  In this paper we present some new results regarding the solvability of
nonlinear Hammerstein integral equations in a special cone of continuous
functions. The proofs are based on a certain fixed point theorem of Leggett and
Williams type. We give an application of the abstract result to prove the
existence of nontrivial solutions of a periodic boundary value problem. We also
investigate, via a version of Krasnosel{\cprime}slki{\u\i}'s theorem for the
sum of two operators, the solvability of perturbed Hammerstein integral
equations in the space of continuous functions of bounded variation in the
sense of Jordan. As an application of these results, we study the solvability
of a boundary value problem subject to integral boundary conditions of
Riemann--Stieltjes type. Some examples are presented in order to illustrate the
obtained results.
