For a given mapF from then-dimensional Euclidean spaceE n into itself, we consider the problem of finding a nonnegative vectorx inE n whose imageF(x) is also nonnegative and such that the two vectors are orthogonal. This problem is refered to in the literature as thecomplemcntarity problcm. The importance of the complementarity problem lies in the fact that it is the unifying mathematical form for a wide range of problems arising in different fields such as mathematical programming, game theory, economics, mechanics, etc ⋯ This paper is concerned mainly with the question of the existence of a solution. Several existence theorems are given under various conditions on the mapF. These theorems cover the cases whenF is nonlinear nondifferentiable, nonlinear but differentiable, and affine.