Brezis pseudomonotonicity is strictly weaker than Ky-Fan hemicontinuity
Abstract
In 1968, H. Brezis introduced a notion of operator pseudomonotonicity which provides a unified approach to monotone and nonmonotone variational inequalities (VIs). 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 VI 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
Cite
@article{arxiv.1810.03995,
title = {Brezis pseudomonotonicity is strictly weaker than Ky-Fan hemicontinuity},
author = {Daniel Steck},
journal= {arXiv preprint arXiv:1810.03995},
year = {2018}
}
Comments
6 pages, 1 figure