*(English)*Zbl 1008.34064

The two-dimensional system

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 $h(t,x,y)$ is a two-dimensional real vector function. It is supposed that the functions ${a}_{jk}$ are absolutely continuous on $[{t}_{0},\infty )$, ${b}_{jk}$ are locally Lebesgue integrable on $[{t}_{0},\infty )$ and the function $h$ satisfies Carathéodory conditions on

with $0<K\le \infty $. 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.