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.

We establish new efficient conditions sufficient for the unique solvability of the initial value problem for two-dimensional systems of linear functional differential equations with monotone operators.

Conjugacy and disconjugacy criteria are established for the equation $$u^{\text{'}\text{'}}+p\left(t\right)u=0,$$ where $p:]-\infty ,+\infty [\to ]-\infty ,+\infty [$ is a locally summable function.

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.