Continuity and maximality properties of pseudomonotone operators

Given a Banach space X, a multivalued operator T: X → 2X ∗ 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