In recent years, the so-called exceptional families of elements (EFE) have been frequently used to study the existence of solutions for the complementarity problem; see for instance [

*G. Isac*,

*V. Bulavski* and

*V. Kalashnikov*, J. Glob. Optim. 10, No. 2, 207–225 (1997;

Zbl 0880.90127)]. The same method has been generalized very recently to variational inequality problems. The present paper introduces a further generalization of EFE to nonlinear projection equations (NPE), i.e., to equations of the form

$h\left(x\right)={{\Pi}}_{K}\left(g\left(x\right)-f\left(x\right)\right)$ where

$f,g,h$ are functions from

${\mathbb{R}}^{n}$ to itself, and

${{\Pi}}_{K}$ is the orthogonal projection to a subset

$K$ of

${\mathbb{R}}^{n}$. It is shown that, under suitable assumptions, either NPE has a solution or there exists an EFE. Applications are then given to special cases of NPE such as the generalized complementarity problem. In case the functions involved are continuous, these results generalize considerably previously known ones.