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Science China Mathematics

, Volume 54, Issue 12, pp 2597–2614 | Cite as

Oscillation of third-order functional dynamic equations on time scales

  • Samir H. SakerEmail author
Articles

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.

Keywords

oscillation third-order dynamic equations time scales 

MSC(2000)

34K11 39A10 39A99 

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Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.College of Science Research CentreKing Saud UniversityRiyadhSaudi Arabia

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