Techniques for maximal monotonicity

The purpose of this paper is to describe three techniques that are useful for the investigation of monotone sets and multifunctions, and give two applications of them. The three techniques use the “fg–theorem”, the “big convexification” of a subset of E × E∗ or a multifunction E 7→ 2E (we suppose throughout that E is a nontrivial real Banach space with dual E∗), and the “convex function associated with a multifunction E 7→ 2E”. The applications that we shall give will be the derivation of various criteria for monotone subset of E × E∗ to be maximal monotone in the special case where E is reflexive, and a slight generalization of Rockafellar’s theorem on the maximal monotonicity of the sum of maximal monotone multifunctions on a reflexive space. Remark 27 at the end of the paper contains pointers to much stronger results that can be proved using these techniques. We use a combination of the one–dimensional Hahn–Banach theorem, the Banach– Alaoglu theorem and a minimax theorem to establish the fg–theorem, Theorem 3. Though monotonicity is not mentioned in it, the fg–theorem is, in fact, an abstraction of results on monotonicity that appeared in our paper [11]. Thus this result is a bridge between functional analysis and monotonicity. The big convexification of a subset G of E × E∗ is a way of embedding G in a large convex set in such a way that monotone sets can be characterized by a different kind of inequality. The details of this are given in the “pqr–lemma”, Lemma 4. We will give the definition of the convex function χS associated with the multifunction S in Definition 21. In Lemma 7, we use the fg–theorem to obtain an equivalence valid for any nonempty subset of E × E∗ when E is reflexive. We apply this equivalence to prove in Theorem 13 that if M is a monotone subset of E × E∗ then