A note on non-autonomous implicit integral equations with discontinuous right-hand side

Let $I:=[0,1]$, $f:I imes[0,sigma] o R$, $g:I imes I o[0,+infty[$ and $h:I imes]0,+infty[ o R$. In this note we prove an existence result for solutions $uin L^s(I)$ of the integral equation $$h(t,u(t))=fBig(t,int_Ig(t,z),u(z),dzBig)quadhbox{for a.a.}quad tin I$$ where, in particular, the continuity of $f$ with respect to the second variable is not assumed. Our result is a partial extension of a previous result of the same authors [1], where the function $h$ was not allowed to depend explicitly on $t$.