Solving discrete systems of nonlinear equations

We study the existence problem of a zero point of a function defined on a finite set of elements of the integer lattice ZnZn of the n  -dimensional Euclidean space RnRn. It is assumed that the set is integrally convex, which implies that the convex hull of the set can be subdivided in simplices such that every vertex is an element of ZnZn and each simplex of the triangulation lies in an n-dimensional cube of size one. With respect to this triangulation we assume that the function satisfies some property that replaces continuity. Under this property and some boundary condition the function has a zero point. To prove this we use a simplicial algorithm that terminates with a zero point within a finite number of iterations. The standard technique of applying a fixed point theorem to a piecewise linear approximation cannot be applied, because the ‘continuity property’ is too weak to assure that a zero point of the piecewise linear approximation induces a zero point of the function itself. We apply the main existence result to prove the existence of a pure Cournot–Nash equilibrium in a Cournot oligopoly model. We further obtain a discrete analogue of the well-known Borsuk–Ulam theorem and a theorem for the existence of a solution for the discrete nonlinear complementarity problem.

► In this paper we solve discrete systems of nonlinear equations. ► The domain of the variables is a regular integrally convex set. ► The underlying function satisfies some discrete continuity condition. ► Applications concern complementarity problems, Borsuk–Ulam theorem, and discrete duopolies.