This contribution is devoted to the problem of asymptotic behaviour of solutions of scalar linear differential equation with variable bounded delay of the form $$\dot{x}\left(t\right)=-c\left(t\right)x(t-\tau \left(t\right)){(}^{*})$$ with positive function $c\left(t\right).$ Results concerning the structure of its solutions are obtained with the aid of properties of solutions of auxiliary homogeneous equation $$\dot{y}\left(t\right)=\beta \left(t\right)[y\left(t\right)-y(t-\tau \left(t\right))]$$ where the function $\beta \left(t\right)$ is positive. A result concerning the behaviour of solutions of Eq. (*) in critical case is given and, moreover, an analogy with behaviour of solutions of...

The paper is concerned with oscillation properties of $n$-th order neutral differential equations of the form $${[x\left(t\right)+cx\left(\tau \left(t\right)\right)]}^{\left(n\right)}+q\left(t\right)f\left(x\left(\sigma \right(t\left)\right)\right)=0,t\ge t_0>0,$$ where $c$ is a real number with $\left|c\right|\ne 1$, $q\in C([t_0,\infty ),ℝ)$, $f\in C(ℝ,ℝ)$, $\tau ,\sigma \in C([t_0,\infty ),{ℝ}_{+})$ with $\tau \left(t\right)<t$ and ${lim}_{t\to \infty }\tau \left(t\right)={lim}_{t\to \infty }\sigma \left(t\right)=\infty$. Sufficient conditions are established for the existence of positive solutions and for oscillation of bounded solutions of the above equation. Combination of these conditions provides necessary and sufficient conditions for oscillation of bounded solutions of the equation. Furthermore, the results are generalized to equations in which...