In this paper, necessary and sufficient conditions are obtained for every bounded solution of
$$\left( * \right)\quad \quad \quad \quad \quad \quad \left {y\left( t \right) - p\left( t \right)y\left( {t - \tau } \right)^{\left( n \right)} } \right] + Q\left( t \right)G\left( {y\left( {t - \sigma } \right)} \right) = f\left( t \right),\quad t \geqslant 0,$$
to oscillate or tend to zero as
t → ∞ for different ranges of
p(
t). It is shown, under some stronger conditions, that every solution of (*) oscillates or tends to zero as
t → ∞. Our results hold for linear, a class of superlinear and other nonlinear equations and answer a conjecture by Ladas and Sficas, Austral. Math. Soc. Ser. B
27 (1986), 502–511, and generalize some known results.