Filling the Gap between Metric Regularity and Fixed Points: The Linear Openness of Compositions

TL;DR

This paper explores the relationship between the openness of compositions of set-valued maps and fixed point theorems, providing a general result that unifies and extends existing findings in the field.

Contribution

It introduces a comprehensive theorem linking metric regularity and fixed points, encompassing and generalizing previous results on openness of compositions.

Findings

Unified framework for openness of compositions

Reobtained fixed point theorem of Dontchev and Frankowska

Extended the understanding of metric regularity and fixed points

Abstract

This paper is devoted to the investigation of an important issue recently brought into attention by a recent paper of Arutyunov: the relation between openness of composition of set-valued maps and fixed point results. More precisely, we prove a general result concerning the openness of compositions and then we show that this result covers and implies most of the known openness results. In particular, we reobtain several recent results in this field, including a fixed point theorem of Dontchev and Frankowska.