Journal of Convex Analysis Volume 10 (2003), No. 2, 459–469 Continuity and Maximality Properties of Pseudomonotone Operators Nicolas Hadjisavvas Department of Product and Systems Design, University of the Aegean, 84100 Hermoupolis, Syros, Greece nhad@aegean.gr Received September 02, 2002 Revised manuscript received March 13, 2003 Given a Banach space X, a multivalued operator T : X → 2 X * is called pseudomonotone (in Karamar- dian’s sense) if for all (x, x * ) and (y,y * ) in its graph, 〈x * ,y − x〉≥ 0 implies 〈y * ,y − x〉≥ 0. We define an equivalence relation on the set of pseudomonotone operators. Based on this relation, we define a notion of “D-maximality” and show that the Clarke subdifferential of a locally Lipschitz pseudoconvex function is D-maximal pseudomonotone. We generalize some well-known results on upper semicontinuity and generic single-valuedness of monotone operators by showing that, under suitable assumptions, a pseu- domonotone operator has an equivalent operator that is upper semicontinuous, generically single-valued etc. Keywords: Maximal monotone operator, pseudomonotone operator, pseudoconvex function 2000 Mathematics Subject Classification: 26B25, 47H04, 47H05 1. Introduction Pseudomonotone operators, as introduced by Karamardian [10], are defined by making use of the order relation in R, without any reference to topological properties. This is in sharp contrast to pseudomonotonicity in Brezis’ sense 1 [1]. Another feature of these operators is that they are closely related to generalized convexity, just like the relation of monotone operators to convex functions; in fact, the subdifferential of a locally Lipschitz function is pseudomonotone if and only if the function is pseudoconvex [12, 13]. Pseudomonotone operators have been the subject of intense study during the last decade. Directions of research include the finding of criteria for pseudomonotonicity of differen- tiable single-valued operators [2, 6] and the pseudomonotone variational inequality prob- lem [14, 5, 7]. However, in contrast to the theory of monotone operators, which is very rich, results on the structure of pseudomonotone operators are rare. For instance, it is known that the subdifferential of a proper, lower semicontinuous convex function is not only monotone, but also maximal monotone. Under some rather weak assumptions, monotone operators are upper semicontinuous in the interior of their domain; also, they are generically single-valued. It is a widely held belief that pseudomonotone operators do not have such properties and that, in particular, maximality is not a relevant property in generalized monotonicity, and in particular for pseudomonotone operators. Let us ten- 1 In order to underline the distinction between pseudomonotone operators in the Karamardian and in the Brezis sense, some authors use the terms order pseudomonotone and topologically pseudomonotone, respectively. ISSN 0944-6532 / $ 2.50 c  Heldermann Verlag