Asymptotic Behaviour of a Two-Dimensional Differential System with a Finite Number of Nonconstant Delays under the Conditions of Instability

Abstract

The asymptotic behaviour of a real two-dimensional differential system $x\prime (t)=\mathsf{A}(t)x(t)+{\sum }_{k=1}^m{\mathsf{B}}_k(t)x({\theta }_k(t))+h(t,x(t),x({\theta }_1(t)),\dots ,x({\theta }_m(t)))$ with unbounded nonconstant delays $t-{\theta }_k(t)\ge 0$ satisfying ${\lim \hspace{0.17em}}_{t\to \infty }{\theta }_k(t)=\infty$ is studied under the assumption of instability. Here, $\mathsf{A}$, ${\mathsf{B}}_{\mathrm{k,}}$ and $h$ are supposed to be matrix functions and a vector function. The conditions for the instable properties of solutions and the conditions for the existence of bounded solutions are given. The methods are based on the transformation of the considered real system to one equation with complex-valued coefficients. Asymptotic properties are studied by means of a Lyapunov-Krasovskii functional and the suitable Ważewski topological principle. The results generalize some previous ones, where the asymptotic properties for two-dimensional systems with one constant or nonconstant delay were studied.