Integrable solutions of highly discontinuous implicit functional-integral equations

Let $I$ be a real compact interval. We deal with the problem of the existence of solutions $u\in L^p(I)$ of the implicit functional-integral equation \begin{equation*} f\Big(t,u(t),\int_Ik(t,s)\,u(\varphi(s))\,ds\Big)=0\quad\hbox{for a.e.}\quad t\in I, \end{equation*} where $Y$ is a closed interval, and $f:I\times Y \times {\bf R}\to{\bf R}$, $k:I\times I\to [0,+\infty[$ and $\varphi:I\to I$ are given functions. Such an equation includes, as special cases, many integral equations studied in the literature. We prove an existence result whose main peculiarity is the following: a function $f(t,y,x)$ satisfying our assumptions can be discontinuous, with respect to the third variable, even at all points $x\in{\bf R}$. As regards the function $y\to f(t,y,x)$, we only require that it is continuous, that it changes its sign over $Y$, and that it is not identically zero over any interval. No assumption of monotonicity is made on $f$. Our result extends and improves several results in the literature. Examples and also counter-examples to possible improvements are presented.