We deal with the integral equation $u\left(t\right)=f\left({\int }_{I}g(t,z)u\left(z\right)dz\right)$, with $t\in I=[0,1]$, $f:{𝐑}^n\to {𝐑}^n$ and $g:I\times I\to [0,+\infty [$. We prove an existence theorem for solutions $u\in L^{\infty }(I,{𝐑}^n)$ where the function $f$ is not assumed to be continuous, extending a result previously obtained for the case $n=1$.