Continuity and Maximality Properties of Pseudomonotone Operators

Given a Banach space X, a multivalued operator $T\colon X \to 2^{X^*}$ is called pseudomonotone (in Karamardian's sense) if for all $(x, x^*)$ and $(y, y^*)$ in its graph, $\langle x^*, y - x\rangle \geq 0$ implies $\langle y^*, y - x\rangle \geq 0$. 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.