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In this paper, necessary and sufficient conditions are obtained for every bounded solution of $${[y\left(t\right)-p\left(t\right)y(t-\tau )]}^{\left(n\right)}+Q\left(t\right)G\left(y(t-\sigma )\right)=f\left(t\right),t\ge 0,(*)$$ to oscillate or tend to zero as $t\to \infty$ for different ranges of $p\left(t\right)$. It is shown, under some stronger conditions, that every solution of $(*)$ oscillates or tends to zero as $t\to \infty$. 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.

We obtain sufficient conditions for every solution of the differential equation $${[y\left(t\right)-p\left(t\right)y\left(r\left(t\right)\right)]}^{\left(n\right)}+v\left(t\right)G\left(y\left(g\left(t\right)\right)\right)-u\left(t\right)H\left(y\left(h\left(t\right)\right)\right)=f\left(t\right)$$ to oscillate or to tend to zero as $t$ approaches infinity. In particular, we extend the results of Karpuz, Rath and Padhy (2008) to the case when $G$ has sub-linear growth at infinity. Our results also apply to the neutral equation $${[y\left(t\right)-p\left(t\right)y\left(r\left(t\right)\right)]}^{\left(n\right)}+q\left(t\right)G\left(y\left(g\left(t\right)\right)\right)=f\left(t\right)$$ when $q\left(t\right)$ has sign changes. Both bounded and unbounded solutions are consideted here; thus some known results are expanded.

In this paper, the oscillation criteria for solutions of the neutral delay differential equation (NDDE) $${\left(y\left(t\right)-p\left(t\right)y\left(t-\tau \right)\right)}^{\left(n\right)}+\alpha Q\left(t\right)G\left(y\left(t-\sigma \right)\right)=f\left(t\right)$$ has been studied where $p\left(t\right)=1$ or $p\left(t\right)\le 0$, $\alpha =\pm 1$, $Q\in C\left([0,\infty ),R^{+}\right)$, $f\in C\left(\right[0,\infty ),R)$, $G\in C(R,R)$. This work improves and generalizes some recent results and answer some questions that are raised in [1].