The present study approaches variational inequalities governed by Fan-hemicontinuous operators. The Fan-hemicontinuity was recently proved (\cite{bogdan:hemicontinuity}) for the gradient of the norm defined on its scalarly-positive, closed convex subdomain of a Hilbert space. The aim of the present study is to establish whether the positive result on Fan-hemicontinuity can be extended to an arbitrary reflexive Banach space. In this matter it is natural to consider the duality map and its topological properties. Based on the weak-weak sequential continuity of the generalized duality map, the discussed property holds on $\ell^p$, $1 < p < +\infty$ and does not hold on $L^p$, $p\neq 2$, thus restricting the space setting where weak compactness results are applicable.