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

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.