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On the oscillation of certain third order nonlinear functional differential equations. (English) Zbl 1154.34368

Summary: We offer some sufficient conditions for the oscillation of all solutions of third order nonlinear functional differential equations of the form
\[ \frac{d}{dt}\left(a(t)\left(\frac{d^2}{dt^2}\;x(t)\right)^\alpha\right)+q(t)f(x[g(t)])=0 \]
and
\[ \frac{d}{dt} \left(a(t)\left(\frac{d^2}{dt^2}\;x(t)\right)^\alpha\right)= q(t)f(x[g(t)])+ p(t)h(x[\sigma(t)]), \]
when \(\int^\infty a^{-1/\alpha}(s)\,ds < \infty\). The case when \(\int^\infty a^{-1/\alpha}(s)\,ds =\infty\) is also included.

MSC:

34K11 Oscillation theory of functional-differential equations
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References:

[1] Agarwal, R. P.; Grace, S. R.; O’Regan, D., Oscillation Theory for Difference and Functional Differential Equations (2000), Kluwer: Kluwer Dordrecht · Zbl 0969.34062
[2] Agarwal, R. P.; Grace, S. R.; O’Regan, D., Oscillation Theory for Second Order Dynamic Equations (2003), Taylor & Francis: Taylor & Francis London · Zbl 1070.34083
[3] Agarwal, R. P.; Grace, S. R.; O’Regan, D., On the oscillation of certain functional differential equations via comparison methods, J. Math. Anal. Appl., 286, 577-600 (2003) · Zbl 1057.34072
[4] Agarwal, R. P.; Grace, S. R.; O’Regan, D., The oscillation of certain higher order functional differential equations, Math. Comput. Modell., 37, 705-728 (2003) · Zbl 1070.34083
[5] Agarwal, R. P.; Grace, S. R.; Smith, T., Oscillation of certain third order functional differential equations, Adv. Math. Sci. Appl., 16, 69-94 (2006) · Zbl 1116.34050
[6] Gyori, I.; Ladas, G., Oscillation Theory of Delay Differential Equations with Applications (1991), Clarendon Press: Clarendon Press Oxford · Zbl 0780.34048
[7] Kitamura, Y., Oscillation of functional differential equations with general deviating arguments, Hiroshima Math. J., 15, 445-491 (1985) · Zbl 0599.34091
[8] Kusano, T.; Lalli, B. S., On oscillation of half-linear functional differential equations with deviating arguments, Hiroshima Math. J., 24, 549-563 (1994) · Zbl 0836.34081
[9] Philos, Ch. G., On the existence of nonoscillatory solutions tending to zero at ∞ for differential equations with positive delays, Arch. Math., 36, 168-178 (1981) · Zbl 0463.34050
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