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摘 要

本文首先介绍Banach空间中的不动点定理、在其他线性拓扑空间中不动点定理的一维推广形式、在一般完备度量空间上的推广形式. 其次,通过分析近几年全国各地高考数学卷中一些试题特点,总结了利用不动点定理求解有关数列的问题.其中包括数列通项、数列的有界性问题.最后介绍了不动点定理中的吸引不动点和排斥不动点在讨论数列的单调性及收敛性方面的应用.

关键词 ：Banach不动点定理,数列通项，有界性，单调性，收敛性.

Abstract

This article firstly introduced the Fixpoint Theorem in Banach space, the HYPERLINK "app:ds:one-dimensional" \t "" one-dimensional extended form of the Fixpoint Theorem in other linear topological space and the extended form in general complete metric space. Then, we summarized the problem on sequence of number using Fixpoint Theorem, analyzing the characteristics of tests emerged on math papers of all parts of our country recent years, including the problem of general term and boundedness of a sequence of number. At last, attractive fix point and rejection fix point in Fixpoint Theorem were introduced which can solve the problem about the monotonicity and astringency of sequence of number.

Keywords:Banach fixed point theorem, Sequence, Boundedness, Monotonicity Convergence.

目 录

TOC \o "1-3" \h \z \u HYPERLINK \l "_Toc261278972" 第1章 绪论 PAGEREF _Toc261278972 \h 1

HYPERLINK \l "_Toc261278973" 1.1导论 PAGEREF _Toc261278973 \h 1

HYPERLINK \l "_Toc261278974" 1.1.1 选题背景 PAGEREF _Toc261278974 \h 1

HYPERLINK \l "_Toc261278975" 1.1.2 选题意义2

HYPERLINK \l "_Toc261278976" 1.1.3 课题研究内容 PAGEREF _Toc261278976 \h 2

HYPERLINK \l "_Toc261278978" 1.2 研究现状2

HYPERLINK \l "_Toc261278982" 1.3本章小结3

HYPERLINK \l "_Toc261278983" 第2章 不动点定理4

HYPERLINK \l "_Toc261278984" 2.1 有关概念4

HYPERLINK \l "_Toc261278987" 2.2 不动点定理和几种推广形式4

HYPERLINK \l "_Toc261278989" 2.3 本章小结7

HYPERLINK \l "_Toc261278990" 第3章 不动点定理在数列中的应用8

HYPERLINK \l "_Toc261278991" 3.1 求数列的通项公式8

HYPERLINK \l "_Toc261278992" 3.2 数列的有界性9

HYPERLINK \l "_Toc261278993" 3.3 数列的单调性及收敛性11

HYPERLINK \l "_Toc261278995" 3.3.1数列的单调性、收敛性的重要结论11

HYPERLINK \l "_Toc261278996" 3.3.2数列的单调性、收敛性的证明14

HYPERLINK \l "_Toc261279001" 3.4 本章小结17

HYPERLINK \l "_Toc261279035" 第6章 结束语18

HYPERLINK \