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Oscillation of third-order functional dynamic equations on time scales

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Abstract

We consider the nonlinear functional dynamic equation

$$(p(t)[(r(t)x^\Delta (t))^\Delta ]^\gamma )^\Delta + q(t)f(x(\tau (t))) = 0, for t \geqslant t_0 ,$$

on a time scale \(\mathbb{T}\), where Γ > 0 is the quotient of odd positive integers, p, r, τ and q are positive rd-continuous functions defined on the time scale \(\mathbb{T}\), and lim t→∞ τ(t) = ∞. The main aim of this paper is to establish some new sufficient conditions which guarantee that the equation has oscillatory solutions or the solutions tend to zero as t →∞. The main investigation depends on the Riccati substitution and the analysis of the associated Riccati dynamic inequality. Our results extend, complement and improve some previously obtained ones. In particular, the results provided substantial improvement for those obtained by Yu and Wang [J Comput Appl Math, 225 (2009), 531–540]. Some examples illustrating the main results are given.

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  1. College of Science Research Centre, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia

    Samir H. Saker

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  1. Samir H. Saker
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Saker, S.H. Oscillation of third-order functional dynamic equations on time scales. Sci. China Math. 54, 2597–2614 (2011). https://doi.org/10.1007/s11425-011-4304-8

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