三阶半线性中立型微分方程的新的振动性结果

摘要

本文的主要目的是获得三阶半线性中立型微分方程所有解振动的判据。本文的主要结果是通过将所研究的方程与两个一阶线性时滞微分方程进行比较，得到了一个振动定理。利用Riccati变换技术得到了另一个结果。举例说明了主要结果的重要性。

1.简介

本文研究了三阶中立型微分方程所有解的振动性

$${<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="(">(</trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src=")">)</trans>}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="y">年</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\ge ">\ge </trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>$$

(1)

哪里$\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>$根据以下假设：

(H（H）₁)

$\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="C">C类</trans>}}^{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans><trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans><trans\; data-src="\infty ">\infty </trans><trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathbb{<trans\; data-src="R">R（右）</trans>}<trans\; data-src=")">)</trans>$具有$\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>$和${<trans\; data-src="lim">林</trans>}_{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="\infty ">\infty </trans>}\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="\infty ">\infty </trans>$;

(H（H）₂)

$\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="C">C类</trans>}<trans\; data-src="(">(</trans><trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans><trans\; data-src="\infty ">\infty </trans><trans\; data-src=")">)</trans><trans\; data-src=",">,</trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\infty ">\infty </trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>$$\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="1">1</trans>}$、和（f）不会完全消失；

(H（H）_三)

$\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}$是奇数正整数的比率；

(H（H）₄)

克,$\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="C">C类</trans>}<trans\; data-src="(">(</trans><trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans><trans\; data-src="\infty ">\infty </trans><trans\; data-src=")">)</trans><trans\; data-src=",">,</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\infty ">\infty </trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>$并满足

$${<trans\; data-src="\int ">\int </trans>}_{{\mathrm{<trans\; data-src="t">t吨</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}}^{<trans\; data-src="\infty ">\infty </trans>}\frac{\mathrm{<trans\; data-src="1">1</trans>}}{{\mathrm{<trans\; data-src="g">克</trans>}}^{\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="=">=</trans>{<trans\; data-src="\int ">\int </trans>}_{{\mathrm{<trans\; data-src="t">t吨</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}}^{<trans\; data-src="\infty ">\infty </trans>}\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="=">=</trans><trans\; data-src="\infty ">\infty </trans><trans\; data-src=".">.</trans>$$

功能$\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$被称为方程的解(1)如果相应的函数$\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="C">C类</trans>}}^{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans><trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="T">T型</trans>}}_{\mathrm{<trans\; data-src="y">年</trans>}}<trans\; data-src=",">,</trans><trans\; data-src="\infty ">\infty </trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>$,${\mathrm{<trans\; data-src="T">T型</trans>}}_{\mathrm{<trans\; data-src="y">年</trans>}}<trans\; data-src="\ge ">\ge </trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$,$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src=")">)</trans>}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="C">C类</trans>}<trans\; data-src="(">(</trans><trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="T">T型</trans>}}_{\mathrm{<trans\; data-src="y">年</trans>}}<trans\; data-src=",">,</trans><trans\; data-src="\infty ">\infty </trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>$,$\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src=")">)</trans>}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="C">C类</trans>}<trans\; data-src="(">(</trans><trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="T">T型</trans>}}_{\mathrm{<trans\; data-src="y">年</trans>}}<trans\; data-src=",">,</trans><trans\; data-src="\infty ">\infty </trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>$、和$\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$满足方程式(1)上的$<trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="T">T型</trans>}}_{\mathrm{<trans\; data-src="y">年</trans>}}<trans\; data-src=",">,</trans><trans\; data-src="\infty ">\infty </trans><trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$我们假设方程式(1)拥有令人满意的解决方案$<trans\; data-src="sup">啜饮</trans><trans\; data-src="\{">\{</trans><trans\; data-src="|">|</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="|">|</trans><trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="\}">\}</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$为所有人$\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="\ge ">\ge </trans>{\mathrm{<trans\; data-src="T">T型</trans>}}_{\mathrm{<trans\; data-src="y">年</trans>}}$即，方程式(1)具有可持续的解决方案。这样的方程解(1)如果其上有无穷多个零，则称为振荡$<trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="T">T型</trans>}}_{\mathrm{<trans\; data-src="y">年</trans>}}<trans\; data-src=",">,</trans><trans\; data-src="\infty ">\infty </trans><trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>$和无振荡否则。我们说一个方程是振荡的，如果它的所有解都是振荡的。

近年来，人们对三阶泛函微分方程的振动性和渐近性进行了研究。此处的结果与[1,2,三,4,5,6,7,8,9,10,11]在所有这些论文中，得到的结果都是解要么振荡要么收敛到零的形式。如果尝试直接证明，这通常是“预期的”结果。在本文中，由于证明技术，我们能够获得所有解的振动性，即通过将所考虑的方程与振动行为已知的不等式进行比较。

在[12]，作者使用了形式的关系

$$\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\ge ">\ge </trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$$

(2)

哪里年是积极的，并且z（z）是积极的，并且在增加。此外，根据中给出的结果([13]，第28页），如果年和z（z）是积极的，并且z（z）然后他们假设年也没有增加。这导致以下关系年和z（z）:

$$\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\ge ">\ge </trans>\left(\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}\right)\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

(3)

然而，在一篇非常好的论文中[14]，作者给出了一个反例，表明如果z（z）正在减少，那么年不需要递减，因此关系式(三)不正确。受此观察的启发，本文首先得出了年和z（z）如果两者都是肯定的并且z（z）减少（参见下面的引理3）。然后利用这个关系，我们提出了方程的一些新的振动准则(1). 因此，本文建立的结果是新的，补充了文献中已经报道的结果。

2.主要成果

我们从以下结果开始，该结果给出了方程正非振荡解的基本性质(1). 一个类似的结果适用于最终的负解。

引理 1

假设$\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$是方程的正解(1)。然后是相应的功能$\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$对于所有足够大的t，满足以下两种情况之一：

(我)

$\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$,${\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="0">0</trans>}$,${<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$、和${<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="(">(</trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src=")">)</trans>}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="0">0</trans>}$;

(二)

$\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$,${\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$,${<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$,${<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="(">(</trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src=")">)</trans>}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="0">0</trans>}$.

证明。 

方程正非振动解的结构(1)在条件（H₄)根据Kiguradze和Chanturia的著名结果[15]. □

在下面的两个引理中，我们获得了函数之间有用的关系年和z（z）。这些将用来代替不正确的不等式(三)在本文的前一节中进行了描述。

引理 2

让$\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$是方程的正解(1)然后让$\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$满足引理1的情形（II）。然后

$$\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\ge ">\ge </trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>$$

(4)

对于所有足够大的t。

证明。 

从定义$\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$，我们有$\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$和

$$\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\ge ">\ge </trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>$$

自从z（z）正在增加。□

为了简化我们的符号，对于任何$\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="\ge ">\ge </trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$，我们设置

$$\begin{array}{ccc}\hfill {\mathrm{<trans\; data-src="G">G公司</trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>& <trans\; data-src="=">=</trans>& {<trans\; data-src="\int ">\int </trans>}_{\mathrm{<trans\; data-src="T">T型</trans>}}^{\mathrm{<trans\; data-src="t">t吨</trans>}}\frac{\mathrm{<trans\; data-src="1">1</trans>}}{{\mathrm{<trans\; data-src="g">克</trans>}}^{\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=",">,</trans>\hfill \\ \hfill {\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>& <trans\; data-src="=">=</trans>& {<trans\; data-src="\int ">\int </trans>}_{\mathrm{<trans\; data-src="T">T型</trans>}}^{\mathrm{<trans\; data-src="t">t吨</trans>}}\frac{{\mathrm{<trans\; data-src="G">G公司</trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=",">,</trans>\hfill \\ \hfill {\mathrm{<trans\; data-src="Q">问</trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>& <trans\; data-src="=">=</trans>& \frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}{<trans\; data-src="\int ">\int </trans>}_{\mathrm{<trans\; data-src="t">t吨</trans>}}^{<trans\; data-src="\infty ">\infty </trans>}{\left(\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}{<trans\; data-src="\int ">\int </trans>}_{\mathrm{<trans\; data-src="s">秒</trans>}}^{<trans\; data-src="\infty ">\infty </trans>}\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="u">u个</trans>}\right)}^{\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}}\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=",">,</trans>\hfill \\ \hfill {\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>& <trans\; data-src="=">=</trans>& <trans\; data-src="exp">经验</trans>\left({<trans\; data-src="\int ">\int </trans>}_{\mathrm{<trans\; data-src="T">T型</trans>}}^{\mathrm{<trans\; data-src="t">t吨</trans>}}{\mathrm{<trans\; data-src="Q">问</trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="s">秒</trans>}\right)\hfill \end{array}$$

为所有人$\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="T">T型</trans>}$.

引理 三。

让$\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$是方程的正解(1)具有$\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$满足引理1的情形（I）并假设$\mathrm{<trans\; data-src="\psi ">\psi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\left(\frac{{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}{{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right)<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$对于$\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="T">T型</trans>}$然后，

$$\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="s">秒</trans>}\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="n">n个</trans>}\mathrm{<trans\; data-src="c">c（c）</trans>}\mathrm{<trans\; data-src="r">第页</trans>}\mathrm{<trans\; data-src="e">e（电子）</trans>}\mathrm{<trans\; data-src="a">一</trans>}\mathrm{<trans\; data-src="s">秒</trans>}\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="n">n个</trans>}\mathrm{<trans\; data-src="g">克</trans>}$$

(5)

和

$$\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="\psi ">\psi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>$$

（6）

对于$\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="T">T型</trans>}$.

证明。 

假设$\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$是方程的正解(1)具有相应的功能$\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$满足引理1的全部情形（I）$\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="T">T型</trans>}$，对于一些$\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="\ge ">\ge </trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$然后，很容易验证${<trans\; data-src="lim">林</trans>}_{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="\infty ">\infty </trans>}\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$和${<trans\; data-src="lim">林</trans>}_{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="\infty ">\infty </trans>}\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="(">(</trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$.方程式的积分(1)然后产生收益

$$\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="(">(</trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src="=">=</trans>{<trans\; data-src="\int ">\int </trans>}_{\mathrm{<trans\; data-src="t">t吨</trans>}}^{<trans\; data-src="\infty ">\infty </trans>}\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="y">年</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="\le ">\le </trans>{<trans\; data-src="\int ">\int </trans>}_{\mathrm{<trans\; data-src="t">t吨</trans>}}^{<trans\; data-src="\infty ">\infty </trans>}\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="\le ">\le </trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="\int ">\int </trans>}_{\mathrm{<trans\; data-src="t">t吨</trans>}}^{<trans\; data-src="\infty ">\infty </trans>}\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=".">.</trans>$$

再次积分，我们得到

$$\begin{array}{c}\hfill \mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\ge ">\ge </trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="\int ">\int </trans>}_{\mathrm{<trans\; data-src="t">t吨</trans>}}^{<trans\; data-src="\infty ">\infty </trans>}{\left(\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}{<trans\; data-src="\int ">\int </trans>}_{\mathrm{<trans\; data-src="s">秒</trans>}}^{<trans\; data-src="\infty ">\infty </trans>}\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="u">u个</trans>}\right)}^{\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}}\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=",">,</trans>\end{array}$$

或

$${\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\ge ">\ge </trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="Q">问</trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

因此，

$${<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="Q">问</trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>$$

所以$\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$正在增加。

从定义z（z），我们有

$$\begin{array}{lll}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>& <trans\; data-src="\ge ">\ge </trans>& \mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\frac{\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}{{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\\ & <trans\; data-src="\ge ">\ge </trans>& \left(\frac{{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}{{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right)\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\end{array}$$

自从$\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$正在增加。这证明了引理。□

我们的最后一个引理提供了一些不等式，包括z（z）和${\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}$和功能${\mathrm{<trans\; data-src="G">G公司</trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}$和${\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}$上述定义。它们用于下面定理1和定理2的证明。

引理 4

假设$\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$是方程的正解(1)和$\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$完全满足引理1的情形（II）$\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="T">T型</trans>}$.然后

$${\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\ge ">\ge </trans>\frac{{\mathrm{<trans\; data-src="g">克</trans>}}^{\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{<trans\; data-src="\prime ">\prime </trans>}{\mathrm{<trans\; data-src="G">G公司</trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>$$

(7)

$$\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\ge ">\ge </trans>{\mathrm{<trans\; data-src="g">克</trans>}}^{\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{<trans\; data-src="\prime ">\prime </trans>}{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>$$

(8)

和

$$\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\ge ">\ge </trans>{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\frac{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}{{\mathrm{<trans\; data-src="G">G公司</trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}$$

(9)

为所有人$\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="T">T型</trans>}$.

证明。 

关系式(7)很容易遵循引理5[11]、和方程式(8)然后积分方程(7)来自T型到t吨和简化。来自方程式(7)，很容易看出

$${\left(\frac{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}{{\mathrm{<trans\; data-src="G">G公司</trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}\right)}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>$$

因此$\frac{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}{{\mathrm{<trans\; data-src="G">G公司</trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}$正在减少$\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="T">T型</trans>}$此外，

$$\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>{<trans\; data-src="\int ">\int </trans>}_{\mathrm{<trans\; data-src="T">T型</trans>}}^{\mathrm{<trans\; data-src="t">t吨</trans>}}\frac{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}{{\mathrm{<trans\; data-src="G">G公司</trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}\frac{{\mathrm{<trans\; data-src="G">G公司</trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="\ge ">\ge </trans>\frac{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}{{\mathrm{<trans\; data-src="G">G公司</trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

因此，

$$\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\ge ">\ge </trans>{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\frac{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}{{\mathrm{<trans\; data-src="G">G公司</trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}<trans\; data-src="\ge ">\ge </trans>{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\frac{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}{{\mathrm{<trans\; data-src="G">G公司</trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}$$

自从$\frac{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}{{\mathrm{<trans\; data-src="G">G公司</trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}$正在减少。这就完成了引理的证明。□

我们现在准备陈述并证明我们的主要成果。

定理 1

让${\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$并假设有一个函数$\mathrm{<trans\; data-src="\xi ">\xi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="C">C类</trans>}}^{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans><trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans><trans\; data-src="\infty ">\infty </trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>$这样的话

$${\mathrm{<trans\; data-src="\xi ">\xi </trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\xi ">\xi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="t">t吨</trans>}\mathrm{<trans\; data-src="a">一</trans>}\mathrm{<trans\; data-src="n">n个</trans>}\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\xi ">\xi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\xi ">\xi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=".">.</trans>$$

(10)

如果两个一阶时滞微分方程

$${\mathrm{<trans\; data-src="w">周</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\left[\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}{<trans\; data-src="\int ">\int </trans>}_{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="\xi ">\xi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}{\left(\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>}{<trans\; data-src="\int ">\int </trans>}_{{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}}^{\mathrm{<trans\; data-src="\xi ">\xi </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>}\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\right)}^{\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}}\mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}\right]\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$$

(11)

和

$${\mathrm{<trans\; data-src="w">周</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="w">周</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$$

（12）

是振荡的，然后是方程(1)是振荡的。

证明。 

让$\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$是方程的正解(1). 然后有一个$\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="\ge ">\ge </trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$这样的话$\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$和$\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$为所有人$\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=".">.</trans>$从定义$\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$，我们有$\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$为所有人$\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="T">T型</trans>}$，其中T型也被选中，因此引理1-4适用于所有$\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="T">T型</trans>}$.

案例（I）。使用方程式(6)在方程式中(1)，我们获得

$${<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="(">(</trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src=")">)</trans>}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>$$

(13)

并从中集成t吨到$\mathrm{<trans\; data-src="\xi ">\xi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$产量

$$\begin{array}{lll}\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="(">(</trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}& <trans\; data-src="\ge ">\ge </trans>& {<trans\; data-src="\int ">\int </trans>}_{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="\xi ">\xi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\\ & <trans\; data-src="\ge ">\ge </trans>& {\mathrm{<trans\; data-src="z">z（z）</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\xi ">\xi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>{<trans\; data-src="\int ">\int </trans>}_{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="\xi ">\xi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=".">.</trans>\end{array}$$

因此，

$$\begin{array}{ccc}\hfill {<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{<trans\; data-src="\prime ">\prime </trans>}& <trans\; data-src="\ge ">\ge </trans>& \mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\xi ">\xi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>{\left(\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}{<trans\; data-src="\int ">\int </trans>}_{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="\xi ">\xi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\right)}^{\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}}<trans\; data-src=".">.</trans>\hfill \end{array}$$

再次从集成t吨到$\mathrm{<trans\; data-src="\xi ">\xi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$，我们获得

$$\begin{array}{ccc}\hfill <trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>& <trans\; data-src="\ge ">\ge </trans>& {<trans\; data-src="\int ">\int </trans>}_{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="\xi ">\xi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\xi ">\xi </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>{\left(\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>}{<trans\; data-src="\int ">\int </trans>}_{{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}}^{\mathrm{<trans\; data-src="\xi ">\xi </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>}\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\right)}^{\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}}\mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=",">,</trans>\hfill \end{array}$$

或

$$\begin{array}{ccc}\hfill <trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>& <trans\; data-src="\ge ">\ge </trans>& \mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\left(\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}{<trans\; data-src="\int ">\int </trans>}_{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="\xi ">\xi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}{\left(\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>}{<trans\; data-src="\int ">\int </trans>}_{{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}}^{\mathrm{<trans\; data-src="\xi ">\xi </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>}\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\right)}^{\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}}\mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}\right)<trans\; data-src=".">.</trans>\hfill \end{array}$$

最后，从t吨到∞给予

$$\begin{array}{ccc}\hfill \mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>& <trans\; data-src="\ge ">\ge </trans>& {<trans\; data-src="\int ">\int </trans>}_{\mathrm{<trans\; data-src="t">t吨</trans>}}^{<trans\; data-src="\infty ">\infty </trans>}\frac{\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src=")">)</trans>}{<trans\; data-src="\int ">\int </trans>}_{{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}}^{\mathrm{<trans\; data-src="\xi ">\xi </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src=")">)</trans>}{\left(\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>}{<trans\; data-src="\int ">\int </trans>}_{{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}}^{\mathrm{<trans\; data-src="\xi ">\xi </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>}\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\right)}^{\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}}\mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}\mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src=".">.</trans>\hfill \end{array}$$

（14）

让我们表示方程的右侧(14)由$\mathrm{<trans\; data-src="w">周</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$.然后$\mathrm{<trans\; data-src="w">周</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$正在减少，$\mathrm{<trans\; data-src="w">周</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$很容易看出$\mathrm{<trans\; data-src="w">周</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$是微分不等式的正解

$$\begin{array}{c}\hfill {\mathrm{<trans\; data-src="w">周</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\left[\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}{<trans\; data-src="\int ">\int </trans>}_{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="\xi ">\xi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}{\left(\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>}{<trans\; data-src="\int ">\int </trans>}_{{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}}^{\mathrm{<trans\; data-src="\xi ">\xi </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>}\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\right)}^{\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}}\mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}\right]\mathrm{<trans\; data-src="w">周</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=".">.</trans>\end{array}$$

中的定理1[16]显示了相应的微分方程(11)也有积极的解决办法，这是一个矛盾。

案例（II）。使用方程式(4)在方程式中(1)，我们有

$${<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="(">(</trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src=")">)</trans>}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=".">.</trans>$$

(15)

来自方程式(8),

$${\mathrm{<trans\; data-src="z">z（z）</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>{<trans\; data-src="[">[</trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="]">]</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=",">,</trans>$$

(16)

所以组合方程式(16)和(15)收益率

$${<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="(">(</trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src=")">)</trans>}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>{<trans\; data-src="[">[</trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="]">]</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=".">.</trans>$$

让$\mathrm{<trans\; data-src="w">周</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="(">(</trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$; 然后我们看到了$\mathrm{<trans\; data-src="w">周</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$是不等式的正解

$${\mathrm{<trans\; data-src="w">周</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="w">周</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=".">.</trans>$$

根据中的定理1[16]，这意味着相应的微分方程(12)也有积极的解决办法，这是一个矛盾。这就完成了定理的证明。□

作为如何使用上述定理获得显式振动准则的示例，我们有以下推论。

推论 1

让定理1的条件成立。如果

$$\underset{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="\infty ">\infty </trans>}{<trans\; data-src="lim\; inf">lim信息</trans>}{<trans\; data-src="\int ">\int </trans>}_{\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="t">t吨</trans>}}\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src=")">)</trans>}{<trans\; data-src="\int ">\int </trans>}_{{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}}^{\mathrm{<trans\; data-src="\xi ">\xi </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src=")">)</trans>}{\left(\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>}{<trans\; data-src="\int ">\int </trans>}_{{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}}^{\mathrm{<trans\; data-src="\xi ">\xi </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>}\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\right)}^{\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}}\mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}\mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src=">">></trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}$$

(17)

和

$$\underset{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="\infty ">\infty </trans>}{<trans\; data-src="lim\; inf">lim信息</trans>}{<trans\; data-src="\int ">\int </trans>}_{\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="t">t吨</trans>}}\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=">">></trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}$$

(18)

满足，则方程式(1)是振荡的。

证明。 

由([17]，定理1），条件方程(17)和(18)暗示方程式(11)和(12)是振荡的。□

我们用以下定理结束这一节。该证明使用了Riccati变换技术。

定理 2

让${\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$然后让$\mathrm{<trans\; data-src="\xi ">\xi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="C">C类</trans>}}^{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans><trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans><trans\; data-src="\infty ">\infty </trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>$满足方程式(10)和(17).如果存在实值不可减微分函数$\mathrm{<trans\; data-src="\rho ">\rho </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$这样的话

$$\underset{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="\infty ">\infty </trans>}{<trans\; data-src="lim\; sup">酸橙酱</trans>}{<trans\; data-src="\int ">\int </trans>}_{{\mathrm{<trans\; data-src="t">t吨</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}}^{\mathrm{<trans\; data-src="t">t吨</trans>}}\left[\mathrm{<trans\; data-src="\rho ">\rho </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}\frac{{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}{{\mathrm{<trans\; data-src="G">G公司</trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="-">-</trans>\frac{\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\rho ">\rho </trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}}{{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\rho ">\rho </trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}\right]\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="=">=</trans><trans\; data-src="\infty ">\infty </trans><trans\; data-src=",">,</trans>$$

（19）

然后是方程式(1)是振荡的。

证明。 

让$\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$是方程的正解(1). 按照定理1的证明进行，我们看到$\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$满足引理1中的一种情况。情况（I）可以通过使用条件方程来消除(17)如定理1的证明。现在考虑案例（II）。定义

$$\mathrm{<trans\; data-src="F">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="\rho ">\rho </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\frac{\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="(">(</trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}}{{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}}\mathrm{<trans\; data-src="f">（f）</trans>}\mathrm{<trans\; data-src="o">o个</trans>}\mathrm{<trans\; data-src="r">第页</trans>}\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=";">;</trans>$$

(20)

然后$\mathrm{<trans\; data-src="F">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$为所有人$\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=".">.</trans>$微分方程(20)使用方程式(15)，并简化，我们得到

$${\mathrm{<trans\; data-src="F">如果</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\frac{{\mathrm{<trans\; data-src="\rho ">\rho </trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="\rho ">\rho </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}\mathrm{<trans\; data-src="F">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\rho ">\rho </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}\frac{{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}{{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}}<trans\; data-src="-">-</trans>\frac{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}{\mathrm{<trans\; data-src="F">如果</trans>}}^{\frac{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}{{\mathrm{<trans\; data-src="\rho ">\rho </trans>}}^{\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="g">克</trans>}}^{\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=".">.</trans>$$

(21)

应用不等式$\mathrm{<trans\; data-src="A">A类</trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="B">B类</trans>}{\mathrm{<trans\; data-src="u">u个</trans>}}^{\frac{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}}<trans\; data-src="\le ">\le </trans>\frac{{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}}{{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}}\frac{{\mathrm{<trans\; data-src="A">A类</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}}{{\mathrm{<trans\; data-src="B">B类</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}}$具有$\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="=">=</trans>\frac{{\mathrm{<trans\; data-src="\rho ">\rho </trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="\rho ">\rho </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}$,$\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src="=">=</trans>\frac{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}{{\mathrm{<trans\; data-src="\rho ">\rho </trans>}}^{\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="g">克</trans>}}^{\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}$、和$\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="F">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$在方程式中(21)，我们获得

$${\mathrm{<trans\; data-src="F">如果</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\rho ">\rho </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}\frac{{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}{{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}}<trans\; data-src="+">+</trans>\frac{\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\rho ">\rho </trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}}{{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\rho ">\rho </trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=".">.</trans>$$

(22)

最后，使用方程式(9)在方程式中(22)然后将所得不等式与T型到t吨，给出

$${<trans\; data-src="\int ">\int </trans>}_{\mathrm{<trans\; data-src="T">T型</trans>}}^{\mathrm{<trans\; data-src="t">t吨</trans>}}\left[\mathrm{<trans\; data-src="\rho ">\rho </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}\frac{{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}{{\mathrm{<trans\; data-src="G">G公司</trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="-">-</trans>\frac{\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\rho ">\rho </trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}}{{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\rho ">\rho </trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}\right]\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="F">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=")">)</trans><trans\; data-src="<"><</trans><trans\; data-src="\infty ">\infty </trans><trans\; data-src=".">.</trans>$$

这与方程式相矛盾(19)并完成了定理的证明。□

3.示例

在本节中，我们提供了一些示例来说明主要结果的重要性。

例子 1

考虑三阶中立型时滞微分方程

$${<trans\; data-src="(">(</trans>\sqrt{\mathrm{<trans\; data-src="t">t吨</trans>}}{<trans\; data-src="(">(</trans>\sqrt{\mathrm{<trans\; data-src="t">t吨</trans>}}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="p">第页</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src=")">)</trans>}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src=")">)</trans>}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="+">+</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=".">.</trans>$$

(23)

这里有$\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\sqrt{\mathrm{<trans\; data-src="t">t吨</trans>}}$,$\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}}$,$\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}$,$\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}\mathrm{<trans\; data-src="t">t吨</trans>}$具有$\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="1">1</trans>}$，我们采取$\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src="<"><</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}$简单计算表明${\mathrm{<trans\; data-src="G">G公司</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="(">(</trans>\sqrt{\mathrm{<trans\; data-src="t">t吨</trans>}}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>$,${\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="2">2</trans>}{<trans\; data-src="(">(</trans>\sqrt{\mathrm{<trans\; data-src="t">t吨</trans>}}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="2">2</trans>}}$,${\mathrm{<trans\; data-src="Q">问</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="t">t吨</trans>}$,${\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}$、和$\mathrm{<trans\; data-src="\psi ">\psi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$。选择$\mathrm{<trans\; data-src="\xi ">\xi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="\beta ">\beta </trans>}\mathrm{<trans\; data-src="t">t吨</trans>}$具有$\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="1">1</trans>}$和$\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}{\mathrm{<trans\; data-src="\beta ">\beta </trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="1">1</trans>}$; 然后$\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}{\mathrm{<trans\; data-src="\beta ">\beta </trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="t">t吨</trans>}$和条件方程(17)成为

$$\begin{array}{c}\hfill \underset{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="\infty ">\infty </trans>}{<trans\; data-src="lim\; inf">lim信息</trans>}{<trans\; data-src="\int ">\int </trans>}_{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}{\mathrm{<trans\; data-src="\beta ">\beta </trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="t">t吨</trans>}}\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\sqrt{{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}}}{<trans\; data-src="\int ">\int </trans>}_{{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}}^{\mathrm{<trans\; data-src="\beta ">\beta </trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}}\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\sqrt{{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}}}{<trans\; data-src="\int ">\int </trans>}_{{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}}^{\mathrm{<trans\; data-src="\beta ">\beta </trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}}\frac{\mathrm{<trans\; data-src="1">1</trans>}}{{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}\mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}\\ \hfill <trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src=")">)</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\sqrt{\mathrm{<trans\; data-src="\beta ">\beta </trans>}}}<trans\; data-src=")">)</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\beta ">\beta </trans>}}<trans\; data-src=")">)</trans><trans\; data-src="ln">在</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}{\mathrm{<trans\; data-src="\beta ">\beta </trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}}<trans\; data-src=".">.</trans>\end{array}$$

条件方程式(18)成为

$$\underset{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="\infty ">\infty </trans>}{<trans\; data-src="lim\; inf">lim信息</trans>}{<trans\; data-src="\int ">\int </trans>}_{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="t">t吨</trans>}}\frac{\mathrm{<trans\; data-src="2">2</trans>}}{{\mathrm{<trans\; data-src="s">秒</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="(">(</trans>\sqrt{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}\mathrm{<trans\; data-src="s">秒</trans>}}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="2">2</trans>}}\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="ln">在</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}<trans\; data-src=".">.</trans>$$

因此，根据推论1，方程式(23)是振荡的，如果$\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="ln">在</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}<trans\; data-src=">">></trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}$和$\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src=")">)</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\sqrt{\mathrm{<trans\; data-src="\beta ">\beta </trans>}}}<trans\; data-src=")">)</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\beta ">\beta </trans>}}<trans\; data-src=")">)</trans><trans\; data-src="ln">在</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}{\mathrm{<trans\; data-src="\beta ">\beta </trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}}<trans\; data-src=">">></trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}$.

例子 2

考虑以下等式

$${<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\frac{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="9">9</trans>}}}{<trans\; data-src="(">(</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="3">三</trans>}}}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="p">第页</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src=")">)</trans>}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src=")">)</trans>}^{\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="3">三</trans>}}}<trans\; data-src=")">)</trans>}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="+">+</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\frac{\mathrm{<trans\; data-src="4">4</trans>}}{\mathrm{<trans\; data-src="3">三</trans>}}}}{\mathrm{<trans\; data-src="y">年</trans>}}^{\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="3">三</trans>}}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=".">.</trans>$$

（24）

我们有$\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\frac{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="9">9</trans>}}}<trans\; data-src=",">,</trans>$$\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="3">三</trans>}}}<trans\; data-src=",">,</trans>$$\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\frac{\mathrm{<trans\; data-src="4">4</trans>}}{\mathrm{<trans\; data-src="3">三</trans>}}}}<trans\; data-src=",">,</trans>$$\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}\mathrm{<trans\; data-src="t">t吨</trans>}$,$\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>$$\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="=">=</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="3">三</trans>}}$，我们采取$\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src="<"><</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{\frac{\mathrm{<trans\; data-src="81">81</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}}$计算表明${\mathrm{<trans\; data-src="G">G公司</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="3">三</trans>}}}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>$,${\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\frac{\mathrm{<trans\; data-src="3">三</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="2">2</trans>}\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="3">三</trans>}{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\frac{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="3">三</trans>}}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>$,${\mathrm{<trans\; data-src="Q">问</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\frac{\mathrm{<trans\; data-src="81">81</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}\mathrm{<trans\; data-src="t">t吨</trans>}}$,${\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\frac{\mathrm{<trans\; data-src="81">81</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}}<trans\; data-src=",">,</trans>$$\mathrm{<trans\; data-src="\psi ">\psi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{\frac{\mathrm{<trans\; data-src="81">81</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$.条件方程(17)和(18)如果

$$\frac{\mathrm{<trans\; data-src="81">81</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{\frac{\mathrm{<trans\; data-src="81">81</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src=")">)</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{{\mathrm{<trans\; data-src="\beta ">\beta </trans>}}^{\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="3">三</trans>}}}}<trans\; data-src=")">)</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{{\mathrm{<trans\; data-src="\beta ">\beta </trans>}}^{\frac{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="3">三</trans>}}}}<trans\; data-src=")">)</trans><trans\; data-src="ln">在</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}{\mathrm{<trans\; data-src="\beta ">\beta </trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}}<trans\; data-src=">">></trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}<trans\; data-src=",">,</trans>$$

和

$${\mathrm{<trans\; data-src="3">三</trans>}}^{\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="3">三</trans>}}}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="3">三</trans>}}}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src=")">)</trans>}^{\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="3">三</trans>}}}<trans\; data-src="ln">在</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}<trans\; data-src=">">></trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}<trans\; data-src=".">.</trans>$$

然后根据推论1，方程式(24)是振荡的。

例子 三。

考虑三阶方程

$$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="p">第页</trans>}\mathrm{<trans\; data-src="y">年</trans>}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}^{<trans\; data-src="‴">‴</trans>}<trans\; data-src="+">+</trans>\frac{\mathrm{<trans\; data-src="2">2</trans>}\mathrm{<trans\; data-src="\gamma ">\gamma </trans>}}{{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="3">三</trans>}}}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=".">.</trans>$$

(25)

在这种情况下，我们有$\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}$,$\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}\mathrm{<trans\; data-src="t">t吨</trans>}$具有$\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="1">1</trans>}$,$\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}$、和$\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\frac{\mathrm{<trans\; data-src="2">2</trans>}\mathrm{<trans\; data-src="\gamma ">\gamma </trans>}}{{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="3">三</trans>}}}$哪里$\mathrm{<trans\; data-src="\gamma ">\gamma </trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$是一个常量。我们采取$\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}$不难看出${\mathrm{<trans\; data-src="G">G公司</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>$,${\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\frac{{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="2">2</trans>}}}{\mathrm{<trans\; data-src="2">2</trans>}}$,${\mathrm{<trans\; data-src="Q">问</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\frac{\mathrm{<trans\; data-src="\gamma ">\gamma </trans>}}{\mathrm{<trans\; data-src="t">t吨</trans>}}$,${\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="\gamma ">\gamma </trans>}}$、和$\mathrm{<trans\; data-src="\psi ">\psi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$.

拿$\mathrm{<trans\; data-src="\xi ">\xi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="\beta ">\beta </trans>}\mathrm{<trans\; data-src="t">t吨</trans>}$具有$\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="1">1</trans>}$和$\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}{\mathrm{<trans\; data-src="\beta ">\beta </trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="1">1</trans>}$，我们有$\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}{\mathrm{<trans\; data-src="\beta ">\beta </trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}\mathrm{<trans\; data-src="t">t吨</trans>}$和条件方程(17)采取形式

$$\begin{array}{c}\hfill \underset{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="\infty ">\infty </trans>}{<trans\; data-src="lim\; inf">lim信息</trans>}{<trans\; data-src="\int ">\int </trans>}_{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}{\mathrm{<trans\; data-src="\beta ">\beta </trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="t">t吨</trans>}}{<trans\; data-src="\int ">\int </trans>}_{{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}}^{\mathrm{<trans\; data-src="\beta ">\beta </trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}}{<trans\; data-src="\int ">\int </trans>}_{{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}}^{\mathrm{<trans\; data-src="\beta ">\beta </trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}}\frac{\mathrm{<trans\; data-src="2">2</trans>}\mathrm{<trans\; data-src="\gamma ">\gamma </trans>}}{{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}^{\mathrm{<trans\; data-src="3">三</trans>}}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}\mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="\gamma ">\gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src=")">)</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{{\mathrm{<trans\; data-src="\beta ">\beta </trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}}<trans\; data-src=")">)</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\beta ">\beta </trans>}}<trans\; data-src=")">)</trans><trans\; data-src="ln">在</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}{\mathrm{<trans\; data-src="\beta ">\beta </trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}}<trans\; data-src=".">.</trans>\end{array}$$

通过选择$\mathrm{<trans\; data-src="\rho ">\rho </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="t">t吨</trans>}$，我们看到了条件方程式(19)成为

$$\begin{array}{l}\underset{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="\infty ">\infty </trans>}{<trans\; data-src="lim\; sup">酸橙酱</trans>}{<trans\; data-src="\int ">\int </trans>}_{\mathrm{<trans\; data-src="1">1</trans>}}^{\mathrm{<trans\; data-src="t">t吨</trans>}}\left[\frac{\mathrm{<trans\; data-src="2">2</trans>}\mathrm{<trans\; data-src="\gamma ">\gamma </trans>}}{{\mathrm{<trans\; data-src="s">秒</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src=")">)</trans>\frac{{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="2">2</trans>}}}{\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="-">-</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="4">4</trans>}\mathrm{<trans\; data-src="s">秒</trans>}}\right]\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="s">秒</trans>}\\ <trans\; data-src=">">></trans>\underset{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="\infty ">\infty </trans>}{<trans\; data-src="lim\; sup">酸橙酱</trans>}{<trans\; data-src="\int ">\int </trans>}_{\mathrm{<trans\; data-src="1">1</trans>}}^{\mathrm{<trans\; data-src="t">t吨</trans>}}\left\{\left[\mathrm{<trans\; data-src="\gamma ">\gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">-</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="4">4</trans>}}\right]\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="s">秒</trans>}}<trans\; data-src="-">-</trans>\frac{\mathrm{<trans\; data-src="2">2</trans>}\mathrm{<trans\; data-src="\gamma ">\gamma </trans>}\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="+">+</trans>\frac{\mathrm{<trans\; data-src="\gamma ">\gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src=")">)</trans>}{{\mathrm{<trans\; data-src="s">秒</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}\right\}<trans\; data-src="\to ">\to </trans><trans\; data-src="\infty ">\infty </trans>\end{array}$$

作为$\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="\infty ">\infty </trans>$假如$\mathrm{<trans\; data-src="\gamma ">\gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="4">4</trans>}$因此，根据定理2，方程式(25)是振荡的，如果

$$\frac{\mathrm{<trans\; data-src="\gamma ">\gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}{{\mathrm{<trans\; data-src="\beta ">\beta </trans>}}^{\mathrm{<trans\; data-src="3">三</trans>}}}<trans\; data-src="ln">在</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}{\mathrm{<trans\; data-src="\beta ">\beta </trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}}<trans\; data-src=">">></trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}\mathrm{<trans\; data-src="a">一</trans>}\mathrm{<trans\; data-src="n">n个</trans>}\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="4">4</trans>}\mathrm{<trans\; data-src="\gamma ">\gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=".">.</trans>$$

4.结论

在本文中，我们获得了方程的一些新的振动准则(1)在以下情况下$\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="1">1</trans>}$和$\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\infty ">\infty </trans><trans\; data-src=")">)</trans>$这些结果确保了所研究方程的所有解都是振荡的。我们的结果是新的，因为论文中的结果[1,2,三,4,5,6,7,8,9,10,11]不会确保方程的所有解(1)是振荡的。

将这里的结果扩展到以下情况也很有趣$<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="0">0</trans>}$或$\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$是振荡的。将这里的结果推广到高阶方程，例如

$${<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="(">(</trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src=")">)</trans>}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="y">年</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\ge ">\ge </trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>$$

哪里$\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>$和$\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="3">三</trans>}$是一个奇数也很有趣。