Regularization of P0-functions in Box Variational Inequality Problems

In two recent papers, Facchinei 7] and Facchinei and Kanzow 8] have shown that for a continuously diierentiable P 0-function f , the nonlinear complementarity problem NCP(f ") corresponding to the regularization f " (x) := f (x) + "x has a unique solution for every " > 0, that dist (x("); SOL(f)) ! 0 as " ! 0 when the solution set SOL(f) of NCP(f) is nonempty and bounded, and NCP(f) is stable if and only if the solution set is nonempty and bounded. They prove these results via the the Fischer function and the Mountain Pass Theorem. In this paper, we generalize these NCP results to a Box Variational Inequality Problem corresponding to a continuous P 0-function where the regularization is described by an integral. We also describe an upper semicontinuity property of the inverse of a weakly univalent function and study its consequences.