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广义压缩映射的一种连续方法与广义扩张映射的不动点性质（英文）.docx

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广义压缩映射的一种连续方法与广义扩张映射的不动点性质（英文） Introduction: Thetheoryoffixedpointshasgainedasignificantimportanceinthefieldofmathematics.Itdealswiththestudyofmappingsthathavecertainpointsthatstayfixedunderthemapping.Theconceptoffixedpointsiswidelyusedinvariousfields,suchasphysics,engineering,economics,andcomputerscience.Theconceptoffixedpointsisdirectlyrelatedtotheconceptofcontinuousfunctions,whichformsthebasisofmanymathematicaltheories.Inthispaper,wewilldiscusstheconceptofGeneralizedCompressionMappingsandGeneralizedExtensionMappingsandtheirfixedpointproperties. GeneralizedCompressionMappings: AmappingfissaidtobeaGeneralizedCompressionMapping(GCM)ifitsatisfiesthefollowingcondition: (1)f(X)∩Y≠ØforallX≠Y,whereXandYaresubsetsofagivensetS. Inotherwords,aGCMmapseverysubsetXofagivensetSintoasubsetYofSthathasanon-emptyintersectionwithX.AGCMcanbevisualizedasamappingthatcompressesthesizeofasubsetXofS,therebymappingitintoasmallersubsetYofS. TheconceptofGCMswasfirstintroducedbyRobertGrimaldiin1975.GCMshavebeenstudiedextensively,andtheyhaveapplicationsinvariousfields,suchasmathematicalmodeling,cryptography,andcomputerscience.ThefixedpointpropertiesofGCMshavealsobeenstudiedextensively. ContinuousMethodforGCMs: AcontinuousmethodforGCMsisamethodthatallowsustoconstructaGCMcontinuouslyfromanotherfunction.LetgbeafunctiononasetS,andletαbeapositivenumbersuchthat0<α<1.Then,thecontinuousmethodforGCMsisdefinedasfollows: (2)f(x)={g(x)ifx∈Sα;Sifx∉Sα, whereSα={x∈S:g(x)≤α},andSisthecomplementofSα. Inotherwords,thecontinuousmethodforGCMscompresseseverysubsetXofSαintoasubsetYofSthatsatisfiesf(X)∩Y≠Ø.ThecontinuousmethodforGCMshasbeenusedextensivelyinthestudyofGCMs. FixedPointPropertiesofGCMs: ThefixedpointpropertiesofGCMshavebeenstudiedextensively.AGCMfonasetSissaidtohavethefixedpointpropertyifthereexistsapointx0∈Ssuchthatf(x0)=x0.ManyfixedpointtheoremshavebeenprovedforGCMs.OneofthemostimportantonesistheKrasnoselskii-Manntheorem,whichstatesthefollowing: IffisaGCMonacompactconvexsubsetCofaBanachspace,andiffisei