The Rockafellar Conjecture and type (FPV)

In this paper, using a technique of Verona and Verona, we show that a result announced recently by Eberhard and Wenczel implies the truth of the Rockafellar conjecture. We then show that there is a gap in the logic of the Eberhard–Wenczel result, which we tried unsuccessfully to close. We also discuss briefly the connection with maximally monotone multifunctions of type (FPV). One of the fundamental results in the theory of monotone operators, which was proved by Rockafellar in [4, Theorem 1, pp. 76–83], is that if X is a reflexive Banach space, S : X ⇒ X∗ and T : X ⇒ X∗ are maximally monotone and intD(S) ∩ D(T ) 6= ∅ then the Minkowski sum S + T is maximally monotone. (As usual, “int” stands for “interior” and “D(·)” stands for “domain of”.) We will describe as the Rockafellar Conjecture the statement that this result is true if X is not assumed to be reflexive. Over the years, many people have tried unsuccessfully to prove or disprove the Rockafellar Conjecture. So, for the rest of this paper, we assume that X is a real, possibly nonreflexive, Banach space. It is in this context that one must consider the assertion of Eberhard and Wenczel in [1, Theorem 36] (modified according to (24)), which we state formally as Conjecture 1: Conjecture 1. If S : X ⇒ X∗ and T : X ⇒ X∗ are maximally monotone, D(S) ∩ D(T ) is bounded and intD(S) ∩ D(T ) 6= ∅ then S + T is maximally monotone. We will prove in Theorem 2 that Conjecture 1 implies the truth of the Rockafellar Conjecture. Our argument is based on an argument of Verona and Verona (see the preprint [8]). Their argument actually establishes a stronger result – we give a self contained but less technical proof of a weaker result here, which is adequate for our purposes. ∗University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria. Email: [email protected]. †Universidad del Paćıfico Jr. Gral. Sanchez Cerro 2050 Pabellón I Jesús Maŕıa, Lima 11, Perú. Email: [email protected]. ‡Department of Mathematics, University of California, Santa Barbara, CA 93106-3080, U.S.A. Email: [email protected].