For the differential equation $$u^{\left(n\right)}\left(t\right)=f(t,u\left({\tau }_1\left(t\right)\right),\cdots ,u^{(n-1)}\left({\tau }_n\left(t\right)\right)),$$ where the vector function $f:]a,b[\times R^{kn}\to R^k$ has nonintegrable singularities with respect to the first argument, sufficient conditions for existence and uniqueness of the Vallée–Poussin problem are established.

Nonimprovable, in a sense sufficient conditions guaranteeing the unique solvability of the problem $$u^{\text{'}}\left(t\right)=\ell \left(u\right)\left(t\right)+q\left(t\right),u\left(a\right)=c,$$ where $\ell C(I,ℝ)\to L(I,ℝ)$ is a linear bounded operator, $q\in L(I,ℝ)$, and $c\in ℝ$, are established.