Acta mathematica scientia,Series B ›› 2021, Vol. 41 ›› Issue (5): 1474-1492.

• Articles •

### SOME OSCILLATION CRITERIA FOR A CLASS OF HIGHER ORDER NONLINEAR DYNAMIC EQUATIONS WITH A DELAY ARGUMENT ON TIME SCALES

Xin WU

1. School of Sciences, East China JiaoTong University, Nanchang 330013, China
• Received:2020-04-21 Revised:2021-04-24 Online:2021-10-25 Published:2021-10-21
• Supported by:
This work was supported by the Jiangxi Provincial Natural Science Foundation (20202BABL211003) and the Science and Technology Project of Jiangxi Education Department (GJJ180354).

Abstract: In this paper, we establish some oscillation criteria for higher order nonlinear delay dynamic equations of the form \begin{align*}[r_n\varphi(\cdots r_2(r_1x^{\Delta})^{\Delta}\cdots)^{\Delta}]^{\Delta}(t)+h(t)f(x(\tau(t)))=0 \end{align*} on an arbitrary time scale $\mathbb{T}$ with $\sup\mathbb{T}=\infty$, where $n\geq 2$, $\varphi(u)=|u|^{\gamma}$sgn$(u)$ for $\gamma>0$, $r_i(1\leq i\leq n)$ are positive rd-continuous functions and $h\in {\mathrm{C}_{\mathrm{rd}}}(\mathbb{T},(0,\infty))$. The function $\tau\in {\mathrm{C}_{\mathrm{rd}}}(\mathbb{T},\mathbb{T})$ satisfies $\tau(t)\leq t$ and $\lim\limits_{t\rightarrow\infty}\tau(t)=\infty$ and $f\in {\mathrm{C}}(\mathbb{R},\mathbb{R})$. 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.

CLC Number:

• 34K11
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