Nonzero solutions of Hammerstein integral equations with discontinuous kernels

Using the theory of fixed point index, we establish new results for the existence of nonzero solutions of integral equations of the form $u(t)=\int_{G}k(t,s)f(s,u(s))\,ds$, where $G$ is a compact set in $\R^{n}$ and $k$ changes sign, so positive solutions may not exist, $f$ satisfies Carath\'{e}odory conditions and $k$ may be discontinuous. We apply our results to prove the existence of nontrivial solutions of some nonlocal boundary value problems.