Continuity and Maximality Properties of Pseudomonotone Operators
Nicolas Hadjisavvas · Journal of convex analysis · 2003
Given a Banach space X, a multivalued operator T\colon X \to 2^{X^*} T : X → 2 X ∗ 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.