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 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.

The asymptotic properties of solutions of the equation $u^{\text{'}\text{'}\text{'}}\left(t\right)=p_1\left(t\right)u\left({\tau }_1\left(t\right)\right)+p_2\left(t\right)u^{\text{'}}\left({\tau }_2\left(t\right)\right)$, are investigated where $p_i:[a,+\infty [\to R(i=1,2)$ are locally summable functions, ${\tau }_i:[a,+\infty [\to R(i=1,2)$ measurable ones and ${\tau }_i\left(t\right)\ge t(i=1,2)$. In particular, it is proved that if $p_1\left(t\right)\le 0$, $p_2^2\left(t\right)\le \alpha \left(t\right)\left|p_1\left(t\right)\right|$, $${\int }_a^{+\infty }{[{\tau }_1\left(t\right)-t]}^2{p}_1\left(t\right)dt<+\infty \text{and}{\int }_a^{+\infty }\alpha \left(t\right)dt<+\infty ,$$ then each solution with the first derivative vanishing at infinity is of the Kneser type and a set of all such solutions forms a one-dimensional linear space.

The problem on the existence of a positive in the interval $]a,b[$ solution of the boundary value problem $$u^{\text{'}\text{'}}=f(t,u)+g(t,u)u^{\text{'}};u(a+)=0,u(b-)=0$$ is considered, where the functions $f$ and $g]a,b[\times ]0,+\infty [\to ℝ$ satisfy the local Carathéodory conditions. The possibility for the functions $f$ and $g$ to have singularities in the first argument (for $t=a$ and $t=b$) and in the phase variable (for $u=0$) is not excluded. Sufficient and, in some cases, necessary and sufficient conditions for the solvability of that problem are established.