Nonzero solutions of perturbed Hammerstein integral equations with deviated arguments and applications

We provide a theory to establish the existence of nonzero so- lutions of perturbed Hammerstein integral equations with deviated argu- ments, being our main ingredient the theory of fixed point index. Our ap- proach is fairly general and covers a variety of cases. We apply our results to a periodic boundary value problem with reflections and to a thermo- stat problem. In the case of reflections we also discuss the optimality of some constants that occur in our theory. Some examples are presented to illustrate the theory.