of dual and self-dual properties

Clarification of the correspondence between mappings and operators

Illustrative examples demonstrating key concepts

Abstract

The notion of a firmly nonexpansive mapping is central in fixed point theory because of attractive convergence properties for iterates and the correspondence with maximal monotone operators due to Minty. In this paper, we systematically analyze the relationship between properties of firmly nonexpansive mappings and associated maximal monotone operators. Dual and self-dual properties are also identified. The results are illustrated through several examples.