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Journal of Convex Analysis 10 (2003), No. 2, 465--475
Copyright Heldermann Verlag 2003
Continuity and Maximality Properties of Pseudomonotone Operators
Nicolas Hadjisavvas
Dept. of Product and Systems Design, University of the Aegean, 84100 Hermoupolis, Syros, Greece,
nhad@aegean.gr
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.
Keywords: maximal monotone operator, pseudomonotone operator, pseudoconvex function.
MSC 2000: 26B25, 47H04, 47H05.
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