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Stability and asymptotic behaviour of a two-dimensional differential system with delay. (English) Zbl 1008.34064

The two-dimensional system

${x}^{\text{'}}\left(t\right)=A\left(t\right)x\left(t\right)+B\left(t\right)x\left(t-r\right)+h\left(t,x\left(t\right),x\left(t-r\right)\right)$

is considered, where $A\left(t\right)=\left({a}_{jk}\left(t\right)\right)$, $B\left(t\right)=\left({b}_{jk}\left(t\right)\right)$, $j,k=1,2$, are real matrices and $h\left(t,x,y\right)$ is a two-dimensional real vector function. It is supposed that the functions ${a}_{jk}$ are absolutely continuous on $\left[{t}_{0},\infty \right)$, ${b}_{jk}$ are locally Lebesgue integrable on $\left[{t}_{0},\infty \right)$ and the function $h$ satisfies Carathéodory conditions on

$\left[{t}_{0},\infty \right)×\left\{\left[{x}_{1},{x}_{2}\right]\in {ℝ}^{2}:{x}_{1}^{2}+{x}_{2}^{2}

with $0. The authors use an original approach for the investigation – with the aid of complex variables the system is rewritten into an equivalent equation with complex-valued coefficients. (This idea was used in a previous paper of the first author, too, see also M. Ráb and J. Kalas [Differ. Integral Equ. 3, No. 1, 127-144 (1990; Zbl 0724.34060)].) Stability and asymptotic stability of the trivial solution, and further asymptotic properties (e.g., the boundedness of all solutions by exponential functions) are studied by means of an appropriate Lyapunov-Krasovskii functional. This approach does not require the uniform stability or uniform asymptotics stability of a corresponding linear system and leads to new, effective and easy applicable results. An illustrative example is considered. The authors discuss possible generalizations, too.

Reviewer: J.Diblík (Brno)
##### MSC:
 34K20 Stability theory of functional-differential equations 34K25 Asymptotic theory of functional-differential equations 34K12 Growth, boundedness, comparison of solutions of functional-differential equations