Continuity and Maximality Properties of Pseudomonotone Operators

Given a Banach space X, a multivalued operator T: X --> 2X* is called pseudomonotone (in Karamardian'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 pseudomonotone operator has an equivalent operator that is upper semicontinuous, generically single-valued etc.