偏序Banach空间上非线性分数阶微分方程的一些存在性结果

摘要

利用锥上的不动点结果，通过给定的边值问题研究了一类非线性分数阶微分方程正解的存在唯一性。为了说明所获得的结果，给出了示例。

1引言

分数阶微分方程领域的理论和应用得到了深入的发展（例如，参见[1–6]以及其中的参考）。值得注意的是，大多数关于分数阶微积分的论文和书籍都致力于研究线性初值分数阶微分方程的特殊函数可解性。有一些论文利用非线性分析技术，如不动点结果、Leray-Shauder定理、稳定性、，等。（例如，请参见[7–19]以及其中的参考）。事实上，分数微分方程出现在许多工程和科学学科中，如物理学、化学、生物学、经济学、控制理论、信号和图像处理、生物物理学、血流现象和空气动力学（例如[20–23]以及其中的参考）。使用分数阶非线性微分方程的主要优点是，我们可以用记忆来描述复杂非局部系统的动力学。在这门课程中，从理论和应用的角度来看，涉及各种分数阶的方程都很重要。我们需要以下概念。

定义1.1([1,4])

对于连续函数$\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src=":">:</trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src=")">)</trans><trans\; data-src="\to ">\to </trans>\mathbb{<trans\; data-src="R">R（右）</trans>}$，分数阶卡普托导数α由定义

$${}^{\mathrm{<trans\; data-src="c">c（c）</trans>}}\mathrm{<trans\; data-src="D">D类</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src=")">)</trans>}{<trans\; data-src="\int ">¦{\rm B}</trans>}_{\mathrm{<trans\; data-src="0">0</trans>}}^{\mathrm{<trans\; data-src="t">t吨</trans>}}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="f">（f）</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=",">,</trans>$$

哪里$\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="n">n个</trans>}$,$\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="=">=</trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="]">]</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}$和$<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="]">]</trans>$表示的整数部分α.

定义1.2([1,4])

Riemann-Liouville分数阶导数α对于连续函数（f）由定义

$${\mathrm{<trans\; data-src="D">D类</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src=")">)</trans>}{<trans\; data-src="(">(</trans>\frac{\mathrm{<trans\; data-src="d">d日</trans>}}{\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="t">t吨</trans>}}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="n">n个</trans>}}{<trans\; data-src="\int ">¦{\rm B}</trans>}_{\mathrm{<trans\; data-src="0">0</trans>}}^{\mathrm{<trans\; data-src="t">t吨</trans>}}\frac{\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}{{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}}\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="=">=</trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="]">]</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>$$

其中右侧是在上逐点定义的$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src=")">)</trans>$.

定义1.3([1,4])

让$<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="b">b条</trans>}<trans\; data-src="]">]</trans>$是一个间隔ℝ和$\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$函数的Riemann-Liouville分数阶积分$\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="L">L（左）</trans>}}^{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="b">b条</trans>}<trans\; data-src="]">]</trans><trans\; data-src=",">,</trans>\mathbb{<trans\; data-src="R">R（右）</trans>}<trans\; data-src=")">)</trans>$由定义

$${\mathrm{<trans\; data-src="I">我</trans>}}_{\mathrm{<trans\; data-src="a">一</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\gamma ">\gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src=")">)</trans>}{<trans\; data-src="\int ">¦{\rm B}</trans>}_{\mathrm{<trans\; data-src="a">一</trans>}}^{\mathrm{<trans\; data-src="t">t吨</trans>}}\frac{\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}{{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}}\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="s">秒</trans>}$$

只要积分存在。

假设E类是由锥部分排序的Banach空间$\mathrm{<trans\; data-src="P">P（P）</trans>}<trans\; data-src="\subseteq ">\subseteq </trans>\mathrm{<trans\; data-src="E">E类</trans>}$也就是说，$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="y">年</trans>}$当且仅当$\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="P">P（P）</trans>}$。我们表示的零元素E类通过θ.一个圆锥体P（P）如果存在常量，则称为正常$\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$这样的话$\mathrm{<trans\; data-src="\theta ">\theta </trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="y">年</trans>}$暗示$<trans\; data-src="\parallel ">\parallel </trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\parallel ">\parallel </trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src="\parallel ">\parallel </trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\parallel ">\parallel </trans>$（请参见[24]). 此外，我们还定义了订购间隔$<trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="]">]</trans><trans\; data-src="=">=</trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="E">E类</trans>}<trans\; data-src="|">|</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\le ">\le </trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="\}">\}</trans>$为所有人${\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="E">E类</trans>}$[24]. 我们说操作员$\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="E">E类</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="E">E类</trans>}$每次都在增加$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="y">年</trans>}$暗示$\mathrm{<trans\; data-src="A">A类</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="A">A类</trans>}\mathrm{<trans\; data-src="y">年</trans>}$。此外，$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\sim ">\sim </trans>\mathrm{<trans\; data-src="y">年</trans>}$意味着存在$\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$和$\mathrm{<trans\; data-src="\mu ">\mu </trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$这样的话$\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="\mu ">\mu </trans>}\mathrm{<trans\; data-src="x">x个</trans>}$（请参见[24]). 最后，把${\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}<trans\; data-src="=">=</trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="E">E类</trans>}<trans\; data-src="|">|</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\sim ">\sim </trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="\}">\}</trans>$为所有人$\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="\theta ">\theta </trans>}$很容易看出${\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}<trans\; data-src="\subseteq ">\subseteq </trans>\mathrm{<trans\; data-src="P">P（P）</trans>}$是凸的，并且$\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}{\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}$为所有人$\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$。我们在结果中回顾了以下内容。让E类成为一个真正的巴拿赫空间，让P（P）成为一个圆锥体E类.让$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="b">b条</trans>}<trans\; data-src=")">)</trans>$作为间隔，让τ和φ是两个正值函数，这样$\mathrm{<trans\; data-src="\phi ">\phi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="\tau ">\tau </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$为所有人$\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="b">b条</trans>}<trans\; data-src=")">)</trans>$和$\mathrm{<trans\; data-src="\tau ">\tau </trans>}<trans\; data-src=":">:</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="b">b条</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\to ">\to </trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>$是一个回注。我们说操作员$\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="P">P（P）</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="P">P（P）</trans>}$是τ-φ-在任何时候$\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\tau ">\tau </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="\phi ">\phi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="A">A类</trans>}\mathrm{<trans\; data-src="x">x个</trans>}$为所有人$\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="b">b条</trans>}<trans\; data-src=")">)</trans>$和$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="P">P（P）</trans>}$[13]. 我们这么说A类是φ-在任何时候$\mathrm{<trans\; data-src="\tau ">\tau </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="t">t吨</trans>}$为所有人t吨[13]. 我们回顾以下结果。

定理1.1([13])

让 E类 成为巴拿赫空间,让 P（P） 成为一个正常的圆锥体 E类,然后让 $\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="P">P（P）</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="P">P（P）</trans>}$ 增加，并且 τ-φ-凹算子.假设存在 $\mathrm{<trans\; data-src="\theta ">\theta </trans>}<trans\; data-src="\ne ">\ne </trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="P">P（P）</trans>}$ 这样的话 $\mathrm{<trans\; data-src="A">A类</trans>}\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}$.那么就有了 ${\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}$ 和 $\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>$ 这样的话 $\mathrm{<trans\; data-src="r">第页</trans>}{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\le ">\le </trans>{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\le ">\le </trans>{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$ 和 ${\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="A">A类</trans>}{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="A">A类</trans>}{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\le ">\le </trans>{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$,算符A有唯一的不动点 ${\mathrm{<trans\; data-src="x">x个</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="]">]</trans>$,和用于 ${\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}$ 和顺序 $<trans\; data-src="\{">\{</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="\}">\}</trans>$ 具有 ${\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="A">A类</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}$,我们有 $<trans\; data-src="\parallel ">\parallel </trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="-">负极</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="\parallel ">\parallel </trans><trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="0">0</trans>}$.

2主要成果

我们研究了分数阶微分方程解的存在唯一性

$${\mathrm{<trans\; data-src="D">D类</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$$

具有两类边界条件和两类分数导数Riemann-Liouville和Caputo的偏序Banach空间。

2.1具有Riemann-Liouville分数阶导数的分数阶微分方程的存在性结果

首先，我们研究了分数阶微分方程正解的存在唯一性

(2.1)

(2.2)

哪里${\mathrm{<trans\; data-src="D">D类</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}$是Riemann-Liouville分数阶导数α.让$\mathrm{<trans\; data-src="E">E类</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="C">C</trans>}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="]">]</trans>$.考虑上连续函数的Banach空间$<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="]">]</trans>$用超范数和集$\mathrm{<trans\; data-src="P">P（P）</trans>}<trans\; data-src="=">=</trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="C">C</trans>}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="]">]</trans><trans\; data-src=":">:</trans>{<trans\; data-src="min">最小值</trans>}_{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="]">]</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="\}">\}</trans>$。那么P（P）是一个正常的圆锥体。

引理2.1 让 $\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="T">T型</trans>}$,$\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="1">1</trans>}$,$\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="]">]</trans>$,$\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$ 和 $\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="1">1</trans>}$.然后是问题 ${\mathrm{<trans\; data-src="D">D类</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$ 具有边值条件 $\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=")">)</trans>$ 有解决方案 ${\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$ 当且仅当 ${\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$ 是分数阶积分方程的解

$$\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{<trans\; data-src="\int ">¦{\rm B}</trans>}_{\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}}^{\mathrm{<trans\; data-src="T">T型</trans>}}\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=",">,</trans>$$

哪里

$$\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="\{">\{</trans>\begin{array}{ll}\frac{{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">负极</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}}{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\eta ">\eta </trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">负极</trans>{\mathrm{<trans\; data-src="T">T型</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="-">负极</trans>\frac{{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}}{\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>& \mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=",">,</trans>\\ \frac{<trans\; data-src="-">负极</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}}{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\eta ">\eta </trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">负极</trans>{\mathrm{<trans\; data-src="T">T型</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="-">负极</trans>\frac{{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}}{\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>& \mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=",">,</trans>\\ \frac{<trans\; data-src="-">负极</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}}{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\eta ">\eta </trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">负极</trans>{\mathrm{<trans\; data-src="T">T型</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>& \mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=".">.</trans>\end{array}$$

证明发件人${\mathrm{<trans\; data-src="D">D类</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$以及边界条件，很容易看出$\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">负极</trans>{\mathrm{<trans\; data-src="c">c（c）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="=">=</trans><trans\; data-src="-">负极</trans>{\mathrm{<trans\; data-src="I">我</trans>}}_{\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>$.通过分数积分的定义，我们得到

$$\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="c">c（c）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">负极</trans>{<trans\; data-src="\int ">¦{\rm B}</trans>}_{\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}}^{\mathrm{<trans\; data-src="t">t吨</trans>}}\frac{{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}}{\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src=")">)</trans>}\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=".">.</trans>$$

因此，$\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="c">c（c）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\eta ">\eta </trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">负极</trans>{<trans\; data-src="\int ">¦{\rm B}</trans>}_{\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}}^{\mathrm{<trans\; data-src="\eta ">\eta </trans>}}\frac{{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}}{\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src=")">)</trans>}\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="s">秒</trans>}$和

$$\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="c">c（c）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="T">T型</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">负极</trans>{<trans\; data-src="\int ">¦{\rm B}</trans>}_{\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}}^{\mathrm{<trans\; data-src="T">T型</trans>}}\frac{{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}}{\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src=")">)</trans>}\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=".">.</trans>$$

自$\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=")">)</trans>$，我们获得

$${\mathrm{<trans\; data-src="c">c（c）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="=">=</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{{\mathrm{<trans\; data-src="\eta ">\eta </trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">负极</trans>{\mathrm{<trans\; data-src="T">T型</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}}{<trans\; data-src="\int ">¦{\rm B}</trans>}_{\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}}^{\mathrm{<trans\; data-src="\eta ">\eta </trans>}}\frac{{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}}{\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </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>\mathrm{<trans\; data-src="u">u个</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="-">负极</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{{\mathrm{<trans\; data-src="\eta ">\eta </trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">负极</trans>{\mathrm{<trans\; data-src="T">T型</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}}{<trans\; data-src="\int ">¦{\rm B}</trans>}_{\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}}^{\mathrm{<trans\; data-src="T">T型</trans>}}\frac{{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}}{\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </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>\mathrm{<trans\; data-src="u">u个</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>$$

因此，

$$\begin{array}{rl}\mathrm{<trans\; data-src="u">u个</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="t">t吨</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}}{{\mathrm{<trans\; data-src="\eta ">\eta </trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">负极</trans>{\mathrm{<trans\; data-src="T">T型</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}}{<trans\; data-src="\int ">¦{\rm B}</trans>}_{\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}}^{\mathrm{<trans\; data-src="\eta ">\eta </trans>}}\frac{{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}}{\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </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>\mathrm{<trans\; data-src="u">u个</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="-">负极</trans>\frac{{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}}{{\mathrm{<trans\; data-src="\eta ">\eta </trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">负极</trans>{\mathrm{<trans\; data-src="T">T型</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}}{<trans\; data-src="\int ">¦{\rm B}</trans>}_{\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}}^{\mathrm{<trans\; data-src="T">T型</trans>}}\frac{{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}}{\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </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>\mathrm{<trans\; data-src="u">u个</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="-">负极</trans>{<trans\; data-src="\int ">¦{\rm B}</trans>}_{\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}}^{\mathrm{<trans\; data-src="t">t吨</trans>}}\frac{{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}}{\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </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>\mathrm{<trans\; data-src="u">u个</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>{<trans\; data-src="\int ">¦{\rm B}</trans>}_{\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}}^{\mathrm{<trans\; data-src="T">T型</trans>}}\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</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>\mathrm{<trans\; data-src="u">u个</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>\end{array}$$

这就完成了证明。□

现在，我们准备陈述并证明我们的第一个主要结果。

定理2.2 让 $\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="T">T型</trans>}$ 被给予并出租 τ 和 φ 上有两个函数 $<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=")">)</trans>$ 这样的话 $\mathrm{<trans\; data-src="\phi ">\phi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="\tau ">\tau </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$ 为所有人 $\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=")">)</trans>$.假设 $\mathrm{<trans\; data-src="\tau ">\tau </trans>}<trans\; data-src=":">:</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\to ">\to </trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>$ 是一个满射 $\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="C">C</trans>}<trans\; data-src="(">(</trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="]">]</trans><trans\; data-src="\times ">\times </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src="]">]</trans><trans\; data-src=")">)</trans>$ 在中增加 u个 对于每个固定 t吨,$\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="0">0</trans>}$ 和 $\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\tau ">\tau </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="\phi ">\phi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>$ 为所有人 $\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=")">)</trans>$ 和 $\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="P">P（P）</trans>}$.假设存在 ${\mathrm{<trans\; data-src="M">M（M）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$,${\mathrm{<trans\; data-src="M">M（M）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$ 和 $\mathrm{<trans\; data-src="\theta ">\theta </trans>}<trans\; data-src="\ne ">\ne </trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="P">P（P）</trans>}$ 这样的话

$${\mathrm{<trans\; data-src="M">M（M）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>{<trans\; data-src="\int ">¦{\rm B}</trans>}_{\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}}^{\mathrm{<trans\; data-src="T">T型</trans>}}\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="\le ">\le </trans>{\mathrm{<trans\; data-src="M">M（M）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$$

为所有人 $\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="]">]</trans>$,哪里 $\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>$ 是引理中定义的格林函数吗2.1.然后是问题(2.1)具有边值条件(2.2)有独特的积极解决方案 ${\mathrm{<trans\; data-src="u">u个</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}$.此外,对于序列 ${\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="=">=</trans>{<trans\; data-src="\int ">¦{\rm B}</trans>}_{\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}}^{\mathrm{<trans\; data-src="T">T型</trans>}}\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="s">秒</trans>}$,我们有 $<trans\; data-src="\parallel ">\parallel </trans>{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="-">负极</trans>{\mathrm{<trans\; data-src="u">u个</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="\parallel ">\parallel </trans><trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="0">0</trans>}$ 为所有人 ${\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}$.

证明通过使用引理2.1，问题等价于积分方程

$$\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{<trans\; data-src="\int ">¦{\rm B}</trans>}_{\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}}^{\mathrm{<trans\; data-src="T">T型</trans>}}\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=",">,</trans>$$

哪里

$$\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="\{">\{</trans>\begin{array}{ll}\frac{{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">负极</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}}{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\eta ">\eta </trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">负极</trans>{\mathrm{<trans\; data-src="T">T型</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="-">负极</trans>\frac{{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}}{\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>& \mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=",">,</trans>\\ \frac{<trans\; data-src="-">负极</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}}{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\eta ">\eta </trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">负极</trans>{\mathrm{<trans\; data-src="T">T型</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="-">负极</trans>\frac{{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}}{\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>& \mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=",">,</trans>\\ \frac{<trans\; data-src="-">负极</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}}{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\eta ">\eta </trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">负极</trans>{\mathrm{<trans\; data-src="T">T型</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>& \mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=".">.</trans>\end{array}$$

定义运算符$\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="P">P（P）</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="E">E类</trans>}$通过$\mathrm{<trans\; data-src="A">A类</trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{<trans\; data-src="\int ">¦{\rm B}</trans>}_{\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}}^{\mathrm{<trans\; data-src="T">T型</trans>}}\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="s">秒</trans>}$。那么u个是问题的解决方案当且仅当$\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="A">A类</trans>}\mathrm{<trans\; data-src="u">u个</trans>}$.很容易检查操作员A类正在上增加P（P）另一方面，

$$\begin{array}{rl}\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\tau ">\tau </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</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 ">¦{\rm B}</trans>}_{\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}}^{\mathrm{<trans\; data-src="T">T型</trans>}}\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</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>\mathrm{<trans\; data-src="\tau ">\tau </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="u">u个</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="\ge ">\ge </trans>\mathrm{<trans\; data-src="\phi ">\phi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="\int ">¦{\rm B}</trans>}_{\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}}^{\mathrm{<trans\; data-src="T">T型</trans>}}\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</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>\mathrm{<trans\; data-src="u">u个</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>\mathrm{<trans\; data-src="\phi ">\phi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="A">A类</trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\end{array}$$

为所有人$\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="]">]</trans>$和$\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="P">P（P）</trans>}$因此，操作员A类是τ-φ-凹面。自

$${\mathrm{<trans\; data-src="M">M（M）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="A">A类</trans>}\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{<trans\; data-src="\int ">¦{\rm B}</trans>}_{\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}}^{\mathrm{<trans\; data-src="T">T型</trans>}}\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="\le ">\le </trans>{\mathrm{<trans\; data-src="M">M（M）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$$

为所有人$\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="]">]</trans>$，我们得到$\mathrm{<trans\; data-src="A">A类</trans>}\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}$现在，通过使用定理1.1A类有独特的积极解决方案${\mathrm{<trans\; data-src="u">u个</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}$。这就完成了证明。□

这里，我们给出以下示例来说明定理2.2。

示例2.1让$\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="1">1</trans>}$给出。考虑周期边值问题

$$\begin{array}{c}{\mathrm{<trans\; data-src="D">D类</trans>}}^{\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="3">三</trans>}}}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>{<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src="\}">\}</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans><trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\hfill \\ \mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\hfill \end{array}$$

哪里$\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$,克持续打开$<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>$和${<trans\; data-src="min">最小值</trans>}_{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>}\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$.放置

$$\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="\{">\{</trans>\begin{array}{ll}\frac{{\mathrm{<trans\; data-src="t">t吨</trans>}}^{<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="-">负极</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}}}{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\eta ">\eta </trans>}}^{<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="-">负极</trans>{\mathrm{<trans\; data-src="1">1</trans>}}^{<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="/">/</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="1">1</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="-">负极</trans>\frac{{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</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="3">三</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>& \mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\\ \frac{<trans\; data-src="-">负极</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}}}{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\eta ">\eta </trans>}}^{<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="-">负极</trans>{\mathrm{<trans\; data-src="1">1</trans>}}^{<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="/">/</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="1">1</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="-">负极</trans>\frac{{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</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="3">三</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>& \mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\\ \frac{<trans\; data-src="-">负极</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}}}{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\eta ">\eta </trans>}}^{<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="-">负极</trans>{\mathrm{<trans\; data-src="1">1</trans>}}^{<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="/">/</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="1">1</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>& \mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=".">.</trans>\end{array}$$

然后${<trans\; data-src="\int ">¦{\rm B}</trans>}_{\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}}^{\mathrm{<trans\; data-src="1">1</trans>}}\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</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>\frac{{\mathrm{<trans\; data-src="t">t吨</trans>}}^{<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="-">负极</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="-">负极</trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\eta ">\eta </trans>}}^{<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="4">4</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=")">)</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\eta ">\eta </trans>}}^{<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}$现在，定义$\mathrm{<trans\; data-src="\tau ">\tau </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="t">t吨</trans>}$,$\mathrm{<trans\; data-src="\phi ">\phi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}}$,${\mathrm{<trans\; data-src="\gamma ">\gamma </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="=">=</trans>{<trans\; data-src="min">最小值</trans>}_{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>}\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$,${\mathrm{<trans\; data-src="\gamma ">\gamma </trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="=">=</trans>{<trans\; data-src="max">最大值</trans>}_{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>}\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$还有$\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="u">u个</trans>}}^{\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}}$为所有人t吨。那么$\mathrm{<trans\; data-src="\tau ">\tau </trans>}<trans\; data-src=":">:</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\to ">\to </trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>$是一个满射$\mathrm{<trans\; data-src="\phi ">\phi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="\tau ">\tau </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$为所有人$\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>$.对于每个$\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="0">0</trans>}$，我们有

$$\begin{array}{rl}\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="\tau ">\tau </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="u">u个</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="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}\mathrm{<trans\; data-src="u">u个</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="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="\lambda ">\lambda </trans>}}^{\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}}{<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="u">u个</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="3">三</trans>}}\\ <trans\; data-src="\ge ">\ge </trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</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="u">u个</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="3">三</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="\phi ">\phi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </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="u">u个</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>\end{array}$$

现在，把$\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="\equiv ">选择</trans>\mathrm{<trans\; data-src="1">1</trans>}$,${\mathrm{<trans\; data-src="M">M（M）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\gamma ">\gamma </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="min">最小值</trans>}_{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>}\frac{<trans\; data-src="-">负极</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="-">负极</trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\eta ">\eta </trans>}}^{<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="4">4</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=")">)</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\eta ">\eta </trans>}}^{<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}$和${\mathrm{<trans\; data-src="M">M（M）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\gamma ">\gamma </trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="max">最大值</trans>}_{\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>}\frac{{\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}}^{<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}}{\mathrm{<trans\; data-src="\eta ">\eta </trans>}}^{\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}}}{\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="4">4</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=")">)</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\eta ">\eta </trans>}}^{<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}$.然后我们得到

$$\begin{array}{r}{<trans\; data-src="\int ">¦{\rm B}</trans>}_{\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}}^{\mathrm{<trans\; data-src="1">1</trans>}}\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</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">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</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="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="3">三</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 ">¦{\rm B}</trans>}_{\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}}^{\mathrm{<trans\; data-src="1">1</trans>}}\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\gamma ">\gamma </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="d">d日</trans>}\mathrm{<trans\; data-src="s">秒</trans>}\\ <trans\; data-src="\le ">\le </trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\gamma ">\gamma </trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>\underset{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>}{<trans\; data-src="max">最大值</trans>}{<trans\; data-src="\int ">¦{\rm B}</trans>}_{\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}}^{\mathrm{<trans\; data-src="1">1</trans>}}\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</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="(">(</trans>{\mathrm{<trans\; data-src="\gamma ">\gamma </trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans><trans\; data-src="(">(</trans>\underset{\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>}{<trans\; data-src="max">最大值</trans>}\frac{{\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}}^{<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}}{\mathrm{<trans\; data-src="\eta ">\eta </trans>}}^{\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}}}{\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="4">4</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=")">)</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\eta ">\eta </trans>}}^{<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="M">M（M）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}\mathrm{<trans\; data-src="h">小时</trans>}\end{array}$$

和

$$\begin{array}{r}{<trans\; data-src="\int ">¦{\rm B}</trans>}_{\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}}^{\mathrm{<trans\; data-src="1">1</trans>}}\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</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">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</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="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="3">三</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><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\gamma ">\gamma </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>\underset{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>}{<trans\; data-src="min">最小值</trans>}{<trans\; data-src="\int ">¦{\rm B}</trans>}_{\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}}^{\mathrm{<trans\; data-src="1">1</trans>}}\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</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><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\gamma ">\gamma </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>\underset{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>}{<trans\; data-src="min">最小值</trans>}\frac{<trans\; data-src="-">负极</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="-">负极</trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\eta ">\eta </trans>}}^{<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="4">4</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=")">)</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\eta ">\eta </trans>}}^{<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="M">M（M）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src=".">.</trans>\end{array}$$

因此，通过使用定理2.2，该问题在${\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}$.

2.2具有Caputo分数阶导数的分数阶微分方程的存在性结果

在这里，我们研究了分数阶微分方程正解的存在唯一性

$${}^{\mathrm{<trans\; data-src="c">c（c）</trans>}}\mathrm{<trans\; data-src="D">D类</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="]">]</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>$$

(2.3)

$$\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="\beta ">\beta </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="\beta ">\beta </trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src=")">)</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="<"><</trans>{\mathrm{<trans\; data-src="\beta ">\beta </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="<"><</trans>{\mathrm{<trans\; data-src="\beta ">\beta </trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>$$

(2.4)

哪里${}^{\mathrm{<trans\; data-src="c">c（c）</trans>}}\mathrm{<trans\; data-src="D">D类</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}$是Caputo分数阶导数α.让$\mathrm{<trans\; data-src="E">E类</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="C">C</trans>}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="]">]</trans>$是上连续函数的Banach空间$<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="]">]</trans>$以超标准和

$$\mathrm{<trans\; data-src="P">P（P）</trans>}<trans\; data-src="=">=</trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="C">C</trans>}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="]">]</trans><trans\; data-src=":">:</trans>\underset{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="]">]</trans>}{<trans\; data-src="min">最小值</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="\}">\}</trans><trans\; data-src=".">.</trans>$$

众所周知P（P）是一个正常的圆锥体。类似于引理2.1的证明，我们可以证明以下结果。

引理2.3 让 $\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="2">2</trans>}$,$\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="1">1</trans>}$,$\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="]">]</trans>$,$\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$ 和 $\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="<"><</trans>{\mathrm{<trans\; data-src="\beta ">\beta </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="<"><</trans>{\mathrm{<trans\; data-src="\beta ">\beta </trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="1">1</trans>}$.然后是问题 ${}^{\mathrm{<trans\; data-src="c">c（c）</trans>}}\mathrm{<trans\; data-src="D">D类</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$ 具有边值条件 $\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="\beta ">\beta </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src=")">)</trans>$ 和 $\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="\beta ">\beta </trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src=")">)</trans>$ 有解决方案 ${\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$ 当且仅当 ${\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$ 是分数阶积分方程的解 $\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{<trans\; data-src="\int ">¦{\rm B}</trans>}_{\mathrm{<trans\; data-src="0">0</trans>}}^{\mathrm{<trans\; data-src="T">T型</trans>}}\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="s">秒</trans>}$,哪里

$$\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="\{">\{</trans>\begin{array}{ll}\frac{<trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="\beta ">\beta </trans>}}_{\mathrm{<trans\; data-src="1">1</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="\beta ">\beta </trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">负极</trans>{\mathrm{<trans\; data-src="\beta ">\beta </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="t">t吨</trans>}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="T">T型</trans>}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}}{\mathrm{<trans\; data-src="T">T型</trans>}\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>& \mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=",">,</trans>\\ \frac{\mathrm{<trans\; data-src="t">t吨</trans>}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="T">T型</trans>}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}}{\mathrm{<trans\; data-src="T">T型</trans>}\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>& \mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=",">,</trans>\\ \frac{\mathrm{<trans\; data-src="t">t吨</trans>}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">负极</trans>\mathrm{<trans\; data-src="1">1</trans>}}}{\mathrm{<trans\; data-src="T">T型</trans>}\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>& \mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=".">.</trans>\end{array}$$

定理2.4 让 $\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="1">1</trans>}$ 被给予并出租 τ 和 φ 两个阳性-上的有值函数 $<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=")">)</trans>$ 这样的话 $\mathrm{<trans\; data-src="\phi ">\phi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="\tau ">\tau </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$ 为所有人 $\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=")">)</trans>$.假设 $\mathrm{<trans\; data-src="\tau ">\tau </trans>}<trans\; data-src=":">:</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\to ">\to </trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>$ 是一个满射 $\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="C">C</trans>}<trans\; data-src="(">(</trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="]">]</trans><trans\; data-src="\times ">\times </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src="]">]</trans><trans\; data-src=")">)</trans>$ 在中增加 u个 对于每个固定 t吨,$\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$ 无论何时 $\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="T">T型</trans>}$ 和 $\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="0">0</trans>}$ 否则,还有 $\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\tau ">\tau </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="\phi ">\phi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>$ 为所有人 $\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=")">)</trans>$ 和 $\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="P">P（P）</trans>}$.假设存在 ${\mathrm{<trans\; data-src="M">M（M）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$,${\mathrm{<trans\; data-src="M">M（M）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$ 和 $\mathrm{<trans\; data-src="\theta ">\theta </trans>}<trans\; data-src="\ne ">\ne </trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="P">P（P）</trans>}$ 这样的话

$${\mathrm{<trans\; data-src="M">M（M）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>{<trans\; data-src="\int ">¦{\rm B}</trans>}_{\mathrm{<trans\; data-src="0">0</trans>}}^{\mathrm{<trans\; data-src="T">T型</trans>}}\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="\le ">\le </trans>{\mathrm{<trans\; data-src="M">M（M）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$$

为所有人 $\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="]">]</trans>$,哪里 $\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>$ 是引理中定义的格林函数2.3.然后是问题(2.3)具有边值条件(2.4)有独特的积极解决方案 ${\mathrm{<trans\; data-src="u">u个</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}$.此外,对于序列 ${\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="=">=</trans>{<trans\; data-src="\int ">¦{\rm B}</trans>}_{\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}}^{\mathrm{<trans\; data-src="T">T型</trans>}}\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="s">秒</trans>}$,我们有 $<trans\; data-src="\parallel ">\parallel </trans>{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="-">负极</trans>{\mathrm{<trans\; data-src="u">u个</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="\parallel ">\parallel </trans><trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="0">0</trans>}$ 为所有人 ${\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}$.

证明定义操作符就足够了$\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="P">P（P）</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="E">E类</trans>}$通过

$$\mathrm{<trans\; data-src="A">A类</trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{<trans\; data-src="\int ">¦{\rm B}</trans>}_{\mathrm{<trans\; data-src="0">0</trans>}}^{\mathrm{<trans\; data-src="T">T型</trans>}}\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=".">.</trans>$$

现在，通过使用定理2.2的类似证明，可以证明$\mathrm{<trans\; data-src="A">A类</trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="0">0</trans>}$为所有人$\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="P">P（P）</trans>}$和$\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="]">]</trans>$，以及操作员A类是τ-φ-凹面。通过使用定理1.1A类有独特的积极解决方案${\mathrm{<trans\; data-src="u">u个</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}$.这通过使用引理2.3完成了证明。□

下面我们给出一个例子来说明定理2.4。

例2.2让$\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="=">=</trans>\frac{\mathrm{<trans\; data-src="3">三</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}$.考虑周期边值问题

$$\begin{array}{c}{}^{\mathrm{<trans\; data-src="c">c（c）</trans>}}\mathrm{<trans\; data-src="D">D类</trans>}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}\mathrm{<trans\; data-src="u">u个</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="u">u个</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="0">0</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans><trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\hfill \\ \mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="3">三</trans>}}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="u">u个</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="2">2</trans>}}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\hfill \end{array}$$

哪里克是上的连续函数$<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>$具有${<trans\; data-src="min">最小值</trans>}_{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>}\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$.放置${\mathrm{<trans\; data-src="\beta ">\beta </trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="2">2</trans>}$,${\mathrm{<trans\; data-src="\beta ">\beta </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}$和

$$\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="\{">\{</trans>\begin{array}{ll}\frac{<trans\; data-src="[">[</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="+">+</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="6">6</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="2">2</trans>}}<trans\; data-src="-">负极</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>}}<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="-">负极</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>}}<trans\; data-src="-">负极</trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="-">负极</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="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>& \mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\\ \frac{\mathrm{<trans\; data-src="t">t吨</trans>}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">负极</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>}}<trans\; data-src="-">负极</trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="-">负极</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="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>& \mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\\ \frac{\mathrm{<trans\; data-src="t">t吨</trans>}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">负极</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="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>& \mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=".">.</trans>\end{array}$$

然后${<trans\; data-src="\int ">¦{\rm B}</trans>}_{\mathrm{<trans\; data-src="0">0</trans>}}^{\mathrm{<trans\; data-src="1">1</trans>}}\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="=">=</trans>\frac{<trans\; data-src="[">[</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="+">+</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="6">6</trans>}}\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="]">]</trans>{<trans\; data-src="(">(</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="-">负极</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="2">2</trans>}}}{\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="5">5</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}$现在，定义$\mathrm{<trans\; data-src="\tau ">\tau </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="t">t吨</trans>}$,$\mathrm{<trans\; data-src="\phi ">\phi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}$,${\mathrm{<trans\; data-src="\gamma ">\gamma </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="=">=</trans>{<trans\; data-src="min">最小值</trans>}_{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>}\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$,${\mathrm{<trans\; data-src="\gamma ">\gamma </trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="=">=</trans>{<trans\; data-src="max">最大值</trans>}_{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>}\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$和$\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="u">u个</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}$那么很容易看出$\mathrm{<trans\; data-src="\tau ">\tau </trans>}<trans\; data-src=":">:</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\to ">\to </trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>$是一张满射图$\mathrm{<trans\; data-src="\phi ">\phi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="\tau ">\tau </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$对于$\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>$。此外，我们还有

$$\begin{array}{rl}\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="\tau ">\tau </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="u">u个</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="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}\mathrm{<trans\; data-src="u">u个</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="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="\lambda ">\lambda </trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}{<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="u">u个</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="\ge ">\ge </trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </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="u">u个</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="=">=</trans>\mathrm{<trans\; data-src="\phi ">\phi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </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="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\end{array}$$

为所有人$\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="0">0</trans>}$现在，把$\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="\equiv ">选择</trans>\mathrm{<trans\; data-src="1">1</trans>}$,${\mathrm{<trans\; data-src="M">M（M）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\gamma ">\gamma </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="min">最小值</trans>}_{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>}\frac{<trans\; data-src="-">负极</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="3">三</trans>}}\mathrm{<trans\; data-src="t">t吨</trans>}{<trans\; data-src="(">(</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">负极</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="2">2</trans>}}}{\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="5">5</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}$还有${\mathrm{<trans\; data-src="M">M（M）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\gamma ">\gamma </trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>\frac{\frac{\mathrm{<trans\; data-src="5">5</trans>}}{\mathrm{<trans\; data-src="6">6</trans>}}{<trans\; data-src="(">(</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="5">5</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}$.那么我们有

$$\begin{array}{r}{<trans\; data-src="\int ">¦{\rm B}</trans>}_{\mathrm{<trans\; data-src="0">0</trans>}}^{\mathrm{<trans\; data-src="1">1</trans>}}\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</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">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</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="s">秒</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>}^{\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="2">2</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 ">¦{\rm B}</trans>}_{\mathrm{<trans\; data-src="0">0</trans>}}^{\mathrm{<trans\; data-src="1">1</trans>}}\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\gamma ">\gamma </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="d">d日</trans>}\mathrm{<trans\; data-src="s">秒</trans>}\\ <trans\; data-src="\le ">\le </trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\gamma ">\gamma </trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>\underset{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>}{<trans\; data-src="max">最大值</trans>}{<trans\; data-src="\int ">¦{\rm B}</trans>}_{\mathrm{<trans\; data-src="0">0</trans>}}^{\mathrm{<trans\; data-src="1">1</trans>}}\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</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="(">(</trans>{\mathrm{<trans\; data-src="\gamma ">\gamma </trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>\frac{\frac{\mathrm{<trans\; data-src="5">5</trans>}}{\mathrm{<trans\; data-src="6">6</trans>}}{<trans\; data-src="(">(</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="5">5</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="M">M（M）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}\mathrm{<trans\; data-src="h">小时</trans>}\end{array}$$

和

$$\begin{array}{rl}{<trans\; data-src="\int ">¦{\rm B}</trans>}_{\mathrm{<trans\; data-src="0">0</trans>}}^{\mathrm{<trans\; data-src="1">1</trans>}}\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</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">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</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="s">秒</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>}^{\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="2">2</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><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\gamma ">\gamma </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>\underset{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>}{<trans\; data-src="min">最小值</trans>}{<trans\; data-src="\int ">¦{\rm B}</trans>}_{\mathrm{<trans\; data-src="0">0</trans>}}^{\mathrm{<trans\; data-src="1">1</trans>}}\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</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><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\gamma ">\gamma </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>\underset{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>}{<trans\; data-src="min">最小值</trans>}\frac{<trans\; data-src="-">负极</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="3">三</trans>}}\mathrm{<trans\; data-src="t">t吨</trans>}{<trans\; data-src="(">(</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">负极</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="2">2</trans>}}}{\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="5">5</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="M">M（M）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src=".">.</trans>\end{array}$$

因此，通过使用定理2.4，该问题在${\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}$.