Functional-integral equations of Hammerstein type associated with discontinuous functions

In this paper we deal with the functional-integral equation \[ u(t)=f\Big(t,\int_I k(t,s)\,g(s,u(\varphi(s)))\,ds\Big)\quad\hbox{for a.e.}\quad t\in I, \] where $I:=[0,1]$, and $f:I\times {\bf R}_0^+\to{\bf R}$, $g:I\times{\bf R}\to{\bf R}_0^+$, $k:I\times I\to {\bf R}_0^+$, and $\varphi:I\to I$ are given functions. We prove an existence result for solutions $u\in L^p(I)$, where the usual Carathéodory conditions on the function $f$ are meaningfully weakened. In particular, a function $f$ satisfying our assumptions can be discontinuous, with respect to the second variable, even at all points $x\in{\bf R}$.