We propose results about sign-constancy of Green's functions to impulsive nonlocal boundary value problems in a form of theorems about differential inequalities. One of the ideas of our approach is to construct Green's functions of boundary value problems for simple auxiliary differential equations with impulses. Careful analysis of these Green's functions allows us to get conclusions about the sign-constancy of Green's functions to given functional differential boundary value problems, using the...

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...

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

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.