Let E be a real Banach space with dual E'. We associate with any nonempty subset H of E x E* a certain compact convex subset of the first quadrant in K*, which we call the picture of H, II(H). In general, n(H) may be empty, but TX(M) is nonempty if M is a nonempty monotone subset of E x E*. If E is reflexive and M is maximal monotone then II(M) is a single point on the diagonal of the first quadrant of K*. On the other hand, we give an example (for E the nonreflexive space Lt[O, 11) of a maximal monotone subset M of E x E' such that (0,l) f fI( M) and (1,l) E n(M) but (1,O) $ II(M). We show that the results for reflexive spaces can be recovered for general Banach spaces by using monotone operator of type '(NI)' -a class of multifunctions from E into E' which includes the subdifferentials of all proper, convex, lower semicontinuous functions on E, all surjective operators and, if E is reflexive, all maximal monotone operators. Our results lead to a simple proof of Rockafellar's result that if E is reflexive and S is maximal monotone on E then S + J is surjective. Our main tool is a classical minimax theorem.
Mathematics
Subject Classification (1991). 47805.