Variational inequality problems with a continuum of solutions: Existence and computation

In this paper three sufficient conditions are provided under each of which an upper semicontinuous point-to-set mapping defined on an arbitrary polyto pe has a connected set of zero points that connect two distinct faces of th e polytope. Furthermore, we obtain an existence theorem of a connected set of solutions to a nonlinear variational inequality problem over arbitrary p olytopes. These results follow in a constructive way by designing a new sim plicial algorithm. The algorithm operates on a triangulation of the polytop e and generates a piecewise linear path of points connecting two distinct f aces of the polytope. Each point on the path is an approximate zero point. As the mesh size of the triangulation goes to zero, the path converges to a connected set of zero points linking the two distinct faces. As a conseque nce, our results generalize Browder's fixed point theorem [ Summa Brasilien sis Mathematicae, 4 (1960), pp. 183-191] and an earlier result by the autho rs [ Math. Oper. Res., 21 (1996), pp. 675-696] on the n-dimensional unit cu be. An application in economics and some numerical examples are also discus sed.