# 一类三阶半线性中立型时滞微分方程的振动性Oscillation of a Class of Third Order Semi-Linear Neutral Delay Differential Equations

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The oscillations of a class of neutral third order semi-linear differential equations are studied. Different functions and classical inequalities are constructed by using Riccati transformation techniques. Some new oscillatory theories of this kind of differential equations are established. The conclusions generalize and improve the relevant results in the literature, and illustrate the application of the new oscillatory conclusions with examples.

1. 引言

${\left[r\left(t\right){|{Z}^{″}\left(t\right)|}^{\alpha -1}{Z}^{″}\left(t\right)\right]}^{\prime }+q\left(t\right){|x\left(\sigma \left(t\right)\right)|}^{\beta -1}x\left(\sigma \left(t\right)\right)=0,\text{\hspace{0.17em}}t\ge {t}_{0}>0,\text{\hspace{0.17em}}\beta >\alpha .$ (E)

(A1) $p\left(t\right),q\left(t\right)\in C\left(\left[{t}_{0},\infty \right),\left(0,\infty \right)\right),0\le p\left(t\right)\le p<1,q\left(t\right)>0$

(A2) $r\left(t\right)\in {C}^{1}\left(\left[{t}_{0},\infty \right),\left(0,\infty \right)\right),r\left(t\right)\ge 0,{r}^{\prime }\left(t\right)\ge 0,{\int }_{{t}_{\text{0}}}^{\infty }{r}^{-\frac{1}{\alpha }}\left(s\right)\text{d}s\le +\infty$

(A3) $\tau \left(t\right),\sigma \left(t\right)\in {C}^{1}\left(\left[{t}_{0},\infty \right),\left(0,\infty \right)\right)$ ，对每一 $t\ge {t}_{0}$ ，都有 $\tau \left(t\right)\le t,\sigma \left(t\right)\le t$

$\sigma \left(t\right)>0,{\sigma }^{\prime }\left(t\right)>0,\underset{t\to \infty }{\mathrm{lim}}\tau \left(t\right)=\underset{t\to \infty }{\mathrm{lim}}\sigma \left(t\right)=\infty .$

${\left(r\left(t\right){|{Z}^{\prime }\left(t\right)|}^{\alpha -1}{Z}^{\prime }\left(t\right)\right)}^{\prime }+q\left(t\right){|x\left(\sigma \left(t\right)\right)|}^{\beta -1}x\left(\sigma \left(t\right)\right)=0.$ (1.1)

${\int }_{{t}_{0}}^{\infty }{r}^{-\frac{1}{\alpha }}\left(s\right)\text{d}s\le +\infty$$\beta >\alpha \text{ },\beta <\alpha ,\beta =\alpha$ 的考虑 $\beta >\alpha$ 。情形新的振动性结论，我们的结果推广和改进了

2. 引理

$\left(A\right)\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }Z\left(t\right)>0,{Z}^{\prime }\left(t\right)>0,{Z}^{″}\left(t\right)>0.$

$\left(B\right)\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }Z\left(t\right)>0,{Z}^{\prime }\left(t\right)<0,{Z}^{″}\left(t\right)>0.$

$u\left(\sigma \left(t\right)\right)\ge \theta \frac{\sigma \left(t\right)}{t}u\left(t\right),\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }t\ge {T}_{\theta }$

$u\left(t\right)\ge \gamma t{u}^{\prime }\left(t\right)\text{ },\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }t\ge {T}_{\gamma }$

$\underset{t\to \infty }{\mathrm{lim}}{\int }_{{t}_{0}}^{t}{\int }_{u}^{\infty }{\left(\frac{1}{r\left(v\right)}{\int }_{v}^{\infty }q\left(s\right)\text{d}s\right)}^{\frac{1}{\alpha }}\text{d}v\text{d}u=+\infty .$ (2.1)

$\underset{t\to +\infty }{\mathrm{lim}}x\left(t\right)=0$

3. 主要结果

$D=\left\{\left(t,s\right)|t\ge s\ge {t}_{0}\right\}$${D}_{0}=\left\{\left(t,s\right)|t>s\ge {t}_{0}\right\}$

i)

ii)

，T充分大。

(3.1)

(3.2)

(3.3)

(3.4)

(3.4)式两边对t进行求导，并利用(3.3)，(3.4)得

(3.5)

(3.6)

(3.7)

(3.8)

，由于单调递减，所以当时，有。又因为，于是有

(3.9)

，根据(3.1)式，则，这与矛盾，故假设不成立，即是方程(E)的振动解。

(3.10)

(3.11)

(3.12)

，使当时，(3.12)式变成

(3.13)

，则，这与矛盾，故假设不成立，即是方程(E)的振动解。

(3.14)

(3.15)

，故有

(3.16)

4. 应用

(E1)

1) 广东省茂名市科技计划项目(2015038)；2) 广东石油化工学院理学院科研扶持基金重点项目(KY2018001)。

NOTES

*通讯作者。

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