非线性泛函分析及其应用(第1卷):不动点定理
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作者[德]宰德勒 著
出版社世界图书出版公司
出版时间2009-08
版次1
装帧精装
货号B923
上书时间2024-05-19
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- 品相描述:九品
图书标准信息
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作者
[德]宰德勒 著
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出版社
世界图书出版公司
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出版时间
2009-08
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版次
1
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ISBN
9787510005190
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定价
99.00元
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装帧
精装
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开本
24开
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纸张
胶版纸
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页数
909页
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正文语种
英语
- 【内容简介】
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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!
- 【目录】
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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
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