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Maximal monotone multifunctions of Brndsted--Rockafellar type
 

Summary: Maximal monotone multifunctions of
Brøndsted--Rockafellar type
by Stephen Simons
1. Introduction
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
sup
x2domf
[f(w) \Gamma f(x) \Gamma hw \Gamma x; w \Lambda i] Ÿ fffi
then there exists (s; s \Lambda ) 2 G(@f) such that ks \Gamma wk Ÿ ff and ks \Lambda \Gamma w \Lambda k Ÿ fi. (Here, ``G''
stands for ``graph of''.) It follows from this that if
inf
(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 Ÿ ff and ks \Lambda \Gamma w \Lambda k Ÿ 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:

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics