If $F : C \rightarrow R^{n}$ is a continuous (nonlinear) mapping on a closed, convex subset $C$ of $R^{n}$, it is shown that very weak coercivity conditins on $F$ guarantee the existence of a solution $x^{} in $C$ to the variational inequality $(x-x^{}, Fx^{}) \geq 0$ for each $x$ in $C$. By restricting the shape of $C$, it is shown that $x^{}$ solves different problems, and in each case we are able to obtain new existence results. If $C$ is a cone $K$, the $x^{}$ is a solution to the complementarity problem: Find an $x^{}$ in $K$ such that $Fx^{}$ belongs to the polar of $K$ and $(x^{}, Fx^{})=0$. In this case it is possible to generalize some of the feasibility results available in the linear theory and to give an iterative scheme for finding $x^{}. If $C$ is similar to a simplex then $x^{}$ turns out to be a solution of nonlinear inequalities in the preorder induced by a cone, while if $C$ or $K$ is $R^{n}$, then $Fx^{} =0$.