A remark on Gwinner's existence theorem on variational inequality problem

Gwinner (1981) proved an existence theorem for a variational inequality problem involving an upper semicontinuous multifunction with compact convex values. The aim of this paper is to solve this problem for a multifunction with open inverse values.

Subgradient representation of multifunctions

We provide necessary and sufficient conditions for a minimal upper semicontinuous multifunction defined on a separable Banach space to be the subdifferential mapping of a Lipschitz function.

Extensions of best approximation and coincidence theorems

LetXbe a Hausdorff compact space,Ea topological vector space on whichE*separates points,F:X→2Ean upper semicontinuous multifunction with compact acyclic values, andg:X→Ea continuous function such thatg(X)is convex andg−1(y)is acyclic for eachy∈g(X). Then either (1) there exists anx0∈Xsuch thatgx0∈Fx0or (2) there exist an(x0,z0)on the graph ofFand a continuous seminormponEsuch that0<p(gx0−z0)≤p(y−z0) for all y∈g(X). A generalization of this result and its application to coincidence theorems are obtained. Our aim in this paper is to unify and improve almost fifty known theorems of others.