  By means of fixed point index theory for multi-valued maps, we provide an
analogue of the classical Birkhoff--Kellogg Theorem in the context of
discontinuous operators acting on affine wedges in Banach spaces. Our theory is
fairly general and can be applied, for example, to eigenvalues and parameter
problems for ordinary differential equations with discontinuities. We
illustrate in details this fact for a class of second order boundary value
problem with deviated arguments and discontinuous terms. In a specific example,
we explicitly compute the terms that occur in our theory.
