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

Authors
• 1 East China JiaoTong University, Nanchang, 330013, China , Nanchang (China)
Type
Published Article
Journal
Acta Mathematica Scientia
Publisher
Springer-Verlag
Publication Date
Jun 29, 2021
Volume
41
Issue
5
Pages
1474–1492
Identifiers
DOI: 10.1007/s10473-021-0505-6
Source
Springer Nature
Keywords
Disciplines
• Article
In this paper, we establish some oscillation criteria for higher order nonlinear delay dynamic equations of the form [rnφ(⋯r2(r1xΔ)Δ⋯)Δ]Δ(t)+h(t)f(x(τ(t)))=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${[{r_n}\varphi {( \cdots {r_2}{({r_1}{x^\Delta })^\Delta } \cdots )^\Delta }]^\Delta }(t) + h(t)f(x(\tau (t))) = 0$$\end{document} on an arbitrary time scale T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{T}$$\end{document} with sup T=∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{T} = \infty$$\end{document}, where n ≥ 2, φ(u) = ∣u∣γsgn(u) for γ > 0, ri(1 ≤ i ≤ n) are positive rd-continuous functions and h∈Crd(T,(0,∞))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h \in {{\rm{C}}_{{\rm{rd}}}}(\mathbb{T},(0,\infty ))$$\end{document}. The function τ∈Crd(T,T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau \in {{\rm{C}}_{{\rm{rd}}}}(\mathbb{T},\mathbb{T})$$\end{document} satisfies τ (t) ≤ t and limt→∞τ(t)=∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathop {\lim }\limits_{t \rightarrow \infty } \tau (t) = \infty$$\end{document} 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.