Consider the delay differential equation $$\dot{x}\left(t\right)=g(x\left(t\right),x(t-r)),\left(1\right)$$ where $r>0$ is a constant and $g{ℝ}^2\to ℝ$ is Lipschitzian. It is shown that if $r$ is small, then the solutions of (1) have the same convergence properties as the solutions of the ordinary differential equation obtained from (1) by ignoring the delay.

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...

We present several results dealing with the asymptotic behaviour of a real two-dimensional system $x^{\text{'}}\left(t\right)=𝖠\left(t\right)x\left(t\right)+{\sum }_{k=1}^m{𝖡}_k\left(t\right)x\left({\theta }_k\left(t\right)\right)+h(t,x\left(t\right),x\left({\theta }_1\left(t\right)\right),\cdots ,x\left({\theta }_m\left(t\right)\right))$ with bounded nonconstant delays $t-{\theta }_k\left(t\right)\ge 0$ satisfying ${lim}_{t\to \infty }{\theta }_k\left(t\right)=\infty$, under the assumption of instability. Here $𝖠$, ${𝖡}_k$ and $h$ are supposed to be matrix functions and a vector function, respectively. The conditions for the instable properties of solutions together with the conditions for the existence of bounded solutions are given. The methods are based on the transformation of the real system considered to one...