  Monotone operators, especially in the form of subdifferential operators, are
of basic importance in optimization. It is well known since Minty, Rockafellar,
and Bertsekas-Eckstein that in Hilbert space, monotone operators can be
understood and analyzed from the alternative viewpoint of firmly nonexpansive
mappings, which were found to be precisely the resolvents of monotone
operators. For example, the proximal mappings in the sense of Moreau are
precisely the resolvents of subdifferential operators. More general notions of
"resolvent", "proximal mapping" and "firmly nonexpansive" have been studied.
One important class, popularized chiefly by Alber and by Kohsaka and Takahashi,
is based on the normalized duality mapping. Furthermore, Censor and Lent
pioneered the use of the gradient of a well behaved convex functions in a
Bregman-distance based framework. It is known that resolvents are firmly
nonexpansive, but the converse has been an open problem for the latter
framework.
  In this note, we build on the very recent characterization of maximal
monotonicity due to Martinez-Legaz to provide a framework for studying
resolvents in which firmly nonexpansive mappings are always resolvents. This
framework includes classical resolvents, resolvents based on the normalized
duality mapping, resolvents based on Bregman distances, and even resolvents
based on (nonsymmetric) rotators. As a by-product of recent work on the
proximal average, we obtain a constructive Kirszbraun-Valentine extension
result for generalized firmly nonexpansive mappings. Several examples
illustrate our results.
