Comments on the Solutions Set of Equilibrium Problems Governed by Topological Pseudomonotone Bifunctions

A recent assertion given by Sadeqi, I., Salehi Paydar, M., in: A comparative study of Ky Fan hemicontinuity and Brezis pseudomonotonicity of mappings and existence results, J. Optim. Theory Appl. (2015), that the set of solutions of the variational inequality problem governed by a pseudomonotone operator is closed, is obtained as a particular case of a result from Bogdan, M., Kolumban, J.: Some regularities for parametric equilibrium problems, J. Global Optim. (2009). An example of a set in a Hilbert space where del parallel to center dot parallel to is not Fan-hemicontinuous is given. A different approach to show that del parallel to center dot parallel to is indeed topologically pseudomonotone, is expressed. From the corresponding definitions, Fanhemicontinuity implies topological pseudomonotonicity, but the reverse implication does not hold in general. This strict relationship is strengthened by del parallel to center dot parallel to. Moreover, another counterexample is given in a Lebesgue space, instead of the Sobolev space W-0(1,3) used in Steck, D.: Brezis pseudomonotonicity is strictly weaker than Ky Fan hemicontinuity. J. Optim. Theory Appl. (2019). Stability of pseudomonotonicity with respect to the composition with a linear operator is formulated as open question.