  Because of Minty's classical correspondence between firmly nonexpansive
mappings and maximally monotone operators, the notion of a firmly nonexpansive
mapping has proven to be of basic importance in fixed point theory, monotone
operator theory, and convex optimization. In this note, we show that if
finitely many firmly nonexpansive mappings defined on a real Hilbert space are
given and each of these mappings is asymptotically regular, which is equivalent
to saying that they have or "almost have" fixed points, then the same is true
for their composition. This significantly generalizes the result by Bauschke
from 2003 for the case of projectors (nearest point mappings). The proof
resides in a Hilbert product space and it relies upon the Brezis-Haraux range
approximation result. By working in a suitably scaled Hilbert product space, we
also establish the asymptotic regularity of convex combinations.
