Absztrakt: | Consider the equation $$x'(t)= b(t) f(x(t- T))- c(t) g(x(t)),\tag{$*$}$$
where $b,c,f,g: \bbfR\mapsto \bbfR$ are continuous functions,
$c(t)\ge 0$ for all $t, xg(x)> 0$ for $x\ne 0$, $f(0)= 0$, and
the positive constant $T$ denotes the time lag. Under the hypotheses:
$(\text{H}_1)$ there exist numbers $\varepsilon_0> 0$, $k> 0$
such that $|x|\le \varepsilon_0$ implies $|f(x)|\le k|g(x)|$
and $(\text{H}_2)$ $c(t)- k|b(t+ T)|\ge 0$ for all $t\in \bbfR_+$,
sufficient conditions are given for the asymptotic stability
of the zero solution by Lyapunov's direct method with Lyapunov
functionals.\par
The effect of the dominating conditions $c(t)-
k|b(t+ T)|\ge \mu\ge 0$, $c(t)- k|b(t)|\ge \nu\ge 0$ for all
$t\ge 0$ with constant $\mu$, $\nu$ is discussed by examples.}
\RV{P.Talpalaru (Ia\c{s}i) |