Regularization of P0-functions in box variational inequality problems

Two recent papers [F. Facchinei, Math. Oper. Res., 23(1998), pp. 735-745 and F. Facchinei and C. Kanzow, SIAM J. Control Optim., 37 (1999), pp. 1150-1161] have shown that for a continuously differentiable P-0-function f, the nonlinear complementarity problem NCP(f(epsilon)) corresponding to the regularization f(epsilon)(x) : = f (x) + epsilonx has a unique solution for every epsilon> 0, that dist (x(epsilon), SOL(f)) --> 0 as epsilon --> 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. These results are proved via the Fischer function and the mountain pass theorem. In this paper, we generalize these nonlinear complementarity 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.