Regularization of P-0-functions in box variational inequality problems

Two recent papers [F. Facchinei, Math. Oper. Res., 23(1998), pp. 735-745 an d F. Facchinei and C. Kanzow, SIAM J. Control Optim., 37 (1999), pp. 1150-1 161] have shown that for a continuously differentiable P-0-function f, the nonlinear complementarity problem NCP(f(epsilon)) corresponding to the regu larization f(epsilon)(x) : = f (x) + epsilonx has a unique solution for eve ry epsilon> 0, that dist (x(epsilon), SOL(f)) --> 0 as epsilon --> 0 when t he solution set SOL(f) of NCP(f) is nonempty and bounded, and NCP(f) is sta ble 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 varia tional inequality problem corresponding to a continuous P-0-function where the regularization is described by an integral. We also describe an upper s emicontinuity property of the inverse of a weakly univalent function and st udy its consequences.