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Keywords:
delayed differential equations; asymptotic behaviour; boundedness of solutions; Lyapunov method; Ważewski topological principle
delayed differential equations; asymptotic behaviour; boundedness of solutions; Lyapunov method; Ważewski topological principle
Summary:
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 $t-\theta _k(t) \ge 0$ 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 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.
References:
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