Abstract. The nonlinear scalar equation $$x'(t)=b(t)f(x(t-T))-c(t)g(x(t)) (c(t)\ge0)$$ is considered under the assumption $|f(x)|\le\kappa |g(x)|$ $(|x|\le\epsilon _0)$ with appropriate constants $\kappa,\epsilon_0>0$. Sufficient conditions are given for the asymptotic stability of the zero solution by Lyapunov's direct method with Lyapunov functionals. The effect of the dominating conditions $$c(t)-\kappa |b(t+T)|\ge\mu \ge0, c(t)- \kappa |b(t)|\ge\nu \ge0$$ for all $t\ge0$ with constant $\mu $, $\nu $ is discussed by examples.

AMS Subject Classification (1991): 34K20

Received November 9, 1994. (Registered under 5639/2009.)