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Brezis Pseudomonotonicity is Strictly Weaker than Ky–Fan Hemicontinuity

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  • Daniel Steck

    (University of Würzburg)

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.

Suggested Citation

  • Daniel Steck, 2019. "Brezis Pseudomonotonicity is Strictly Weaker than Ky–Fan Hemicontinuity," Journal of Optimization Theory and Applications, Springer, vol. 181(1), pages 318-323, April.
    • Handle: RePEc:spr:joptap:v:181:y:2019:i:1:d:10.1007_s10957-018-1435-x
    DOI: 10.1007/s10957-018-1435-x
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    References listed on IDEAS

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    1. I. Sadeqi & M. Salehi Paydar, 2015. "A Comparative Study of Ky Fan Hemicontinuity and Brezis Pseudomonotonicity of Mappings and Existence Results," Journal of Optimization Theory and Applications, Springer, vol. 165(2), pages 344-358, May.
    2. B. T. Kien & M. M. Wong & N. C. Wong & J. C. Yao, 2009. "Solution Existence of Variational Inequalities with Pseudomonotone Operators in the Sense of Brézis," Journal of Optimization Theory and Applications, Springer, vol. 140(2), pages 249-263, February.
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    Cited by:

    1. Bijaya Kumar Sahu & Ouayl Chadli & Ram N. Mohapatra & Sabyasachi Pani, 2020. "Existence Results for Mixed Equilibrium Problems Involving Set-Valued Operators with Applications to Quasi-Hemivariational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 184(3), pages 810-823, March.

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