  By means of classical fixed point index, we prove new results on the
existence, non-existence, localization and multiplicity of nontrivial solutions
for systems of Hammerstein integral equations where the nonlinearities are
allowed to depend on the first derivative. As a byproduct of our theory we
discuss the existence of positive solutions of a system of third order ODEs
subject to nonlocal boundary conditions. Some examples are provided in order to
illustrate the applicability of the theoretical results.
