Abstract

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

Author information

Contact details are reproduced from the original publication and may be historical.

Nicolas Hadjisavvas

Dept. of Product and Systems Design, University of the Aegean, 84100 Hermoupolis, Syros, Greece

nhad@aegean.gr

Keywords

  • maximal monotone operator
  • pseudomonotone operator
  • pseudoconvex function

Mathematics Subject Classification

26B25, 47H04, 47H05

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Suggested citation

N. Hadjisavvas. “Continuity and Maximality Properties of Pseudomonotone Operators.” Journal of Convex Analysis 10 (2003), No. 2, 465–475. https://doi.org/10.68381/jca1028

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Published by Heldermann Verlag, 2003. Rights now held by Banach Press LLC.