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 $n=1$.

We deal with the integral equation $u\left(t\right)=f(t,{\int }_{I}g(t,z)u\left(z\right)dz)$, with $t\in I:=[0,1]$, $f:I\times {ℝ}^n\to {ℝ}^n$ and $g:I\times I\to [0,+\infty [$. We prove an existence theorem for solutions $u\in L^s(I,{ℝ}^n)$, $s\in ]1,+\infty ]$, where $f$ is not assumed to be continuous in the second variable. Our result extends a result recently obtained for the special case where $f$ does not depend explicitly on the first variable $t\in I$.