On the Solutions of Discrete Nonlinear Complementarity and Related Problems

This paper concerns the existence of solutions to the discrete nonlinear complementarity problem. The problem is that of finding an integer vector x in the n-dimensional Euclidean space such that both x and f(x) are nonnegative, and the inner product of x and f(x) is equal to zero, where f is a nonlinear function mapping from the n-dimensional Euclidean space into itself. Several sufficient conditions are introduced to show the existence of solutions or a unique solution to the problem. Meanwhile, we study the closely related discrete fixed point problem and provide sufficient conditions for the existence of discrete fixed points. In addition, two economic applications are discussed.