  We prove new results on the existence, non-existence, localization and
multiplicity of nontrivial solutions for perturbed Hammerstein integral
equations. Our approach is topological and relies on the classical fixed point
index. Some of the criteria involve a comparison with the spectral radius of
some related linear operators. We apply our results to some boundary value
problems with local and nonlocal boundary conditions of Neumann type. We
illustrate in some examples the methodologies used.
