Josef Kalas, Department of Mathematics and Statistics, Masaryk University, Kotlářská 2, 611 37 Brno, Czech Republic, e-mail: kalas@math.muni.cz; Josef Rebenda, Department of Mathematics and Statistics, Masaryk University, Kotlářská 2, 611 37 Brno, Czech Republic, e-mail: rebenda@math.muni.cz
Abstract: We present several results dealing with the asymptotic behaviour of a real two-dimensional system $x'(t)={\mathsf A}(t)x(t)+\sum_{k=1}^m{\mathsf B}_k(t)x(\theta_k(t)) +h(t,x(t),x(\theta_1(t)),\dots,x(\theta_m(t)))$ with bounded nonconstant delays $t-\theta_k(t) \ge0$ satisfying $\lim_{t \to\infty} \theta_k(t)=\infty$, under the assumption of instability. Here $\sf A$, ${\mathsf B}_k$ 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 complex-valued coefficients. Asymptotic properties are studied by means of a suitable Lyapunov-Krasovskii functional and the Wazewski topological principle. The results generalize some previous ones, where the asymptotic properties for two-dimensional systems with one constant or nonconstant delay were studied.
Keywords: delayed differential equations, asymptotic behaviour, boundedness of solutions, Lyapunov method, Wazewski topological principle
Classification (MSC 2010): 34K12, 34K20
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