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On oscillation of solutions of forced nonlinear neutral differential equations of higher order. (English) Zbl 1080.34522
Summary: 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 \geq 0, \tag{\(*\)} \] to oscillate or tend to zero as \(t \to \infty \) for different ranges of \(p (t)\). It is shown, under some stronger conditions, that every solution of  \((*)\) oscillates or tends to zero as \(t \to \infty \). Our results hold for linear, a class of superlinear and other nonlinear equations and answer a conjecture by G. Ladas and Y. G. Sficas [J. Aust. Math. Soc., Ser. B 27, 502-511 (1986; Zbl 0566.34055)], and generalize some known results.

MSC:
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34K40 Neutral functional-differential equations
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References:
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