Continuity and Maximality Properties of Pseudomonotone Operators

Given a Banach space X, a multivalued operator

is called pseudomonotone (in Karamardian's sense) if for all

(x, x^*) ( x , x ∗ )

and

(y, y^*) ( y , y ∗ )

in its graph,

\langle x^*, y - x\rangle \geq 0 ⟨ x ∗ , y − x ⟩ ≥ 0

implies

\langle y^*, y - x\rangle \geq 0 ⟨ 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.