Brezis pseudomonotonicity is strictly weaker than Ky-Fan hemicontinuity

TL;DR

This paper clarifies the relationship between Brezis pseudomonotonicity and Ky-Fan hemicontinuity, demonstrating that the two concepts are not equivalent by providing a counterexample.

Contribution

It disproves the claimed equivalence between Brezis pseudomonotonicity and Ky-Fan hemicontinuity by constructing a specific counterexample.

Findings

Brezis pseudomonotonicity does not imply Ky-Fan hemicontinuity.

Ky-Fan hemicontinuity is strictly stronger than Brezis pseudomonotonicity.

The paper provides a concrete example of an operator that is pseudomonotone but not hemicontinuous.

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.