The paper contains an existence theorem and a Kneser-type theorem for the problem $x'=A(t)x+f(t,x)$, $x(0)=x_0$, where $\{A(t)\}_{t\in[0,d]}$ is a family of linear operator generating an evolution operator $U(t,s)$, and $f$ is a continuous function satisfying a Kamke condition with respect to the measure of noncompactness.