Brezis Pseudomonotonicity is Strictly Weaker than Ky–Fan Hemicontinuity

Daniel Steck ()
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Daniel Steck: University of Würzburg

Journal of Optimization Theory and Applications, 2019, vol. 181, issue 1, No 16, 318-323

Abstract: Abstract In 1968, H. Brezis introduced a notion of operator pseudomonotonicity which provides a unified approach to monotone and nonmonotone variational inequalities. A closely related notion is that of Ky–Fan hemicontinuity, a continuity property which arises if the famous Ky–Fan minimax inequality is applied to the variational inequality framework. It is clear from the corresponding definitions that Ky–Fan hemicontinuity implies Brezis pseudomonotonicity, but quite surprisingly, a recent publication by Sadeqi and Paydar (J Optim Theory Appl 165(2):344–358, 2015) claims the equivalence of the two properties. The purpose of the present note is to show that this equivalence is false; this is achieved by providing a concrete example of a nonlinear operator which is Brezis pseudomonotone but not Ky–Fan hemicontinuous.

Keywords: Brezis pseudomonotonicity; Ky–Fan hemicontinuity; Counterexample; Variational inequality; Equilibrium problem; 46B; 46T; 47H; 47J (search for similar items in EconPapers)
Date: 2019
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10957-018-1435-x

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