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Keywords:
oscillation; nonoscillation; neutral equations; asymptotic behaviour
Summary:
In this paper, necessary and sufficient conditions are obtained for every bounded solution of \[ [y (t) - p (t) y (t - \tau )]^{(n)} + Q (t) G \bigl (y (t - \sigma )\bigr ) = f (t), \quad t \ge 0, \qquad \mathrm{(*)}\] to oscillate or tend to zero as $t \rightarrow \infty $ for different ranges of $p (t)$. It is shown, under some stronger conditions, that every solution of $(*)$ oscillates or tends to zero as $t \rightarrow \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.
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