We discuss the asymptotic behaviour of all solutions of the functional differential equation $$y^{\text{'}}\left(x\right)=\sum _{i=1}^m{a}_i\left(x\right)y\left({\tau }_i\left(x\right)\right)+b\left(x\right)y\left(x\right),$$ where $b\left(x\right)<0$. The asymptotic bounds are given in terms of a solution of the functional nondifferential equation $$\sum _{i=1}^m\left|a_i\left(x\right)\right|\omega \left({\tau }_i\left(x\right)\right)+b\left(x\right)\omega \left(x\right)=0.$$

The main result of the present paper is obtaining new inequalities for solutions of scalar equation $\dot{x}\left(t\right)=b\left(t\right)x(t-\tau \left(t\right))$. Except this the interval of transient process is computed, i.e. the time is estimated, during which the given solution $x\left(t\right)$ reaches an $\epsilon$ - neighbourhood of origin and remains in it.

The asymptotic behaviour of the solutions is studied for a real unstable two-dimensional system $x^{\text{'}}\left(t\right)=𝖠\left(t\right)x\left(t\right)+𝖡\left(t\right)x(t-r)+h(t,x\left(t\right),x(t-r))$, where $r>0$ is a constant delay. It is supposed that $𝖠$, $𝖡$ and $h$ are matrix functions and a vector function, respectively. Our results complement those of Kalas [Nonlinear Anal. 62(2) (2005), 207–224], where the conditions for the existence of bounded solutions or solutions tending to the origin as $t\to \infty$ are given. The method of investigation is based on the transformation of the real system considered to one...

The paper discusses the asymptotic properties of solutions of the scalar functional differential equation $$y^{\text{'}}\left(x\right)=ay\left(\tau \left(x\right)\right)+by\left(x\right),x\in [x_0,\infty )$$ of the advanced type. We show that, given a specific asymptotic behaviour, there is a (unique) solution $y\left(x\right)$ which behaves in this way.

We study the asymptotic behavior of the solutions of a differential equation with unbounded delay. The results presented are based on the first Lyapunov method, which is often used to construct solutions of ordinary differential equations in the form of power series. This technique cannot be applied to delayed equations and hence we express the solution as an asymptotic expansion. The existence of a solution is proved by the retract method.

In the paper we study the existence of nonoscillatory solutions of the system $x_i^{\left(n\right)}\left(t\right)={\sum }_{j=1}^2{p}_{ij}\left(t\right)f_{ij}\left(x_j\left(h_{ij}\left(t\right)\right)\right),n\ge 2,i=1,2$, with the property $li{m}_{t\to \infty }{x}_i\left(t\right)/t^{k_i}=const\ne 0$ for some $k_i\in \{1,2,...,n-1\},i=1,2$. Sufficient conditions for the oscillation of solutions of the system are also proved.

This paper establishes existence of nonoscillatory solutions with specific asymptotic behaviors of second order quasilinear functional differential equations of neutral type. Then sufficient, sufficient and necessary conditions are proved under which every solution of the equation is either oscillatory or tends to zero as $t\to \infty$.