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) = ∣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.