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 equation with...

In the paper the fourth order nonlinear differential equation $y^{\left(4\right)}+{\left(q\left(t\right)y^{\text{'}}\right)}^{\text{'}}+r\left(t\right)f\left(y\right)=0$, where $q\in C^1\left([0,\infty )\right)$, $r\in C^0\left([0,\infty )\right)$, $f\in C^0\left(R\right)$, $r\ge 0$ and $f\left(x\right)x>0$ for $x\ne 0$ is considered. We investigate the asymptotic behaviour of nonoscillatory solutions and give sufficient conditions under which all nonoscillatory solutions either are unbounded or tend to zero for $t\to \infty$.

Asymptotic behaviour of oscillatory solutions of the fourth-order nonlinear differential equation with quasiderivates $y^{\left[4\right]}+r\left(t\right)f\left(y\right)=0$ is studied.

Sufficient conditions are given under which the sequence of the absolute values of all local extremes of $y^{\left[i\right]}$, $i\in \{0,1,\cdots ,n-2\}$ of solutions of a differential equation with quasiderivatives $y^{\left[n\right]}=f(t,y^{\left[0\right]},\cdots ,y^{[n-1]})$ is increasing and tends to $\infty$. The existence of proper, oscillatory and unbounded solutions is proved.

This paper deals with property A and B of a class of canonical linear homogeneous delay differential equations of $n$-th order.

Inequalities for some positive solutions of the linear differential equation with delay ẋ(t) = -c(t)x(t-τ) are obtained. A connection with an auxiliary functional nondifferential equation is used.

In this article, stability and asymptotic properties of solutions of a real two-dimensional system $x^{\text{'}}\left(t\right)=𝐀\left(t\right)x\left(t\right)+𝐁\left(t\right)x\left(\tau \left(t\right)\right)+𝐡(t,x\left(t\right),x\left(\tau \left(t\right)\right))$ are studied, where $𝐀$, $𝐁$ are matrix functions, $𝐡$ is a vector function and $\tau \left(t\right)\le t$ is a nonconstant delay which is absolutely continuous and satisfies $\underset{t\to \infty }{lim}\tau \left(t\right)=\infty$. Generalization of results on stability of a two-dimensional differential system with one constant delay is obtained using the methods of complexification and Lyapunov-Krasovskii functional and some new corollaries and examples are presented.

In this paper we investigate the asymptotic properties of all solutions of the delay differential equation y’(x)=a(x)y((x))+b(x)y(x),      xI=[x0,). We set up conditions under which every solution of this equation can be represented in terms of a solution of the differential equation z’(x)=b(x)z(x),      xI and a solution of the functional equation |a(x)|((x))=|b(x)|(x),      xI.

In the paper we consider the difference equation of neutral type $${\Delta }^3[x\left(n\right)-p\left(n\right)x\left(\sigma \left(n\right)\right)]+q\left(n\right)f\left(x\left(\tau \left(n\right)\right)\right)=0,n\in \mathbb{N}\left(n_0\right),$$ where $p,q:\mathbb{N}\left(n_0\right)\to {ℝ}_{+}$; $\sigma ,\tau :\mathbb{N}\to \mathbb{Z}$, $\sigma$ is strictly increasing and $\underset{n\to \infty }{lim}\sigma \left(n\right)=\infty ;$$\tau$ is nondecreasing and $\underset{n\to \infty }{lim}\tau \left(n\right)=\infty$, $f:ℝ\to ℝ$, $xf\left(x\right)>0$. We examine the following two cases: $$0<p\left(n\right)\le {\lambda }^{*}<1,\sigma \left(n\right)=n-k,\tau \left(n\right)=n-l,$$ and $$1<{\lambda }_{*}\le p\left(n\right),\sigma \left(n\right)=n+k,\tau \left(n\right)=n+l,$$ where $k$, $l$ are positive integers. We obtain sufficient conditions under which all nonoscillatory solutions of the above equation tend to zero as $n\to \infty$ with a weaker assumption on $q$ than the...

Sufficient conditions are established for the oscillation of proper solutions of the system $$\begin{array}{cc}\hfill u_1^{\text{'}}\left(t\right)& =p\left(t\right)u_2\left(\sigma \left(t\right)\right),\hfill \\ \hfill u_2^{\text{'}}\left(t\right)& =-q\left(t\right)u_1\left(\tau \left(t\right)\right),\hfill \end{array}$$ where $p,q:R_{+}\to R_{+}$ are locally summable functions, while $\tau$ and $\sigma :R_{+}\to R_{+}$ are continuous and continuously differentiable functions, respectively, and $\underset{t\to +\infty }{lim}\tau \left(t\right)=+\infty$, $\underset{t\to +\infty }{lim}\sigma \left(t\right)=+\infty$.

We study asymptotic properties of solutions of the system of differential equations of neutral type.