We study the integral equation $x=F(x)$ in a Banach space $E$, where $F(x)(t)=\int_Df(t,s,x(s))ds$ and $f$ satisfies usual conditions which guarantee that $F$ continuously maps the space $L^P(D,E)$ into itself. We show that if $f$ satisfies a Kamke condition with respect to the Kuratowski measure of noncompactness, then the above equation has a solution in $L^P(D,E)$.