压缩映射原理的性质和应用

I压缩映射原理的性质和应用摘要本文较有系统的研究了压缩映射原理及其一些应用，由于压缩映射原理是属于不动点理论中的一类原理，所以有许多不同的形式，本文主要利用在常规度量空间中讨论压缩映射原理的方法，在概率度量空间中讨论压缩映射原理。主要内容如下：第一章，是绪论部分，首先讲了我之所以写这篇文章的原因，然后是本文所研究问题的历史背景和发展情况。第二章，介绍压缩映射原理的最基本的形式，即Banach压缩映射原理，通过对其定理内容和证明方法的分析，深刻认识了Picard迭代方法在证明中起到的重要作用，总结出了一套通用的方法证明这类定理，还找了一个例子，用总结出的方法进行了证明。第三章，用第一章总结出的方法研究了压缩映射原理更复杂的形式，随着研究问题的复杂，也使第一章总结出的方法变得更加完善。第四章，把前几章得到的结论和方法应用到了微分方程和微分方程组的解的存在唯一性上。虽然只有两个例子，但是获得方法和思想可以用到许多其他的例子上。第五章，引入概率度量空间的概念，和其中一系列与压缩映射原理有关的概念，结合概率度量空间的一些特殊性质，用前几章的讨论方法，在概率度量空间上讨论压缩映射原理，依次讨论了含随机数的压缩映射原理，在概率度量空间上添加一些条件后的基本压缩映射原理，非线性的压缩映射原理及应用等。关键词：压缩映射；不动点；概率度量空间；非线性微分方程IIABSTRACTInthispaper,asystematicstudyofthecompressionmappingprincipleandsomeapplications,becauseofthecontractionmappingtheoryisoneoftheprincipleinbelongtothetheoryoffixedpoint,sotherearemanydifferentforms,thispapermainlydiscussedusedinconventionalmetricspacecompressionmappingprinciple,themethodofcontractivemappingprincipleinprobabilisticmetricspace.Themaincontentsareasfollows:Thefirstchapteristheintroductionpart,firstofalltellthereasonwhyIwritethisarticle,andthenthispaperstudiesthehistoricalbackgroundanddevelopmentoftheproblem.Thesecondchapter,thispaperintroducesthebasicformofcompressionmappingprinciple,namelythecontractionmappingtheory,throughtheanalysisofitsproofcontentandmethods,understandingtheiterationmethodplaysanimportantroleinproof,summarizesasetofgenericmethodstoprovethistheorem,stilllookingforanexample,summarizesthewayhascarriedontheproof.Thethirdchapter,inthefirstchaptersummarizesthemethodofcompressionmappingprincipleisstudiedintheformofmorecomplex,astheresearchproblemofcomplex,alsomadethefirstchaptersummarizesthemethodsbecomemoreperfect.Thefourthchapter,inthepreviouschapterconclusionandmethodisappliedtotheexistenceanduniquenessofsolutionofdifferentialequationanddifferentialequations.Althoughonlytwoexamples,methodsandthoughtscanbeusedonmanyotherexamples.Thefifthchapter,theintroductionoftheconceptofprobabilisticmetricSpaces,andaseriesofconceptsrelatedtothecontractionmappingtheory,combinedwithsomespecialpropertiesoftheprobabilisticmetricSpaces,theuseofthepreviouschaptersdiscussmethod,compressionmappingsinprobabilisticmetricspaceprinciple,inordertodiscussthecompressionmappingprinciple,containingtherandomnumberafteraddingsomeconditionsinprobabilisticmetricspacebasiccompressionmappingprinciple,theprincipleandapplicationofthecompressionofnonlinearmapping,etc.Keywords:compressionmapping;Thefixedpoint.Probabilisticmetricspace;ThenonlineardifferentialequationIII目录摘要................................................................................................................................IABSTRACT..................................................................................................................II第一章绪论...................................................................................................................11.1写作动机..........................................................................................................11.2不动点理论背景知识，历史渊源..................................................................21.3压缩映射原理的简介......................................................................................3第二章Banach压缩映射定理的证明思路探究..........................................................62.1定理内容和证明..............................................................................................62.2一个例子..........................................................................................................62.3本章总结..........................................................................................................8第三章Banach压缩映射原理的推广........................................................................103.1推广的背景：................................................................................................103.2压缩映射原理的一种推广形式及其证明....................................................103.3本章总结.........................................................................................................12第四章压缩映射原理的应用举例.............................................................................134.1一类简单积分方程的解的存在与唯一性的证明........................................134.2积分方程组的解的存在与唯一性证明........................................................144.3本章总结........................................................................................................16第五章概率度量空间中的压缩映射原理.................................................................175.1基本概念的构造............................................................................................175.2随机压缩映射原理的构造............................................................................175.3概率度量空间的背景知识............................................................................195.4概率度量空间中的基本概念........................................................................195.5：t范数的概念及其性质...........................................................................215.6概率度量空间上的压缩映射原理................................................................215.7概率度量空间上非线性的压缩映射原理.............................................