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Article

Keywords:
delayed differential equation; asymptotic behaviour; boundedness of solutions; two-dimensional systems; Lyapunov method; Wa.zewski topological principle
Summary:
The asymptotic behaviour of the solutions is studied for a real unstable two-dimensional system $x^{\prime }(t)={\mathsf A}(t)x(t)+{\mathsf B}(t)x(t-r)+h(t,x(t),x(t-r))$, where $r>0$ is a constant delay. It is supposed that $\mathsf A$, $\mathsf {B}$ and $h$ are matrix functions and a vector function, respectively. Our results complement those of Kalas [Nonlinear Anal. 62(2) (2005), 207–224], where the conditions for the existence of bounded solutions or solutions tending to the origin as $t\rightarrow \infty $ are given. The method of investigation is based on the transformation of the real system considered to one equation with complex-valued coefficients. Asymptotic properties of this equation are studied by means of a suitable Lyapunov-Krasovskii functional and by virtue of the Wa.zewski topological principle. Stability and asymptotic behaviour of the solutions for the stable case of the equation considered were studied in Kalas and Baráková [J. Math. Anal. Appl. 269(1) (2002), 278–300].
References:
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