Sufficient conditions are obtained so that every solution of ${[y\left(t\right)-p\left(t\right)y(t-\tau )]}^{\left(n\right)}+Q\left(t\right)G\left(y(t-\sigma )\right)=f\left(t\right)$ where n ≥ 2, p,f ∈ C([0,∞),ℝ), Q ∈ C([0,∞),[0,∞)), G ∈ C(ℝ,ℝ), τ > 0 and σ ≥ 0, oscillates or tends to zero as $t\to \infty$. Various ranges of p(t) are considered. In order to accommodate sublinear cases, it is assumed that ${\int }_0^{\infty }Q\left(t\right)dt=\infty$. Through examples it is shown that if the condition on Q is weakened, then there are sublinear equations whose solutions tend to ±∞ as t → ∞.

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.

This paper deals with the system of functional-differential equations $$\frac{dx\left(t\right)}{dt}=p\left(x\right)\left(t\right)+q\left(t\right),$$ where $p:C_{\omega }\left(R^n\right)\to L_{\omega }\left(R^n\right)$ is a linear bounded operator, $q\in L_{\omega }\left(R^n\right)$, $\omega >0$ and $C_{\omega }\left(R^n\right)$ and $L_{\omega }\left(R^n\right)$ are spaces of $n$-dimensional $\omega$-periodic vector functions with continuous and integrable on $[0,\omega ]$ components, respectively. Conditions which guarantee the existence of a unique $\omega$-periodic solution and continuous dependence of that solution on the right hand side of the system considered are established.