Vector solutions of implicit functional-integral equations with highly discontinuous operators

Let $I\subseteq{\bf R}$ be a compact interval. In this paper we prove an existence result for solutions $u\in L^p(I,{\bf R}^n)$, with $p\in]1,+\infty]$, of the implicit functional-integral equation \[ f\big(t,u(t),\int_I\xi(t,s)\,u(\varphi(s))\,ds\big)=0\quad\hbox{for almost every}\quad t\in I, \] where $f:I\times S\times{\bf R}^n\to{\bf R}$, $\varphi:I\to I$ and $\xi:I\times I\to[0,+\infty[$ are given functions, and $S\subseteq{\bf R}^n$ is a suitable closed connected and locally connected set. The main peculiarity of our result is the regularity assumption on $f$ with respect to the third variable, considerably weaker than the usual continuity required in the literature. A function $f$ satisfying the assumptions of our result can be discontinuous, with respect to the third variable, even at each point $x\in{\bf R}^n$. Our result extends a very recent result proved in the scalar case $n=1$. Such an extension is not trivial and requires more articulated assumptions, together with a more articulated and delicate technical construction.