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Some Oscillation Criteria for a Class of Higher Order Nonlinear Dynamic Equations with a Delay Argument on Time Scales

Abstract

In this paper, we establish some oscillation criteria for higher order nonlinear delay dynamic equations of the form

$${[{r_n}\varphi {( \cdots {r_2}{({r_1}{x^\Delta })^\Delta } \cdots )^\Delta }]^\Delta }(t) + h(t)f(x(\tau (t))) = 0$$

on an arbitrary time scale \(\mathbb{T}\) with sup \(\mathbb{T} = \infty \), where n ≥ 2, φ(u) = ∣uγsgn(u) for γ > 0, ri(1 ≤ i ≤ n) are positive rd-continuous functions and \(h \in {{\rm{C}}_{{\rm{rd}}}}(\mathbb{T},(0,\infty ))\). The function \(\tau \in {{\rm{C}}_{{\rm{rd}}}}(\mathbb{T},\mathbb{T})\) satisfies τ (t) ≤ t and \(\mathop {\lim }\limits_{t \rightarrow \infty } \tau (t) = \infty \) and f ∈ C(ℝ, ℝ). By using a generalized Riccati transformation, we give sufficient conditions under which every solution of this equation is either oscillatory or tends to zero. The obtained results are new for the corresponding higher order differential equations and difference equations. In the end, some applications and examples are provided to illustrate the importance of the main results.

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Correspondence to Xin Wu.

Additional information

This work was supported by the Jiangxi Provincial Natural Science Foundation (20202BABL211003) and the Science and Technology Project of Jiangxi Education Department (GJJ180354).

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Wu, X. Some Oscillation Criteria for a Class of Higher Order Nonlinear Dynamic Equations with a Delay Argument on Time Scales. Acta Math Sci 41, 1474–1492 (2021). https://doi.org/10.1007/s10473-021-0505-6

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  • DOI: https://doi.org/10.1007/s10473-021-0505-6

Key words

  • oscillation
  • nonlinear dynamic equations
  • higher order equation
  • delay dynamic equations
  • time scale

2010 MR Subject Classification

  • 34K11
  • 39A10
  • 39A99