格结构有序Banach空间中减算子的不动点定理及其应用

摘要

本文给出了格结构Banach空间中递减算子不动点的一些定理。将结果应用于非线性二阶椭圆方程。

MSC公司：47H10，34B15。

1简介和前言

在过去的30年中，有序Banach空间中单调算子的不动点理论得到了广泛的研究[1–8]. 在非线性压缩条件下，利用锥定理和单调迭代技术证明了许多新的不动点定理。这些结果已用于研究常微分方程、偏微分方程和积分方程。

本文研究具有格结构的有序Banach空间中的减算子。利用著名的算子Schauder不动点定理，推广了不动点的理论结果。我们减弱了Schauder不动点定理的条件。本文的结果不需要算子域的闭有界性和凸性。为了证明我们的结果的适用性，我们在本文的最后一节将其应用于研究一个非线性二阶椭圆方程问题，并获得了解的存在性。

让E类成为巴拿赫空间P（P）是一个圆锥体E类.我们定义了关于P（P）通过$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="y">年</trans>}$如果仅当$\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="P">P（P）</trans>}$.一个圆锥体$\mathrm{<trans\; data-src="P">P（P）</trans>}<trans\; data-src="\subset ">\subset </trans>\mathrm{<trans\; data-src="E">E类</trans>}$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$<trans\; data-src="\parallel ">\parallel </trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\parallel ">\parallel </trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src="\parallel ">\parallel </trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\parallel ">\parallel </trans>$为所有人$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="E">E类</trans>}$.最小正常数N个满足上述不等式称为P（P）.

让E类是一个偏序集。我们打电话给E类偏序≤的格。对于任意$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="E">E类</trans>}$,$<trans\; data-src="sup">啜饮</trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\}">\}</trans>$和$<trans\; data-src="inf">inf公司</trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\}">\}</trans>$存在。人们可以看到[7]关于格的定义和性质。

让$\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="\subset ">\subset </trans>\mathrm{<trans\; data-src="E">E类</trans>}$，操作员$\mathrm{<trans\; data-src="A">一</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="⟶">⟶</trans>\mathrm{<trans\; data-src="E">E类</trans>}$称为递增运算符，如果$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="D">D类</trans>}$,$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="y">年</trans>}$，表示$\mathrm{<trans\; data-src="A">一</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="A">一</trans>}\mathrm{<trans\; data-src="y">年</trans>}$; 操作员$\mathrm{<trans\; data-src="A">一</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="⟶">⟶</trans>\mathrm{<trans\; data-src="E">E类</trans>}$称为递减运算符，如果$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="D">D类</trans>}$,$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="y">年</trans>}$，暗示$\mathrm{<trans\; data-src="A">一</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="A">一</trans>}\mathrm{<trans\; data-src="x">x个</trans>}$.

引理1.1[9]

让 E类 成为一个真正的巴拿赫空间,$\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="\subset ">\subset </trans>\mathrm{<trans\; data-src="E">E类</trans>}$ 非空的,闭有界凸,和 $\mathrm{<trans\; data-src="A">一</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="D">D类</trans>}$ 正在冷凝.然后 一 在中有一个固定点 D类.

引理1.2[10]

让 E类 成为一个真正的巴拿赫空间,$\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="\subset ">\subset </trans>\mathrm{<trans\; data-src="E">E类</trans>}$ 非空的,闭有界凸,和 $\mathrm{<trans\; data-src="A">一</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="D">D类</trans>}$ 完全连续.然后 一 在中有一个固定点 D类.

引理1.3[11]

让 E类 成为一个真正的巴拿赫空间,$\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="\subset ">\subset </trans>\mathrm{<trans\; data-src="E">E类</trans>}$ 非空的,闭有界凸,和 $\mathrm{<trans\; data-src="A">一</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="D">D类</trans>}$ 严格-设置-收缩映射.然后 一 在中有一个固定点 D类.

备注1引理1.1是著名的Sadovskii不动点定理；引理1.2是著名的Schauder不动点定理；引理1.3是著名的Darbo不动点定理。

2主要成果

定理2.1 让 E类 是一个具有格结构的有序Banach空间,$\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="\subset ">\subset </trans>\mathrm{<trans\; data-src="E">E类</trans>}$ 被限定,和 $\mathrm{<trans\; data-src="A">一</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="⟶">⟶</trans>\mathrm{<trans\; data-src="D">D类</trans>}$ 做一个减少和冷凝的操作员.然后是操作员 一 在中有一个固定点 D类.

证明对于任何$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="D">D类</trans>}$，自$\mathrm{<trans\; data-src="A">一</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="⟶">⟶</trans>\mathrm{<trans\; data-src="D">D类</trans>}$，我们有$\mathrm{<trans\; data-src="A">一</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="D">D类</trans>}$.

自E类是具有晶格结构的Banach空间$\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="\subset ">\subset </trans>\mathrm{<trans\; data-src="E">E类</trans>}$有界，存在${\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="D">D类</trans>}$这样的话

$$<trans\; data-src="inf">inf公司</trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="A">一</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\}">\}</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=".">.</trans>$$

那就是，

$${\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="A">一</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=".">.</trans>$$

(2.1)

自一是一个递减运算符，我们有

$${\mathrm{<trans\; data-src="A">一</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="A">一</trans>}{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="A">一</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="A">一</trans>}{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=".">.</trans>$$

(2.2)

（2.1）和（2.2）表明

$${\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="A">一</trans>}{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=".">.</trans>$$

(2.3)

与（2.3）的证明类似，存在${\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="D">D类</trans>}$这样的话

$$<trans\; data-src="sup">啜饮</trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="A">一</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\}">\}</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=".">.</trans>$$

那就是，

$$\mathrm{<trans\; data-src="A">一</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\le ">\le </trans>{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\le ">\le </trans>{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=".">.</trans>$$

(2.4)

自一是一个递减算子，我们有

$$\mathrm{<trans\; data-src="A">一</trans>}{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\le ">\le </trans>{\mathrm{<trans\; data-src="A">一</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="A">一</trans>}{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="A">一</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=".">.</trans>$$

(2.5)

（2.4）和（2.5）表明

$$\mathrm{<trans\; data-src="A">一</trans>}{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\le ">\le </trans>{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=".">.</trans>$$

(2.6)

（2.3）和（2.6）以及${\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\le ">\le </trans>{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$向大家展示

$${\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="A">一</trans>}{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="A">一</trans>}{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\le ">\le </trans>{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=".">.</trans>$$

(2.7)

对于任何$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="]">]</trans>$，自一是一个递减算子，我们有

$$\mathrm{<trans\; data-src="A">一</trans>}{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="A">一</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="A">一</trans>}{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=".">.</trans>$$

通过（2.7），我们已经

$$\mathrm{<trans\; data-src="A">一</trans>}<trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="]">]</trans><trans\; data-src="⟶">⟶</trans><trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="]">]</trans><trans\; data-src=".">.</trans>$$

很容易知道$<trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="]">]</trans>$是一个闭凸集。自$\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="\subset ">\subset </trans>\mathrm{<trans\; data-src="E">E类</trans>}$有界，我们有$<trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="]">]</trans>$有界。因此，$<trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="]">]</trans>$是一个闭的有界凸集。因此，引理1.1意味着算子一在中有一个固定点D类. □

定理2.2 让 E类 是具有格结构的有序Banach空间,$\mathrm{<trans\; data-src="P">P（P）</trans>}<trans\; data-src="\subset ">\subset </trans>\mathrm{<trans\; data-src="E">E类</trans>}$ 是一个正常的圆锥体,和 $\mathrm{<trans\; data-src="A">一</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="E">E类</trans>}<trans\; data-src="⟶">⟶</trans>\mathrm{<trans\; data-src="E">E类</trans>}$ 做一个减少和冷凝的操作员.然后是操作员 一 在中有一个固定点 E类.

证明对于任何$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="E">E类</trans>}$，自$\mathrm{<trans\; data-src="A">一</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="E">E类</trans>}<trans\; data-src="⟶">⟶</trans>\mathrm{<trans\; data-src="E">E类</trans>}$，我们有$\mathrm{<trans\; data-src="A">一</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="E">E类</trans>}$.

自E类是一个格子结构的Banach空间，存在${\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="E">E类</trans>}$这样的话

$$<trans\; data-src="inf">inf公司</trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="A">一</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\}">\}</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=".">.</trans>$$

那就是，

$${\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="A">一</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=".">.</trans>$$

(2.8)

自一是一个递减算子，我们有

$${\mathrm{<trans\; data-src="A">一</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="A">一</trans>}{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="A">一</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="A">一</trans>}{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=".">.</trans>$$

(2.9)

（2.8）和（2.9）表明

$${\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="A">一</trans>}{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=".">.</trans>$$

(2.10)

与（2.10）的证明类似，存在${\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="E">E类</trans>}$这样的话

$$<trans\; data-src="sup">啜饮</trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="A">一</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\}">\}</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=".">.</trans>$$

那就是，

$$\mathrm{<trans\; data-src="A">一</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\le ">\le </trans>{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\le ">\le </trans>{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=".">.</trans>$$

(2.11)

自一是一个递减算子，我们有

$$\mathrm{<trans\; data-src="A">一</trans>}{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\le ">\le </trans>{\mathrm{<trans\; data-src="A">一</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="A">一</trans>}{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="A">一</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=".">.</trans>$$

(2.12)

（2.11）和（2.12）表明

$$\mathrm{<trans\; data-src="A">一</trans>}{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\le ">\le </trans>{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=".">.</trans>$$

(2.13)

（2.10）和（2.13）以及${\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\le ">\le </trans>{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$向大家展示

$${\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="A">一</trans>}{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="A">一</trans>}{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\le ">\le </trans>{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=".">.</trans>$$

(2.14)

对于任何$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="]">]</trans>$，自一是一个递减算子，我们有

$$\mathrm{<trans\; data-src="A">一</trans>}{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="A">一</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="A">一</trans>}{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=".">.</trans>$$

截至（2.14），我们已经

$$\mathrm{<trans\; data-src="A">一</trans>}<trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="]">]</trans><trans\; data-src="⟶">⟶</trans><trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="]">]</trans><trans\; data-src=".">.</trans>$$

很容易知道$<trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="]">]</trans>$是一个闭凸集。自P（P）是E类，我们有$<trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="]">]</trans>$有界。因此，$<trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="]">]</trans>$是一个闭的有界凸集。因此，引理1.1意味着算子一在中有一个固定点D类. □

3推论和相关结果

与定理2.1的证明类似，通过引理1.2和引理1.3，我们可以得到以下推论和相关结果。

推论3.1 让 E类 是具有格结构的有序Banach空间,$\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="\subset ">\subset </trans>\mathrm{<trans\; data-src="E">E类</trans>}$ 被限定,和 $\mathrm{<trans\; data-src="A">一</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="⟶">⟶</trans>\mathrm{<trans\; data-src="D">D类</trans>}$ 是一个递减且完全连续的算子.然后是操作员 一 在中有一个固定点 D类.

推论3.2 让 E类 是具有格结构的有序Banach空间,$\mathrm{<trans\; data-src="P">P（P）</trans>}<trans\; data-src="\subset ">\subset </trans>\mathrm{<trans\; data-src="E">E类</trans>}$ 是一个正常的圆锥体,和 $\mathrm{<trans\; data-src="A">一</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="E">E类</trans>}<trans\; data-src="⟶">⟶</trans>\mathrm{<trans\; data-src="E">E类</trans>}$ 是一个递减且完全连续的算子.然后是操作员 一 在中有一个固定点 E类.

推论3.3 让 E类 是具有格结构的有序Banach空间,$\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="\subset ">\subset </trans>\mathrm{<trans\; data-src="E">E类</trans>}$ 被限定,和 $\mathrm{<trans\; data-src="A">一</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="⟶">⟶</trans>\mathrm{<trans\; data-src="D">D类</trans>}$ 减少和严格-设置-收缩映射.然后是映射 一 在中有一个固定点 D类.

推论3.4 让 E类 是具有格结构的有序Banach空间,$\mathrm{<trans\; data-src="P">P（P）</trans>}<trans\; data-src="\subset ">\subset </trans>\mathrm{<trans\; data-src="E">E类</trans>}$ 是一个正常的圆锥体,和 $\mathrm{<trans\; data-src="A">一</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="E">E类</trans>}<trans\; data-src="⟶">⟶</trans>\mathrm{<trans\; data-src="E">E类</trans>}$ 减少和严格-设置-收缩映射.然后是映射 一 在中有一个固定点 E类.

4应用

在本节中，我们使用定理2.1来证明一致椭圆微分问题解的存在性。设Ω是中的有界凸域${\mathrm{<trans\; data-src="R">R（右）</trans>}}^{\mathrm{<trans\; data-src="n">n个</trans>}}$($\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="2">2</trans>}$)其边界∂假设Ω足够平滑。考虑上的一致椭圆微分算子$\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}$

$$\mathrm{<trans\; data-src="L">L（左）</trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="=">=</trans><trans\; data-src="-">-</trans>\underset{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="n">n个</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="j">j个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\frac{{\mathrm{<trans\; data-src="\partial ">\partial </trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}\mathrm{<trans\; data-src="u">u个</trans>}}{\mathrm{<trans\; data-src="\partial ">\partial </trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="j">j个</trans>}}}<trans\; data-src="+">+</trans>\underset{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="n">n个</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\frac{\mathrm{<trans\; data-src="\partial ">\partial </trans>}\mathrm{<trans\; data-src="u">u个</trans>}}{\mathrm{<trans\; data-src="\partial ">\partial </trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="c">c（c）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="u">u个</trans>}$$

即，存在一个正常数${\mathrm{<trans\; data-src="\mu ">\mu </trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$这样的话${<trans\; data-src="\sum ">\sum </trans>}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}^{\mathrm{<trans\; data-src="n">n个</trans>}}{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="j">j个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="\xi ">\xi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}\mathrm{<trans\; data-src="\xi ">\xi </trans>}\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src="\ge ">\ge </trans>{\mathrm{<trans\; data-src="\mu ">\mu </trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}{<trans\; data-src="|">|</trans>\mathrm{<trans\; data-src="\xi ">\xi </trans>}<trans\; data-src="|">|</trans>}^{\mathrm{<trans\; data-src="2">2</trans>}}$对于任何$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}$和$\mathrm{<trans\; data-src="\xi ">\xi </trans>}<trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\xi ">\xi </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\xi ">\xi </trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\xi ">\xi </trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="R">R（右）</trans>}}^{\mathrm{<trans\; data-src="n">n个</trans>}}$，其中${\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="j">j个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="j">j个</trans>}\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$,$\mathrm{<trans\; data-src="c">c（c）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="0">0</trans>}$。为了简单起见，我们假设所有函数${\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="j">j个</trans>}\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$,${\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$,$\mathrm{<trans\; data-src="c">c（c）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$足够平滑。

考虑Dirichlet问题

$$\mathrm{<trans\; data-src="L">L（左）</trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}{<trans\; data-src="|">|</trans>}_{\mathrm{<trans\; data-src="\partial ">\partial </trans>}\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>$$

(4.1)

我们得出以下结论。

定理4.1 假设 $\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="C">C类</trans>}<trans\; data-src="(">(</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}<trans\; data-src="\times ">\times </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>$,其在以下日期减少 u个,然后是问题(4.1)有一个积极的解决方案.

证明很容易知道$\mathrm{<trans\; data-src="E">E类</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="C">C类</trans>}<trans\; data-src="(">(</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}<trans\; data-src=")">)</trans>$是具有最大范数的Banach空间$<trans\; data-src="\parallel ">\parallel </trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src="\parallel ">\parallel </trans>$它也是一个格子。让$\mathrm{<trans\; data-src="P">P（P）</trans>}<trans\; data-src="=">=</trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="E">E类</trans>}<trans\; data-src="\mid ">\mid </trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\forall ">\forall </trans>}\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}<trans\; data-src="\}">\}</trans>$和P（P）成为一个正常的圆锥体E类众所周知（参见[1,10])Dirichlet问题（4.1）的解等价于积分算子的不动点一

$$\mathrm{<trans\; data-src="A">一</trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{<trans\; data-src="\int ">\int </trans>}_{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}}\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=",">,</trans>$$

哪里$\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans>$表示带边界条件的微分算子L的格林函数$\mathrm{<trans\; data-src="u">u个</trans>}{<trans\; data-src="|">|</trans>}_{\mathrm{<trans\; data-src="\partial ">\partial </trans>}\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$众所周知$\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans>$满足以下不等式：

$$\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="<"><</trans><trans\; data-src="\{">\{</trans>\begin{array}{cc}{\mathrm{<trans\; data-src="K">K（K）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}{<trans\; data-src="|">|</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="|">|</trans>}^{\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\hfill & \mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\hfill \\ {\mathrm{<trans\; data-src="K">K（K）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="|">|</trans><trans\; data-src="ln">自然对数</trans><trans\; data-src="|">|</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="|">|</trans><trans\; data-src="|">|</trans><trans\; data-src=",">,</trans>\hfill & \mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="2">2</trans>}\hfill \end{array}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\ne ">\ne </trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

因此，线性积分算子

$$\mathrm{<trans\; data-src="B">B类</trans>}\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{<trans\; data-src="\int ">\int </trans>}_{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}}\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="y">年</trans>}$$

是来自E类进入之内E类显然，叠加算子$\mathrm{<trans\; data-src="F">F类</trans>}\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>$那张地图P（P）进入之内P（P）连续且有界。因此，操作员$\mathrm{<trans\; data-src="A">一</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="B">B类</trans>}\mathrm{<trans\; data-src="F">F类</trans>}$那张地图P（P）进入之内P（P）是完全连续的，因此一正在冷凝。

此外，映射一正在减少u个事实上，根据假设$\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="v">v（v）</trans>}$,

$$\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>$$

意味着

$$\begin{array}{rl}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">一</trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>& <trans\; data-src="=">=</trans>{<trans\; data-src="\int ">\int </trans>}_{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}}\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="y">年</trans>}\\ <trans\; data-src="\le ">\le </trans>{<trans\; data-src="\int ">\int </trans>}_{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}}\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">一</trans>}\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=")">)</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}<trans\; data-src=",">,</trans>\end{array}$$

所以一正在减少。

因此，定理2.1的条件成立，定理4.1得到了证明。□