Vector implicit functional-integral equations associated with discontinuous functions

Let $I:=[0,1]$. In this paper we deal with the implicit vector functional-integral equation % $$h(t,u(t))=g(t)+f(t,\int_Ik(t,s)\,u(\varphi(s))\,ds)\quad\hbox{for a.e.}\quad t\in I,$$ % where $h:I\times{\bf R}^n\to{\bf R}$, $\varphi:I\to I$, $g:I\to{\bf R}$, $k:I\times I\to[0,+\infty[$ and $f:I\times{\bf R}^n\to{\bf R}$ are given. We prove an existence theorem for solutions $u\in L^p(I,{\bf R}^n)$ (with $p\in]1,+\infty]$), which extends a very recent result proved for the case $n=1$. Such an extension is not immediate and requires a more articulated technical construction. The main peculiarity of our result is the regularity assumption on $f$, considerably weaker than the usual Carath\`eodory condition required in the literature. As a matter of fact, a function $f$ satisfying the assumptions of our main result could be discontinuous, with respect to the second variable, even at each point $x\in{\bf R}^n$.