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1、NotesChapter1Section1.1containsthebasicdifferentialcalculusonBanachspaces;thema-terialcanbefoundinanynonlinearfunctionalanalysisbook,forinstance,Schwartz[Scw],Nirenberg[Ni1],Deimling[De],Zeidler[Zei],Berger[Ber1]etc.ThediscussiononNemytsckioperatorcanbefoundinVainberg[Va],buttheproof
2、ismuchsimplerthanthatinthereference.Thepresentationsoftheimplicitfunctiontheoremandtheinversefunc-tiontheoremarestandard.Interestingapplicationsarescatteredintheliter-ature,forinstance,Nirenberg[Ni4],Kazdan[Ka],Chow,Hale[CH],Mawhin[Maw3]etc.Thecontinuitymethodisextensivelyusedinthee
3、xistenceproofofdifferentialequations.TheglobalimplicitfunctiontheoremisduetoHadamard[Ha],Caccioppoli[Cac1];extensionscanbefoundinBrowder[Bd1]andPlastock[Pl].However,theproofpresentedhereisverydifferent.Theapplicationofthecontinuitymethodtoanaprioriboundasasimplerproofforsemi-linearell
4、ipticequationswithquadraticgrowthwasgivenbyAmannandCrandall[AC];theresultforquasi-linearellipticequationscanbefoundinLadyzenskayaandUral’zeva[LU].TheLyapunov–Schmidtreductionisextensivelyusedinnonlinearprob-lems.TheapplicationtothestudyofbifurcationproblemsintroducedhereisduetoCrand
5、allandRabinowitz[CR1],[CR2].ThereferencesonbifurcationtheoryarerecommendedtoChowandHale[CH].Sections1.3.3–1.3.4provideexamplesongluing,whichisanimportanttechniqueinsymplecticgeometry.ThematerialhereistakenfromFloerandWeinstein[FW]andOh[Oh].FurtherreferencesareFloer[Fl1][Fl2],Hoferan
6、dZehnder[HZ2].Paralleltotheimplicitfunctiontheoremmethod,aglu-ingtechniqueviathevariationalmethodwasdevelopedbySere[Se];ithasbeenappliedtothehomoclinicorbitsinHamiltoniansystems,multi-bumpsolutions,andmulti-peaksolutionsforellipticdifferentialequations,seealsoCotiZelatiandRabinowitz[
7、CR],Li[Li1],Gui[Gu]etc.Section1.3.5isan-otherimportanttechniqueinapplyingtheimplicitfunctiontheorem.The420Notesnotionoftransversalityistakenfromdifferentialgeometry.CombiningwiththeSardtheorem,itprovidesamethodforprovingvariousgenerictypere-sults.Thefinite-dimensionalformofthetransver
8、salitytheoremcanbef