Fan-hemicontinuity for the gradient of the norm in several reflexive Banach spaces - Bogdan, Marcel (Topological Methods in Nonlinear Analysis, 2025)
The present study approaches variational inequalities governed by Fan-hemicontinuous operators. The Fan-hemicontinuity was recently proved (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 ^p, 1 < p < +∞ and does not hold on L^p, p eq 2, thus restricting the space setting where weak compactness results are applicable.