Geraghty压缩的最佳逼近点定理

摘要

本文的目的是为Geraghty压缩存在唯一最佳邻近点提供充分条件。

我们的论文扩展了Geraghty（Proc.Am.Math.Soc.40:604-6081973）的一个结果。

1引言

让A类和B类是度量空间的非空子集$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src=")">)</trans>$.

操作员$\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="B">B类</trans>}$如果存在，则称为k收缩$\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>$这样的话$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans>$对于任何$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="A">A类</trans>}$巴纳赫收缩原理指出，当A类是的完整子集X（X）和T型是映射的k收缩A类那么就变成了它自己T型在中具有唯一的固定点A类.

文献中出现了大量对这一原则的概括。特别是，以下对巴拿赫收缩原理的概括是由于杰拉蒂[1].

首先，我们介绍一下这个类ℱ这些功能$\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src=":">:</trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src=")">)</trans><trans\; data-src="\to ">\to </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>$满足以下条件：

$$\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="1">1</trans>}\text{<trans data-src="implies">暗示</trans>}{\mathrm{<trans\; data-src="t">t吨</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=".">.</trans>$$

定理1.1([1])

让 $<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src=")">)</trans>$ 是一个完整的度量空间，并且 $\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="X">X（X）</trans>}$ 成为操作员.假设存在 $\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src="\in ">\in </trans>\mathcal{<trans\; data-src="F">如果</trans>}$ 这样，对于任何 $\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="X">X（X）</trans>}$,

$$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\cdot ">\cdot </trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

(1)

然后 T型 具有唯一的固定点.

由于常量函数$\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="k">k个</trans>}$，其中$\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>$，属于ℱ定理1.1扩展了巴拿赫收缩原理。

备注1.1由于函数属于ℱ严格小于1，条件（1）意味着

$$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans>\text{<trans data-src="for any ">对于任何</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="X">X（X）</trans>}\text{<trans data-src=" with ">带有</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\ne ">\ne </trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=".">.</trans>$$

因此，任何操作员$\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="X">X（X）</trans>}$满足（1）是一个连续算子。

本文的目的是通过考虑非自映射来推广定理1.1T型.

首先，我们简要讨论了最佳接近点。

让A类是度量空间的非空子集$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src=")">)</trans>$和$\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="X">X（X）</trans>}$成为一个映射。方程的解$\mathrm{<trans\; data-src="T">T型</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="x">x个</trans>}$是的固定点T型。因此，$\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\cap ">\cap </trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="\ne ">\ne </trans>\mathrm{<trans\; data-src="\varnothing ">\varnothing </trans>}$是算子存在不动点的必要条件T型。如果此必要条件不成立，则$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$对于任何$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="A">A类</trans>}$和映射$\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="X">X（X）</trans>}$没有任何固定点。在这种情况下，我们的目标是找到一个元素$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="A">A类</trans>}$这样的话$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$在某种意义上是最小值。最佳逼近理论和最佳逼近点分析都是在这个方向上发展起来的。

在我们的上下文中，我们考虑两个非空子集A类和B类完备度量空间和映射$\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="B">B类</trans>}$.

一个自然的问题是能否找到元素${\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="A">A类</trans>}$这样的话$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="min">最小值</trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="\}">\}</trans>$.自$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src=")">)</trans>$对于任何$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="A">A类</trans>}$，此问题的最佳解决方案将是$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src=")">)</trans>$由实值函数获得$\mathrm{<trans\; data-src="\phi ">\phi </trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="\to ">\to </trans>\mathbb{<trans\; data-src="R">R（右）</trans>}$由提供$\mathrm{<trans\; data-src="\phi ">\phi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$.

有关最佳邻近点的一些结果可以在中找到[2–9].

2符号和基本事实

让A类和B类是度量空间的两个非空子集$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src=")">)</trans>$.

我们表示为${\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$和${\mathrm{<trans\; data-src="B">B类</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$以下集合：

哪里$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="inf">inf公司</trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="A">A类</trans>}\text{<trans data-src=" and ">和</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src="\}">\}</trans>$.

在[8]，作者提出了确定集合何时${\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$和${\mathrm{<trans\; data-src="B">B类</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$是非空的。

现在，我们给出以下定义。

定义2.1让A类,B类是度量空间的两个非空子集$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src=")">)</trans>$.A映射$\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="B">B类</trans>}$如果存在，则称为Geraghty收缩$\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src="\in ">\in </trans>\mathcal{<trans\; data-src="F">如果</trans>}$这样的话

$$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\cdot ">\cdot </trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans>\text{<trans data-src="for any ">对于任何</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=".">.</trans>$$

请注意，自$\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src=":">:</trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src=")">)</trans><trans\; data-src="\to ">\to </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>$，我们有

$$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\cdot ">\cdot </trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans>\text{<trans data-src="for any ">对于任何</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="A">A类</trans>}\text{<trans data-src=" with ">带有</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\ne ">\ne </trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=".">.</trans>$$

因此，每个Geraghty收缩都是一个收缩映射。

在[10]，作者介绍了以下定义。

定义2.2([10])

让$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src=")">)</trans>$是度量空间的一对非空子集$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src=")">)</trans>$具有${\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\ne ">\ne </trans>\mathrm{<trans\; data-src="\varnothing ">\varnothing </trans>}$然后是这对$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src=")">)</trans>$据说有P（P）-属性当且仅当用于任何${\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$和${\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="B">B类</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$,

$$\begin{array}{lcr}\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans>& <trans\; data-src="=">=</trans>& \mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src=")">)</trans>\\ \mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>& <trans\; data-src="=">=</trans>& \mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src=")">)</trans>\end{array}<trans\; data-src="\}">\}</trans><trans\; data-src="\Rightarrow ">\Rightarrow </trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

很容易看出，对于任何非空子集A类属于$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src=")">)</trans>$，这对$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>$有P（P）-属性。

在[10]，作者证明任何一对$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src=")">)</trans>$实Hilbert空间的非空闭凸子集H（H）满足P（P）-属性。

3主要结果

我们从这一节开始介绍我们的主要结果。

定理3.1 让 $<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src=")">)</trans>$ 是完备度量空间的一对非空闭子集 $<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src=")">)</trans>$ 这样的话 ${\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$ 非空.让 $\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="B">B类</trans>}$ 成为杰拉蒂-收缩满足 $\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\subseteq ">\subseteq </trans>{\mathrm{<trans\; data-src="B">B类</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$.假设这对 $<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src=")">)</trans>$ 有 P（P）-财产.然后存在一个唯一的 ${\mathrm{<trans\; data-src="x">x个</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}$ 在里面 A类 这样的话 $\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src=")">)</trans>$.

证明自${\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$我们认为它不是空的${\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="A">A类</trans>}$.

作为$\mathrm{<trans\; data-src="T">T型</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\subseteq ">\subseteq </trans>{\mathrm{<trans\; data-src="B">B类</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$，我们可以找到${\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$这样的话$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src=")">)</trans>$同样，由于$\mathrm{<trans\; data-src="T">T型</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\subseteq ">\subseteq </trans>{\mathrm{<trans\; data-src="B">B类</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$，存在${\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$这样的话$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src=")">)</trans>$重复这个过程，我们可以得到一个序列$<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans>$在里面${\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$令人满意的

$$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src=")">)</trans>\text{<trans data-src="for any ">对于任何</trans>}\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\in ">\in </trans>\mathbb{<trans\; data-src="N">N个</trans>}<trans\; data-src=".">.</trans>$$

自$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src=")">)</trans>$有P（P）-财产，我们有

$$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans>\text{<trans data-src="for any ">对于任何</trans>}\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\in ">\in </trans>\mathbb{<trans\; data-src="N">N个</trans>}<trans\; data-src=".">.</trans>$$

考虑到这一点T型对任何人来说都是Geraghty的缩写$\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\in ">\in </trans>\mathbb{<trans\; data-src="N">N个</trans>}$，我们有

$$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x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="=">=</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}{\mathrm{<trans\; data-src="x">x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>\mathrm{<trans\; data-src="T">T型</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x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>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\cdot ">\cdot </trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x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>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x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>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

(2)

假设存在${\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\in ">\in </trans>\mathbb{<trans\; data-src="N">N个</trans>}$这样的话$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$.

在这种情况下，

$$\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>$$

因此，$\mathrm{<trans\; data-src="T">T型</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="T">T型</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}}$.

因此，

$$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}}<trans\; data-src=")">)</trans>$$

这就是所期望的结果。

在相反的情况下，假设$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$对于任何$\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\in ">\in </trans>\mathbb{<trans\; data-src="N">N个</trans>}$.

由（2），$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>$是非负实数的递减序列，因此存在$\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="0">0</trans>}$这样的话

$$\underset{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}}{<trans\; data-src="lim">林</trans>}\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src=".">.</trans>$$

在续集中，我们证明了$\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$.

假设$\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$，然后从（2）开始

$$\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="<"><</trans>\frac{\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="1">1</trans>}\text{<trans data-src="for any ">对于任何</trans>}\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\in ">\in </trans>\mathbb{<trans\; data-src="N">N个</trans>}<trans\; data-src=".">.</trans>$$

最后一个不等式意味着${<trans\; data-src="lim">林</trans>}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}}\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}$从那以后$\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src="\in ">\in </trans>\mathcal{<trans\; data-src="F">如果</trans>}$，我们获得$\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$这与我们的假设相矛盾。

因此，

$$\underset{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}}{<trans\; data-src="lim">林</trans>}\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=".">.</trans>$$

(3)

请注意，自$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src=")">)</trans>$对于任何$\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\in ">\in </trans>\mathbb{<trans\; data-src="N">N个</trans>}$，用于$\mathrm{<trans\; data-src="p">对</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="q">q个</trans>}<trans\; data-src="\in ">\in </trans>\mathbb{<trans\; data-src="N">N个</trans>}$修复了，我们有$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="p">对</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="p">对</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="q">q个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="q">q个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src=")">)</trans>$，自$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src=")">)</trans>$满足P（P）-属性，$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="p">对</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="q">q个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="p">对</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="q">q个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans>$.

在下文中，我们证明了$<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans>$是一个柯西序列。

在相反的情况下，我们有

$$\underset{\mathrm{<trans\; data-src="m">米</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}}{<trans\; data-src="lim">林</trans><trans\; data-src="\hspace{0.17em}">\hspace{0.17em}</trans><trans\; data-src="sup">啜饮</trans>}\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=".">.</trans>$$

利用三角不等式，

$$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

由（2）及之后$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src=")">)</trans>$，根据上述评论，我们

$$\begin{array}{rl}\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src=")">)</trans>& <trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x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="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src=")">)</trans>\\ <trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x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="+">+</trans>\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\cdot ">\cdot </trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\end{array}$$

这给了我们

$$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans><trans\; data-src=".">.</trans>$$

自${<trans\; data-src="lim">林</trans><trans\; data-src="\hspace{0.17em}">\hspace{0.17em}</trans><trans\; data-src="sup">啜饮</trans>}_{\mathrm{<trans\; data-src="m">米</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}}\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$以及（3），${<trans\; data-src="lim">林</trans><trans\; data-src="\hspace{0.17em}">\hspace{0.17em}</trans><trans\; data-src="sup">啜饮</trans>}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}}\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$从上一个不等式可以看出

$$\underset{\mathrm{<trans\; data-src="m">米</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}}{<trans\; data-src="lim">林</trans><trans\; data-src="\hspace{0.17em}">\hspace{0.17em}</trans><trans\; data-src="sup">啜饮</trans>}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src=".">.</trans>$$

因此，${<trans\; data-src="lim">林</trans><trans\; data-src="\hspace{0.17em}">\hspace{0.17em}</trans><trans\; data-src="sup">啜饮</trans>}_{\mathrm{<trans\; data-src="m">米</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}}\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}$.

考虑到这一点$\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src="\in ">\in </trans>\mathcal{<trans\; data-src="F">如果</trans>}$，我们得到${<trans\; data-src="lim">林</trans><trans\; data-src="\hspace{0.17em}">\hspace{0.17em}</trans><trans\; data-src="sup">啜饮</trans>}_{\mathrm{<trans\; data-src="m">米</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}}\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$这与我们的假设相矛盾。

因此，$<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans>$是一个柯西序列。

自$<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\subset ">\subset </trans>\mathrm{<trans\; data-src="A">A类</trans>}$和A类是完整度量空间的闭子集$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src=")">)</trans>$，我们可以找到${\mathrm{<trans\; data-src="x">x个</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="A">A类</trans>}$这样的话${\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="x">x个</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}$.

由于任何Geraghty收缩都是收缩映射，因此是连续的，我们有$\mathrm{<trans\; data-src="T">T型</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="T">T型</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}$.

这意味着$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans>$.

考虑到顺序$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>$是具有值的常量序列$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src=")">)</trans>$，我们推断

$$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

这意味着${\mathrm{<trans\; data-src="x">x个</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}$是的最佳接近点T型.

这证明了我们定理存在的部分。

对于唯一性，假设${\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}$和${\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}$是两个最佳接近点T型具有${\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="\ne ">\ne </trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}$.

这意味着

$$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src=")">)</trans>\text{<trans data-src="for ">的</trans>}\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=".">.</trans>$$

使用P（P）-财产，我们有

$$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

利用以下事实T型是Geraghty收缩

$$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\cdot ">\cdot </trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>$$

这是一个矛盾。

因此，${\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}$.

这就完成了证明。 □

4个示例

为了说明我们的结果，我们给出了一些例子。

示例4.1考虑$\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src="=">=</trans>{\mathbb{<trans\; data-src="R">R（右）</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}$使用通常的度量标准。

让A类和B类是的子集X（X）由定义

$$\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="=">=</trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="\}">\}</trans><trans\; data-src="\times ">\times </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src=")">)</trans>\text{<trans data-src="and">和</trans>}\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src="=">=</trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\}">\}</trans><trans\; data-src="\times ">\times </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

显然，$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}$和A类,B类是的非空闭子集X（X）.

此外，很容易看出${\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="A">A类</trans>}$和${\mathrm{<trans\; data-src="B">B类</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="B">B类</trans>}$.

让$\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="B">B类</trans>}$映射定义为

$$\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans><trans\; data-src="ln">自然对数</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\text{<trans data-src="for any ">对于任何</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=".">.</trans>$$

在续集中，我们检查了一下T型是Geraghty收缩。

事实上，对于$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="A">A类</trans>}$具有$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\ne ">\ne </trans>\mathrm{<trans\; data-src="y">年</trans>}$，我们有

$$\begin{array}{rl}\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</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="T">T型</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>& <trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans><trans\; data-src="ln">自然对数</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</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><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans><trans\; data-src="ln">自然对数</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\\ <trans\; data-src="=">=</trans><trans\; data-src="|">|</trans><trans\; data-src="ln">自然对数</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans><trans\; data-src="ln">自然对数</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="|">|</trans>\\ <trans\; data-src="=">=</trans><trans\; data-src="|">|</trans><trans\; data-src="ln">自然对数</trans><trans\; data-src="(">(</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}<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="y">年</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="|">|</trans><trans\; data-src=".">.</trans>\end{array}$$

(4)

现在，我们证明

$$<trans\; data-src="|">|</trans><trans\; data-src="ln">自然对数</trans><trans\; data-src="(">(</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="x">x个</trans>}}{\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="y">年</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="|">|</trans><trans\; data-src="\le ">\le </trans><trans\; data-src="ln">自然对数</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="+">+</trans><trans\; data-src="|">|</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="|">|</trans><trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

(5)

假设$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="y">年</trans>}$（同样的推理适用于$\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="x">x个</trans>}$).

然后，因为$\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="ln">自然对数</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$严格来说是在增加$<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src=")">)</trans>$，我们有

$$<trans\; data-src="ln">自然对数</trans><trans\; data-src="(">(</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="x">x个</trans>}}{\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="y">年</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="ln">自然对数</trans><trans\; data-src="(">(</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="y">年</trans>}}{\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="y">年</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="ln">自然对数</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="+">+</trans>\frac{\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="y">年</trans>}}{\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="y">年</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans><trans\; data-src="ln">自然对数</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="ln">自然对数</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="+">+</trans><trans\; data-src="|">|</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="|">|</trans><trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

这证明了（5）。

考虑到（4）和（5），我们有

$$\begin{array}{rl}\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</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="T">T型</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>& <trans\; data-src="=">=</trans><trans\; data-src="|">|</trans><trans\; data-src="ln">自然对数</trans><trans\; data-src="(">(</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}<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="y">年</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="|">|</trans>\\ <trans\; data-src="\le ">\le </trans><trans\; data-src="ln">自然对数</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="+">+</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>\frac{<trans\; data-src="ln">自然对数</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="+">+</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>\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="\cdot ">\cdot </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>\frac{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</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>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</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>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}<trans\; data-src="\cdot ">\cdot </trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</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>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\\ <trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</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>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\cdot ">\cdot </trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</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>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\end{array}$$

（6）

哪里$\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="ln">自然对数</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$对于$\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="0">0</trans>}$、和$\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\frac{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="t">t吨</trans>}}$对于$\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$和$\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$.

显然，当$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="y">年</trans>}$，满足不等式（6）。

很容易看出$\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\frac{<trans\; data-src="ln">自然对数</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="t">t吨</trans>}}<trans\; data-src="\in ">\in </trans>\mathcal{<trans\; data-src="F">如果</trans>}$通过使用基本微积分。

因此，T型是Geraghty的缩写。

注意这对$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src=")">)</trans>$满足P（P）-属性。

的确，如果

然后${\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}$和${\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}$因此，

$$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="|">|</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="|">|</trans><trans\; data-src="=">=</trans><trans\; data-src="|">|</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="|">|</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

根据定理3.1，T型有一个独特的最佳接近点。

显然，这一点是$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="A">A类</trans>}$.

条件A类和B类是度量空间的非空闭子集$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src=")">)</trans>$不是Geraghty收缩存在唯一最佳邻近点的必要条件$\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="B">B类</trans>}$正如下面的例子所证明的那样。

示例4.2考虑$\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src="=">=</trans>{\mathbb{<trans\; data-src="R">R（右）</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}$具有通常的度量和X（X）由提供

$$\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="=">=</trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="\}">\}</trans><trans\; data-src="\times ">\times </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src=")">)</trans>\text{<trans data-src="and">和</trans>}\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src="=">=</trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\}">\}</trans><trans\; data-src="\times ">\times </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\frac{\mathrm{<trans\; data-src="\pi ">\pi </trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

显然，$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}$和B类不是的闭子集X（X）.

请注意${\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="\times ">\times </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\frac{\mathrm{<trans\; data-src="\pi ">\pi </trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>$和${\mathrm{<trans\; data-src="B">B类</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="B">B类</trans>}$.

我们考虑映射$\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="B">B类</trans>}$定义为

$$\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans><trans\; data-src="arctan">阿卡坦</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\text{<trans data-src="for any ">对于任何</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=".">.</trans>$$

现在，我们检查一下T型是Geraghty收缩。

事实上，对于$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="A">A类</trans>}$具有$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\ne ">\ne </trans>\mathrm{<trans\; data-src="y">年</trans>}$，我们有

$$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans><trans\; data-src="arctan">阿卡坦</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans><trans\; data-src="arctan">阿卡坦</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="|">|</trans><trans\; data-src="arctan">阿卡坦</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="-">-</trans><trans\; data-src="arctan">阿卡坦</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="|">|</trans><trans\; data-src=".">.</trans>$$

(7)

在下文中，我们需要证明

$$<trans\; data-src="|">|</trans><trans\; data-src="arctan">阿卡坦</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="-">-</trans><trans\; data-src="arctan">阿卡坦</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="|">|</trans><trans\; data-src="\le ">\le </trans><trans\; data-src="arctan">阿卡坦</trans><trans\; data-src="|">|</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="|">|</trans><trans\; data-src=".">.</trans>$$

(8)

事实上，假设$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="y">年</trans>}$（同样的论点适用于$\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="x">x个</trans>}$).

放置$<trans\; data-src="arctan">阿卡坦</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}$和$<trans\; data-src="arctan">阿卡坦</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="\beta ">\beta </trans>}$（请注意$\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="\beta ">\beta </trans>}$自从函数$\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="arctan">阿卡坦</trans>\mathrm{<trans\; data-src="t">t吨</trans>}$对于$\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="0">0</trans>}$正在严格增加）。

考虑到这一点

$$<trans\; data-src="tan">棕褐色的</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\frac{<trans\; data-src="tan">棕褐色的</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">-</trans><trans\; data-src="tan">棕褐色的</trans>\mathrm{<trans\; data-src="\beta ">\beta </trans>}}{\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="+">+</trans><trans\; data-src="tan">棕褐色的</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="\cdot ">\cdot </trans><trans\; data-src="tan">棕褐色的</trans>\mathrm{<trans\; data-src="\beta ">\beta </trans>}}$$

从那以后$\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\frac{\mathrm{<trans\; data-src="\pi ">\pi </trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>$，我们有$<trans\; data-src="tan">棕褐色的</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src=",">,</trans><trans\; data-src="tan">棕褐色的</trans>\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src=")">)</trans>$因此，从上一个不等式可以看出

$$<trans\; data-src="tan">棕褐色的</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans><trans\; data-src="tan">棕褐色的</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">-</trans><trans\; data-src="tan">棕褐色的</trans>\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src=".">.</trans>$$

应用ϕ（请注意$\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="arctan">阿卡坦</trans>\mathrm{<trans\; data-src="t">t吨</trans>}$)最后一个不等式，并考虑到ϕ，我们有

$$\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src="\le ">\le </trans><trans\; data-src="arctan">阿卡坦</trans><trans\; data-src="(">(</trans><trans\; data-src="tan">棕褐色的</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">-</trans><trans\; data-src="tan">棕褐色的</trans>\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>$$

或同等标准，

$$<trans\; data-src="arctan">阿卡坦</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="-">-</trans><trans\; data-src="arctan">阿卡坦</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src="\le ">\le </trans><trans\; data-src="arctan">阿卡坦</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>$$

这证明了（8）。

通过（7）和（8），我们得到

$$\begin{array}{rl}\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</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="T">T型</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>& <trans\; data-src="=">=</trans><trans\; data-src="|">|</trans><trans\; data-src="arctan">阿卡坦</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="-">-</trans><trans\; data-src="arctan">阿卡坦</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="|">|</trans>\\ <trans\; data-src="\le ">\le </trans><trans\; data-src="arctan">阿卡坦</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>\frac{<trans\; data-src="arctan">阿卡坦</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>\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="\cdot ">\cdot </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>\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</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="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\cdot ">\cdot </trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</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>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\end{array}$$

(9)

哪里$\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\frac{<trans\; data-src="arctan">阿卡坦</trans>\mathrm{<trans\; data-src="t">t吨</trans>}}{\mathrm{<trans\; data-src="t">t吨</trans>}}$对于$\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$和$\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$显然，不等式（9）满足$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="A">A类</trans>}$具有$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="y">年</trans>}$.

现在，我们证明$\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src="\in ">\in </trans>\mathcal{<trans\; data-src="F">如果</trans>}$.

事实上，如果$\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\frac{<trans\; data-src="arctan">阿卡坦</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}}{{\mathrm{<trans\; data-src="t">t吨</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="1">1</trans>}$，然后是序列$<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans>$是一个有界序列，因为在相反的情况下，${\mathrm{<trans\; data-src="t">t吨</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}$因此$\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="0">0</trans>}$。假设${\mathrm{<trans\; data-src="t">t吨</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="\nrightarrow ">\nrightarrow </trans>\mathrm{<trans\; data-src="0">0</trans>}$。这意味着存在$\mathrm{<trans\; data-src="ϵ">ϵ</trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$这样，对于每个$\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\in ">\in </trans>\mathbb{<trans\; data-src="N">N个</trans>}$，存在${\mathrm{<trans\; data-src="p">对</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="n">n个</trans>}$具有${\mathrm{<trans\; data-src="t">t吨</trans>}}_{{\mathrm{<trans\; data-src="p">对</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="ϵ">ϵ</trans>}$.的有界字符$<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans>$给出了子序列的存在性$<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}_{{\mathrm{<trans\; data-src="k">k个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}}<trans\; data-src=")">)</trans>$属于$<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}_{{\mathrm{<trans\; data-src="p">对</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}}<trans\; data-src=")">)</trans>$具有$<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}_{{\mathrm{<trans\; data-src="k">k个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}}<trans\; data-src=")">)</trans>$收敛的。假设${\mathrm{<trans\; data-src="t">t吨</trans>}}_{{\mathrm{<trans\; data-src="k">k个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="a">一</trans>}$.来自$\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="1">1</trans>}$，我们获得$\frac{<trans\; data-src="arctan">阿卡坦</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}_{{\mathrm{<trans\; data-src="k">k个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}}}{{\mathrm{<trans\; data-src="t">t吨</trans>}}_{{\mathrm{<trans\; data-src="k">k个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}}}<trans\; data-src="\to ">\to </trans>\frac{<trans\; data-src="arctan">阿卡坦</trans>\mathrm{<trans\; data-src="a">一</trans>}}{\mathrm{<trans\; data-src="a">一</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}$并且，作为$<trans\; data-src="arctan">阿卡坦</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="x">x个</trans>}$是${\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$，我们获得$\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$.

因此，${\mathrm{<trans\; data-src="t">t吨</trans>}}_{{\mathrm{<trans\; data-src="k">k个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="0">0</trans>}$这与事实相矛盾${\mathrm{<trans\; data-src="t">t吨</trans>}}_{{\mathrm{<trans\; data-src="k">k个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="ϵ">ϵ</trans>}$对于任何$\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\in ">\in </trans>\mathbb{<trans\; data-src="N">N个</trans>}$.

因此，${\mathrm{<trans\; data-src="t">t吨</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="0">0</trans>}$这证明了$\mathrm{<trans\; data-src="\beta ">\beta </trans>}<trans\; data-src="\in ">\in </trans>\mathcal{<trans\; data-src="F">如果</trans>}$.

与示例4.1中使用的参数类似的参数证明了$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src=")">)</trans>$有P（P）-属性。

另一方面，关键是$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="A">A类</trans>}$是最佳接近点T型自从

$$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans><trans\; data-src="arctan">阿卡坦</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

此外，$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans>$是唯一的最佳接近点T型.

的确，如果$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="A">A类</trans>}$是最佳接近点T型，然后

$$\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="(">(</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans><trans\; data-src="arctan">阿卡坦</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\sqrt{\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="+">+</trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="-">-</trans><trans\; data-src="arctan">阿卡坦</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="2">2</trans>}}}<trans\; data-src=",">,</trans>$$

这给了我们

$$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="=">=</trans><trans\; data-src="arctan">阿卡坦</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=".">.</trans>$$

考虑到这个方程的唯一解是$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$，我们已经证明了T型具有唯一的最佳接近点，即$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans>$.

注意，在这种情况下B类未关闭。

由于对于任何非空子集A类属于X（X），这对$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>$满足P（P）-属性，我们有以下推论。

推论4.1 让 $<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src=")">)</trans>$ 是一个完整的度量空间，并且 A类 是X的非空闭子集.让 $\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="A">A类</trans>}$ 成为杰拉蒂-收缩.然后 T型 具有唯一的固定点.

证明当$\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="B">B类</trans>}$，预期结果如下。 □

请注意，当$\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="X">X（X）</trans>}$，推论4.1是由Gerahty引起的定理1.1[1].