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

on an arbitrary time scale \(\mathbb{T}\) with sup \(\mathbb{T} = \infty \), where n ≥ 2, φ(u) = ∣u∣^γsgn(u) for γ > 0, r_i(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.

在本文中，我们为以下形式的高阶非线性延迟动力学方程建立了一些振荡准则

在任意时间尺度\(\mathbb{T}\)与 sup \(\mathbb{T} = \infty \)，其中n ≥ 2，φ ( u ) = ∣ u ∣ ^γ sgn( u ) for γ > 0, r _i (1 ≤ i ≤ n ) 是正 rd 连续函数和\(h \in {{\rm{C}}}_{{\rm{rd}}}}(\mathbb{T},( 0,\infty ))\)。函数\(\tau \in {{\rm{C}}_{{\rm{rd}}}}(\mathbb{T},\mathbb{T})\)满足 τ ( t ) ≤ t并且\(\mathop {\lim }\limits_{t \rightarrow \infty } \tau (t) = \infty \)和f∈ C(ℝ, ℝ). 通过使用广义 Riccati 变换，我们给出了这个方程的每个解都是振荡的或趋于零的充分条件。得到的结果对于相应的高阶微分方程和差分方程是新的。最后，提供了一些应用和例子来说明主要结果的重要性。