非线性泛函分析及其应用（第1卷）：不动点定理

首先，这部书讲清楚了泛函分析理论对数学其他领域的应用。例如，第2A卷讲述线性单调算子。他从椭圆型方程的边值问题出发，讲问题的古典解，由于具体物理背景的需要，问题须作进一步推广，而需要讨论问题的广义解。这种方法背后的分析原理是什么?其实就是完备化思想的一个应用！将古典问题所依赖的连续函数空间，完备化成为Sobolev空间，则可讨论问题的广义解。在这种讨论中间，我们可以看到Hilbert空间的作用。书中不仅有这种理论讨论，而且还讲了怎样计算问题的近似解（Ritz方法）。

其次，这部书讲清楚了分析理论在诸多领域（如物理学、化学、生物学、工程技术和经济学等等）的广泛应用。例如，第3卷讲解变分方法和优化，它从函数极值问题开始，讲到变分问题及其对于Euler微分方程和Hammerstein积分方程的应用；讲到优化理论及其对于控制问题（如庞特里亚金极大值原理）、统计优化、博弈论、参数识别、逼近论的应用；讲了凸优化理论及应用；讲了极值的各种近似计算方法。比如第4卷，讲物理应用，写作原理是：由物理事实到数学模型；由数学模型到数学结果；再由数学结果到数学结果的物理解释；最后再回到物理事实。

再次，该书由浅人深地讲透了基本理论的发展历程及走向，它既讲清楚了所涉及学科的具体问题，也讲清楚了其背后的数学原理及其作用。数学理论讲得也非常深入，例如，不动点理论，就从Banach不动点定理讲到Schauder不动点定理，以及Bourbaki—Kneser不动点定理等等。

这套书的写作起点很低，具备本科数学水平就可以读；应用都是从最简单情形入手，应用领域的读者也可以读；全书材料自足，各部分又尽可能保持独立；书后附有极其丰富的参考文献及一些文献评述；该书文字优美，引用了许多大师的格言，读之你会深受启发。这套书的优点不胜枚举，每个与数理学科相关的人，搞理论的，搞应用的，搞研究的，搞教学的，都可读该书，哪怕只是翻一翻，都不会空手而返！

PrefacetotheSecondCorrectedPrinting
PrefacetotheFirstPrinting
Introduction
FUNDAMENTALFIXED-POINTPRINCIPLES
CHAPTER1
TheBanachFixed-PointTheoremandlterativeMethods
1.1.TheBanachFixed-PointTheorem
1.2.ContinuousDependenceonaParameter
1.3.TheSignificanceoftheBanachFixed-PointTheorem
1.4.ApplicationstoNonlinearEquations
1.5.AcceleratedConvergenceandNewtonsMethod
1.6.ThePicard-Lindel6fTheorem
1.7.TheMainTheoremforIterativeMethodsforLinearOperator
Equations
1.8.ApplicationstoSystemsofLinearEquations
1.9.ApplicationstoLinearIntegralEquations

CHAPTER2
TheSchauderFixed-PointTheoremandCompactness
2.1.ExtensionTheorem
2.2.Retracts
2.3.TheBrouwerFixed-PointTheorem
2.4.ExistencePrincipleforSystemsofEquations
2.5.CompactOperators
2.6.TheSchauderFixed-PointTheorem
2.7.PeanosTheorem
2.8.IntegralEquationswithSmallParameters
2.9.SystemsofIntegralEquationsandSemilinearDifferential
Equations
2.10.AGeneralStrategy
2.11.ExistencePrincipleforSystemsofInequalities
APPLICATIONSOFTHEFUNDAMENTAL
FIXED-POINTPRINCIPLES

CHAPTER3
OrdinaryDifferentialEquationsinB-spaces
3.1.IntegrationofVectorFunctionsofOneRealVariablet
3.2.DifferentiationofVectorFunctionsofOneRealVariablet
3.3.GeneralizedPicard-LindeltfTheorem
3.4.GeneralizedPeanoTheorem
3.5.GronwalrsLemma
3.6.StabilityofSolutionsandExistenceofPeriodicSolutions
3.7.StabilityTheoryandPlaneVectorFields,ElectricalCircuits,
LimitCycles
3.8.Perspectives

CHAPTER4
DifferentialCalculusandtheImplicitFunctionTheorem
4.1.FormalDifferentialCalculus
4.2.TheDerivativesofFrtchetandGiteaux
4.3.SumRule,ChainRule,andProductRule
4.4.PartialDerivatives
4.5.HigherDifferentialsandHigherDerivatives
4.6.GeneralizedTaylorsTheorem
4.7.TheImplicitFunctionTheorem
4.8.ApplicationsoftheImplicitFunctionTheorem
4.9.AttractingandRepellingFixedPointsandStability
4.10.ApplicationstoBiologicalEquilibria
4.11.TheContinuouslyDifferentiableDependenceoftheSolutionsof
OrdinaryDifferentialEquationsinB-spacesontheInitialValues
andontheParameters
4.12.TheGeneralizedFrobeniusTheoremandTotalDifferential
Equations
4.13.DiffeomorphismsandtheLocalInverseMappingTheorem
4.14.ProperMapsandtheGlobalInverseMappingTheorem
4.15.TheSurjectiveImplicitFunctionTheorem
4.16.NonlinearSystemsofEquations,Subimmersions,andtheRank
Theorem
4.17.ALookatManifolds
4.18.SubmersionsandaLookattheSard-SmaleTheorem
4.19.TheParametrizedSardTheoremandConstructiveFixed-Point
Theory

CHAPTER5
NewtonsMethod
5.1.ATheoremonLocalConvergence
5.2.TheKantoroviSemi-LocalConvergenceTheorem

CHAPTER6
ContinuationwithRespecttoaParameter
6.1.TheContinuationMethodforLinearOperators
6.2.B-spacesofH61derContinuousFunctions
6.3.ApplicationstoLinearPartialDifferentialEquations
6.4.Functional-AnalyticInterpretationoftheExistenceTheoremand
itsGeneralizations
6.5.ApplicationstoSemi-linearDifferentialEquations
6.6.TheImplicitFunctionTheoremandtheContinuationMethod
6.7.OrdinaryDifferentialEquationsinB-spacesandtheContinuation
Method
6.8.TheLeray-SchauderPrinciple
6.9.ApplicationstoQuasi-linearEllipticDifferentialEquations

CHAPTER7
PositiveOperators
7.I.OrderedB-spaces
7.2.MonotoneIncreasingOperators
7.3.TheAbstractGronwallLemmaanditsApplicationstoIntegral
Inequalities
7.4.Supersolutions,Subsolutions,IterativeMethods,andStability
7.5.Applications
7.6.MinorantMethodsandPositiveEigensolutions
7.7.Applications
7.8.TheKrein-RutmanTheoremanditsApplications
7.9.AsymptoticLinearOperators
7.10.MainTheoremforOperatorsofMonotoneType
7.11.ApplicationtoaHeatConductionProblem
7.12.ExistenceofThreeSolutions
7.13.MainTheoremforAbstractHammersteinEquationsinOrdered
B-spaces
7.14.EigensolutionsofAbstractHammersteinEquations,Bifurcation,
Stability,andtheNonlinearKrein-RutmanTheorem
7.15.ApplicationstoHammersteinIntegralEquations
7.16.ApplicationstoSemi-linearEllipticBoundary-ValueProblems
7.17.ApplicationtoEllipticEquationswithNonlinearBoundary
Conditions
7.18.ApplicationstoBoundaryInitial-ValueProblemsforParabolic
DifferentialEquationsandStability

CHAPTER8
AnalyticBifurcationTheory
8.1.ANecessaryConditionforExistenceofaBifurcationPoint
8.2.AnalyticOperators
8.3.AnAnalyticMajorantMethod
8.4.FredholmOperators
8.5.TheSpectrumofCompactLinearOperators
（Riesz-SchauderTheory）
8.6.TheBranchingEquationsofLjapunov-Schmidt
8.7.TheMainTheoremontheGenericBifurcationFromSimpleZeros
8.8.ApplicationstoEigenvalueProblems
8.9.ApplicationstoIntegralEquations
8.10.ApplicationtoDifferentialEquations
8.11.TheMainTheoremonGenericBifurcationforMultiparametric
OperatorEquations——TheBunchTheorem
8.12.MainTheoremforRegularSemi-linearEquations
8.13.Parameter-InducedOscillation
8.14.Self-InducedOscillationsandLimitCycles
8.15.HopfBifurcation
8.16.TheMainTheoremonGenericBifurcationfromMultipleZeros
8.17.StabilityofBifurcationSolutions
8.18.GenericPointBifurcation

CHAPTER9
FixedPointsofMultivaluedMaps
9.1.GeneralizedBanachFixed-PointTheorem
9.2.UpperandLowerSemi-continuityofMultivaluedMaps
9.3.GeneralizedSchauderFixed-PointTheorem
9.4.VariationalInequalitiesandtheBrowderFixed-PointTheorem
9.5.AnExtremalPrinciple
9.6.TheMinimaxTheoremandSaddlePoints
9.7.ApplicationsinGameTheory
9.8.SelectionsandtheMarriageTheorem
……
CHAPTER10
CHAPTER11
CHAPTER12
CHAPTER13
CHAPTER14
CHAPTER15
CHAPTER16
CHAPTER17
Index