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.AbstractThisarticlefirstlyintroducedtheFixpointTheoreminBanachspace,theone-dimensionalextendedformoftheFixpointTheoreminotherlineartopologicalspaceandtheextendedformingeneralcompletemetricspace.Then,wesummarizedtheproblemonsequenceofnumberusingFixpointTheorem,analyzingthecharacteristicsoftestsemergedonmathpapersofallpartsofourcountryrecentyears,includingtheproblemofgeneraltermandboundednessofasequenceofnumber.Atlast,attractivefixpointandrejectionfixpointinFixpointTheoremwereintroducedwhichcansolvetheproblemaboutthemonotonicityandastringencyofsequenceofnumber.Keywords:Banachfixedpointtheorem,Sequence,Boundedness,MonotonicityConvergence.I目录第1章绪论..............................................................................................................................11.1导论.............................................................................................................................11.1.1选题背景..........................................................................................................11.1.2选题意义..........................................................................................................21.1.3课题研究内容..................................................................................................21.2研究现状....................................................................................................................21.3本章小结.....................................................................................................................3第2章不动点定理..................................................................................................................42.1有关概念....................................................................................................................42.2不动点定理和几种推广形式.....................................................................................42.3本章小结....................................................................................................................7第3章不动点定理在数列中的应用......................................................................................83.1求数列的通项公式.....................................................................................................83.2数列的有界性............................................................................................................93.3数列的单调性及收敛性...........................................................................................113.3.1数列的单调性、收敛性的重要结论............................................................113.3.2数列的单调性、收敛性的证明....................................................................143.4本章小结..................................................................................................................17第6章结束语.........................................................................................................
