Non-autonomous implicit integral equations with discontinuous right-hand side

summary:We deal with the implicit integral equation $$ h(u(t))=f(\,t\,,\int_Ig(t,z)\,u(z)\,dz) \hbox{ for a.a. } t\in I, $$ where $I:=[0,1]$ and where $f:I\times [0,\lambda]\to{\Bbb R}$, $g:I\times I\to[0,+\infty[$ and $h:\,]\,0,+\infty\,[\,\to {\Bbb R}$. We prove an existence theorem for solutions $u\in L^s(I)$ where the contituity of $f$ with respect to the second variable is not assumed