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On oscillatory fourth order nonlinear neutral differential equations I. (English) Zbl 1099.34061
Summary: Here, oscillatory and asymptotic properties of solutions of the classes of fourth order neutral differential equations \[ \bigl (r(t)\bigl (y(t)+p(t) y(t-\tau)\bigr)''\bigr)'' + q(t) G \bigl (y(t - \sigma)\bigr) = f(t) \tag{*} \] and \[ \bigl (r(t)\bigl (y(t)+p(t)y(t-\tau)\bigr)''\bigr)'' + q(t) G \bigl (y(t - \sigma)\bigr) = 0 \] are studied under the assumption \(\int _0^\infty \frac t{r(t)}\,\operatorname {d}\!t<\infty \) for various ranges of \(p(t)\). Sufficient conditions are obtained for the existence of bounded positive solutions of equation (*).

MSC:
34K11 Oscillation theory of functional-differential equations
34K25 Asymptotic theory of functional-differential equations
34K12 Growth, boundedness, comparison of solutions to functional-differential equations
34K40 Neutral functional-differential equations
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References:
[1] GYORI I.-LADAS G.: Oscillation Theory of Delay Differential Equations with Applications. Clarendon Press, Oxford, 1991.
[2] KUSANO T.-NAITO M.: Non linear oscillation of fourth order differential equations. Canad. J. Math. 4 (1976), 840-852. · Zbl 0432.34022
[3] PARHI N.-RATH R. N.: On oscillation of solutions of forced non linear neutral differential equations of higher order. Czechoslovak Math. J. · Zbl 1080.34522
[4] PARHI N.-RATH R. N.: On oscillation criteria for forced nonlinear higher order neutral differential equations. · Zbl 1099.34060
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