## 时间轴上二阶非线性非自治延迟动力系统的振动性

1 邵阳学院理学院 湖南邵阳 422004

2 梧州学院大数据与软件工程学院 广西梧州 543002

## Oscillation of Second-Order Nonlinear Nonautonomous Delay Dynamic Equations on Time Scales

Zhang Ping,1, Yang Jiashan,2, Qin Guijiang,2

1 School of Science, Shaoyang University, Hunan Shaoyang 422004

2 School of Data Science and Software Engineering, Wuzhou University, Guangxi Wuzhou 543002

 基金资助: 国家自然科学基金.  51765060湖南省教育厅一般项目.  20C1683广西自然科学基金.  2020JJA110021广西高校中青年教师基础能力提升项目.  2020KY17007梧州学院校级科研重点项目.  2020B005邵阳市科技计划项目.  2021005ZD

 Fund supported: the NSFC.  51765060the Science Project of Hunan Province Education Department.  20C1683the NSF of Guangxi.  2020JJA110021the Basic Ability Improvement Project of Young and Middle-aged Teachers in Guangxi Universities.  2020KY17007the Key Project of Wuzhou University.  2020B005The Science and Technology Project of Shaoyang City.  2021005ZD

Abstract

The oscillatory behavior of a class of second-order nonlinear nonautonomous variable delay damped dynamic equations are studied on a time scale T, where the equations are noncanonical form. By using the generalized Riccati transformation, and the time scales theory and the classical inequality, we establish some new oscillation criteria for the equation. The results fully reflect the influential actions of delay functions and damping terms in system oscillation. Finally, some examples are given to show that our results extend, improve and enrich those reported in the literature, and that they have good effectiveness and practicability.

Keywords： Oscillation ; Time scales ; Emden-Fowler dynamic equations ; Riccati transformation

Zhang Ping, Yang Jiashan, Qin Guijiang. Oscillation of Second-Order Nonlinear Nonautonomous Delay Dynamic Equations on Time Scales. Acta Mathematica Scientia[J], 2022, 42(3): 839-850 doi:

## 1 引言

$$$\left[A(t) \phi_{1}\left(y^{\Delta}(t)\right)\right]^{\Delta}+b(t) \phi_{1}\left(y^{\Delta}(t)\right)+P(t) F\left(\phi_{2}(x(\delta(t)))\right)=0, t \in {\bf T}, t \geq t_{0}$$$

$\left(\mathrm{H}_{5}\right) $$A(t)>0 , 且 -b/A\in {{\Re }^{+}} . 关于系统(1.1)的解及其振动性定义, 英文文献可参见[1-3, 6, 7, 9, 13, 16], 中文文献可参见[4, 5, 8, 10-12, 14, 15].笔者仅关注系统(1.1)的不最终恒为零的解.系统(1.1)包含了多种类型的方程, 如 $$\left[r(t)\left(x^{\Delta}(t)\right)^{\gamma}\right]^{\Delta}+p(t) x^{\gamma}(t)=0,$$ $$\left(a(t) x^{\Delta}(t)\right)^{\Delta}+b(t) x^{\Delta}(t)+p(t) f(x(\delta(t)))=0,$$ $$\left(a(t)\left|x^{\Delta}(t)\right|^{\lambda-1} x^{\Delta}(t)\right)^{\Delta}+b(t)\left|x^{\Delta}(t)\right|^{\lambda-1} x^{\Delta}(t)+p(t)|x(\delta(t))|^{\lambda-1} x(\delta(t))=0,$$ 等.对正则系统即系统满足条件 $$\int_{t_{0}}^{+\infty}\left[A^{-1}(s) e_{-b / A}\left(s, t_{0}\right)\right]^{1 / \lambda} \Delta s=+\infty$$ 的情形, 其振动结果较多.如对正则系统(1.1), 文献[4]得到了其振动的若干准则, 其中一个基本定理如下: 定理A[4] 设条件 \left(\mathrm{H}_{1}\right)$$ \left(\mathrm{H}_{5}\right)$及(1.5)成立, 若存在函数$\varphi \in C_{{}}^{1}({\bf T}, (0, +\infty ))$, 使得当$\lambda \leqslant \beta$

$$$\limsup\limits_{t \rightarrow +\infty} \int_{t_{0}}^{t} \varphi(s)\left\{k^{\beta} \Psi(s)-\frac{[A(\sigma(s))]^{\lambda+1}}{M A^{\lambda}(s)}\left|\frac{\varphi^{\Delta}(s)}{\varphi(s)}-\frac{b(s)}{A(\sigma(s))}\right|^{\lambda+1}\right\} \Delta s=+\infty,$$$

$\lambda>\beta$

$$$\limsup\limits_{t \rightarrow +\infty} \int_{t_{0}}^{t} \varphi(s)\left\{k^{\beta} \Psi(s)-\frac{[A(\sigma(s))]^{\beta+1}}{M A^{\beta}(s)}\left|\frac{\varphi^{\Delta}(s)}{\varphi(s)}-\frac{b(s)}{A(\sigma(s))}\right|^{\beta+1}\right\} \Delta s=+\infty,$$$

$$$\limsup\limits_{t \rightarrow +\infty} \int_{t_{0}}^{t} \varphi(s)\left\{\Psi(s)-\frac{A^{\omega_{1}}(s)}{k}\left|\frac{\varphi^{\Delta}(s)}{\varphi(s)} -\frac{b(s)}{A(s)}\right|^{\omega_{2}+1}\right\} \Delta s=+\infty,$$$

$$$\begin{array}{ll} { } \Psi(t)=L P(t)[1-\eta B(\delta(t))]^{\beta} \xi^{\beta}\left(t, \delta_{0}\right), \\ { } \xi\left(t, \delta_{0}\right)= \bigg(\int_{\delta_{0}}^{\sigma(t)} \frac{1}{A^{1 / \lambda}(s)} \Delta s \bigg)^{-1} \bigg(\int_{\delta_{0}}^{\delta(t)} \frac{1}{A^{1 / \lambda}(s)} \Delta s\bigg), \end{array}$$$

$$$\int_{t_{0}}^{+\infty}\left[A^{-1}(s) e_{-b / A}\left(s, t_{0}\right)\right]^{1 / \lambda} \Delta s<+\infty$$$

$$$\int_{t_{0}}^{+\infty} a^{-1 / \lambda}(s) \Delta s<+\infty$$$

$$$\int_{t_{0}}^{+\infty}\left[a^{-1}(t) \int_{t_{0}}^{t} p(s)\left(\int_{s}^{+\infty} a^{-1 / \gamma}(u) \Delta u\right)^{\gamma} \Delta s\right]^{1 / \gamma} \Delta t=+\infty.$$$

$$$\left(t^{2} x^{\prime}(t)\right)^{\prime}+q_{0} x(t)=0,$$$

$$$\int_{t_{0}}^{+\infty}\left[a^{-1}(s) e_{-b / a}\left(s, t_{0}\right)\right]^{1 / \lambda} \Delta s<+\infty$$$

$$$\limsup\limits_{t \rightarrow +\infty} \int_{t_{0}}^{t}\left[p(s)-\frac{a(\delta(s))}{(\lambda+1)^{\lambda+1}\left(\delta^{\Delta}(s)\right)^{\lambda}}\left|\frac{\varphi^{\Delta}(s)}{\varphi(s)}-\frac{b(s)}{a(s)}\right|^{\lambda+1}\right] \varphi(s) \Delta s=+\infty$$$

$$$\int_{t_{0}}^{+\infty}\left[a^{-1}(t) \int_{t_{0}}^{t} e_{-b / a}(t, \sigma(s)) p(s) \Delta s\right]^{1 / \lambda} \Delta t=+\infty,$$$

$$$\int_{t_{1}}^{+\infty}\left[a^{-1}(t) \int_{t_{1}}^{t} e_{-b / a}(t, \sigma(s)) \theta^{\lambda}(s) p(s) \Delta s\right]^{1 / \lambda} \Delta t=+\infty,$$$

## 2 系统(1.1)的振动准则

$$$(f \circ g)^{\Delta}(t)=\left[\int_{0}^{1} f^{\prime}\left(g(t)+h \mu(t) g^{\Delta}(t)\right) \mathrm{d} h\right] g^{\Delta}(t).$$$

$$$\left(x^{\alpha}(t)\right)^{\Delta}=\alpha \int_{0}^{1}\left[h x^{\sigma}+(1-h) x\right]^{\alpha-1} x^{\Delta}(t) \mathrm{d} h.$$$

$$$\frac{A(s)\left(-y^{\Delta}(s)\right)^{\lambda-1} y^{\Delta}(s)}{e_{-b / A}\left(s, t_{0}\right)} \leq \frac{A(t)\left(-y^{\Delta}(t)\right)^{\lambda-1} y^{\Delta}(t)}{e_{-b / A}\left(t, t_{0}\right)} \leq \frac{A(\tau(t))\left(-y^{\Delta}(\tau(t))\right)^{\lambda-1} y^{\Delta}(\tau(t))}{e_{-b / A}\left(\tau(t), t_{0}\right)},$$$

${{y}^{\Delta }}(s)\le \frac{{{A}^{1/\lambda }}(t){{y}^{\Delta }}(t)}{{{[{{e}_{-b/A}}(t, {{t}_{0}})]}^{1/\lambda }}}\frac{{{[{{e}_{-b/A}}(s, {{t}_{0}})]}^{1/\lambda }}}{{{A}^{1/\lambda }}(s)}$, 两边积分, 得

$u\to +\infty$, 并注意到函数$\theta (t)$的定义, 由上式则可导出

$$$y(t)+\frac{A^{1 / \lambda}(t) y^{\Delta}(t)}{\left[e_{-b / A}\left(t, t_{0}\right)\right]^{1 / \lambda}} \theta(t) \geq 0.$$$

$$$\limsup\limits_{t \rightarrow +\infty} \int_{t_{1}}^{t} H(t, \sigma(s))\left[L P(\tau(s)) \bar{Q}^{\beta}(s) \pi(s)-\frac{A(s) \bar{B}^{\lambda+1}(t, s)}{(\lambda+1)^{\lambda+1} e_{-b / A}(s, \tau(s))}\right] \Delta s>0,$$$

同定理2.3的证明, 可得(2.16)式, 即当$s\ge {{t}_{1}}\ge {{t}_{0}}$时, 有

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