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...

A sufficient integral condition for the absence of eventually positive solutions of a first order stable type differential inequality with one nondecreasing retarded argument is given. In the special case of equality the result becomes an oscillation criterion.

This paper provides an introduction to delay differential equations together with a short survey on state-dependent delay differential equations arising in population dynamics. Our main goal is to examine how the delays emerge from inner mechanisms in the model, how they induce oscillations and stability switches in the system and how the qualitative behaviour of a biological model depends on the form of the delay.

Several recent oscillation criteria are obtained for nonlinear delay impulsive differential equations by relating them to linear delay impulsive differential equations or inequalities, and then comparison and oscillation criteria for the latter are applied. However, not all nonlinear delay impulsive differential equations can be directly related to linear delay impulsive differential equations or inequalities. Moreover, standard oscillation criteria for linear equations cannot be applied directly...

Consider the forced higher-order nonlinear neutral functional differential equation $$\frac{{\mathrm{d}}^n}{\mathrm{d}{t}^n}[x\left(t\right)+C\left(t\right)x(t-\tau )]+\sum _{i=1}^m{Q}_i\left(t\right)f_i\left(x(t-{\sigma }_i)\right)=g\left(t\right),t\ge t_0,$$ where $n,m\ge 1$ are integers, $\tau ,{\sigma }_i\in {ℝ}^{+}=[0,\infty )$, $C,Q_i,g\in C([t_0,\infty ),ℝ)$, $f_i\in C(ℝ,ℝ)$, $(i=1,2,\cdots ,m)$. Some sufficient conditions for the existence of a nonoscillatory solution of above equation are obtained for general $Q_i\left(t\right)$$(i=1...$

In this paper, we study the existence of oscillatory and nonoscillatory solutions of neutral differential equations of the form $x\left(t\right)-cx(t-r)$’$P\left(t\right)x(t-\theta )-Q\left(t\right)x(t-\delta )$=0 where $c>0$, $r>0$, $\theta >\delta \ge 0$ are constants, and $P$, $Q\in C({ℝ}^{+},{ℝ}^{+})$. We obtain some sufficient and some necessary conditions for the existence of bounded and unbounded positive solutions, as well as some sufficient conditions for the existence of bounded and unbounded oscillatory solutions.

The higher order neutral functional differential equation $$\frac{{\mathrm{d}}^n}{\mathrm{d}{t}^n}\left[x\left(t\right)+h\left(t\right)x\left(\tau \right(t\left)\right)\right]+\sigma f\left(t,x\left(g\right(t\left)\right)\right)=0\left(1\right)$$ is considered under the following conditions: $n\ge 2$, $\sigma =\pm 1$, $\tau \left(t\right)$ is strictly increasing in $t\in [t_0,\infty )$, $\tau \left(t\right)<t$ for $t\ge t_0$, ${lim}_{t\to \infty }\tau \left(t\right)=\infty$, ${lim}_{t\to \infty }g\left(t\right)=\infty$, and $f(t,u)$ is nonnegative on $[t_0,\infty )\times (0,\infty )$ and nondecreasing in $u\in (0,\infty )$. A necessary and sufficient condition is derived for the existence of certain positive solutions of (1).