Existence and limiting behavior of trajectories associated with P-0-equations

Given a continuous P-0-function F : R-n --> R-n, we describe a method of co nstructing trajectories associated with the P-0-equation F(x) = 0. Various well known equation-based reformulations of the nonlinear complementarity p roblem and the box variational inequality problem corresponding to a contin uous P-0-function lead to P-0-equations. In particular, reformulations via (a) the Fischer function for the NCP, (b) the min function for the NCP, (c) the fixed point map for a BVI, and (d) the normal map for a BVI give raise to P-0-equations when the underlying function is P-0. To generate the traj ectories, we perturb the given P-0-function F to a P-function F(x, epsilon) ; unique solutions of F(x, epsilon) = 0 as epsilon varies over an interval in (0, infinity) then define the trajectory. We prove general results on th e existence and limiting behavior of such trajectories. As special cases we study the interior point trajectory, trajectories based on the fixed point map of a BVI, trajectories based on the normal map of a BVI, and a traject ory based on the aggregate function of a vertical nonlinear complementarity problem.