  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.
