The nonimprovable sufficient conditions for 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, $q\in L(I;ℝ)$, $c\in ℝ$, are established which are different from the previous results. More precisely, they are interesting especially in the case where the operator $\ell$ is not of Volterra’s type with respect to the point $a$.