Conditions for the existence and uniqueness of a solution of the Cauchy problem $$u^{\text{'}}\left(t\right)=p\left(t\right)u\left(\tau \left(t\right)\right)+q\left(t\right),u\left(a\right)=c,$$ established in [2], are formulated more precisely and refined for the special case, where the function $\tau$ maps the interval $]a,b[$ into some subinterval $[{\tau }_0,{\tau }_1]\subseteq [a,b]$, which can be degenerated to a point.

Nonimprovable, in a certain sense, sufficient conditions for the unique solvability of the boundary value problem $$u^{\text{'}}\left(t\right)=\ell \left(u\right)\left(t\right)+q\left(t\right),u\left(a\right)+\lambda u\left(b\right)=c$$ are established, where $\ell :C\left(\right[a,b];R)\to L\left(\right[a,b];R)$ is a linear bounded operator, $q\in L\left(\right[a,b];R)$, $\lambda \in R_{+}$, and $c\in R$. The question on the dimension of the solution space of the homogeneous problem $$u^{\text{'}}\left(t\right)=\ell \left(u\right)\left(t\right),u\left(a\right)+\lambda u\left(b\right)=0$$ is discussed as well.