Summary: Maximal monotone multifunctions of
by Stephen Simons
Let E be a (nonzero) real Banach space with dual E \Lambda , ff; fi ? 0 and (w; w \Lambda ) 2 E \Theta E \Lambda .
The Brøndsted--Rockafellar theorem states that if f : E 7! IR[f1g is proper, convex and
lower semicontinuous, with subdifferential @f , and
[f(w) \Gamma f(x) \Gamma hw \Gamma x; w \Lambda i] fffi
then there exists (s; s \Lambda ) 2 G(@f) such that ks \Gamma wk ff and ks \Lambda \Gamma w \Lambda k fi. (Here, ``G''
stands for ``graph of''.) It follows from this that if
(t;t \Lambda )2G(@f)
ht \Gamma w; t \Lambda \Gamma w \Lambda i \Gammafffi
then there exists (s; s \Lambda ) 2 G(@f) such that ks \Gamma wk ff and ks \Lambda \Gamma w \Lambda k fi.
These inequality--splitting properties of subdifferentials have turned out to be abso
lutely fundamental in convex analysis. Now in the second of the results mentioned above,
f does not appear explicitly, it only appears in the form of @f , so it makes sense to ask if
a similar inequality--splitting property is true for any maximal monotone operator. Thus: