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非線性泛函分析及其應(yīng)用·第1卷:不動(dòng)點(diǎn)定理

非線性泛函分析及其應(yīng)用·第1卷:不動(dòng)點(diǎn)定理

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作 者: (德)宰德勒 著
出版社: 世界圖書(shū)出版公司
叢編項(xiàng):
標(biāo) 簽: 函數(shù)

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ISBN: 9787510005190 出版時(shí)間: 2009-08-01 包裝: 精裝
開(kāi)本: 24開(kāi) 頁(yè)數(shù): 909 字?jǐn)?shù):  

內(nèi)容簡(jiǎn)介

  首先,這部書(shū)講清楚了泛函分析理論對(duì)數(shù)學(xué)其他領(lǐng)域的應(yīng)用。例如,第2A卷講述線性單調(diào)算子。他從橢圓型方程的邊值問(wèn)題出發(fā),講問(wèn)題的古典解,由于具體物理背景的需要,問(wèn)題須作進(jìn)一步推廣,而需要討論問(wèn)題的廣義解。這種方法背后的分析原理是什么?其實(shí)就是完備化思想的一個(gè)應(yīng)用!將古典問(wèn)題所依賴的連續(xù)函數(shù)空間,完備化成為Sobolev空間,則可討論問(wèn)題的廣義解。在這種討論中間,我們可以看到Hilbert空間的作用。書(shū)中不僅有這種理論討論,而且還講了怎樣計(jì)算問(wèn)題的近似解(Ritz方法)。其次,這部書(shū)講清楚了分析理論在諸多領(lǐng)域(如物理學(xué)、化學(xué)、生物學(xué)、工程技術(shù)和經(jīng)濟(jì)學(xué)等等)的廣泛應(yīng)用。例如,第3卷講解變分方法和優(yōu)化,它從函數(shù)極值問(wèn)題開(kāi)始,講到變分問(wèn)題及其對(duì)于Euler微分方程和Hammerstein積分方程的應(yīng)用;講到優(yōu)化理論及其對(duì)于控制問(wèn)題(如龐特里亞金極大值原理)、統(tǒng)計(jì)優(yōu)化、博弈論、參數(shù)識(shí)別、逼近論的應(yīng)用;講了凸優(yōu)化理論及應(yīng)用;講了極值的各種近似計(jì)算方法。比如第4卷,講物理應(yīng)用,寫(xiě)作原理是:由物理事實(shí)到數(shù)學(xué)模型;由數(shù)學(xué)模型到數(shù)學(xué)結(jié)果;再由數(shù)學(xué)結(jié)果到數(shù)學(xué)結(jié)果的物理解釋;最后再回到物理事實(shí)。再次,該書(shū)由淺人深地講透了基本理論的發(fā)展歷程及走向,它既講清楚了所涉及學(xué)科的具體問(wèn)題,也講清楚了其背后的數(shù)學(xué)原理及其作用。數(shù)學(xué)理論講得也非常深入,例如,不動(dòng)點(diǎn)理論,就從Banach不動(dòng)點(diǎn)定理講到Schauder不動(dòng)點(diǎn)定理,以及Bourbaki—Kneser不動(dòng)點(diǎn)定理等等。這套書(shū)的寫(xiě)作起點(diǎn)很低,具備本科數(shù)學(xué)水平就可以讀;應(yīng)用都是從最簡(jiǎn)單情形入手,應(yīng)用領(lǐng)域的讀者也可以讀;全書(shū)材料自足,各部分又盡可能保持獨(dú)立;書(shū)后附有極其豐富的參考文獻(xiàn)及一些文獻(xiàn)評(píng)述;該書(shū)文字優(yōu)美,引用了許多大師的格言,讀之你會(huì)深受啟發(fā)。這套書(shū)的優(yōu)點(diǎn)不勝枚舉,每個(gè)與數(shù)理學(xué)科相關(guān)的人,搞理論的,搞應(yīng)用的,搞研究的,搞教學(xué)的,都可讀該書(shū),哪怕只是翻一翻,都不會(huì)空手而返!

作者簡(jiǎn)介

暫缺《非線性泛函分析及其應(yīng)用·第1卷:不動(dòng)點(diǎn)定理》作者簡(jiǎn)介

圖書(shū)目錄

Preface to the Second Corrected Printing
Preface to the First Printing
Introduction
FUNDAMENTAL FIXED-POINT PI~INCIPLES
 CHAPTER I The Banach Fixed-Point Theorem and Iterative Methods
 1.1. The Banach Fixed-Point Theorem
  1.2. Continuous Dependence on a Parameter
  1.3. The Significance of the Banach Fixed-Point Theorem
  1.4. Applications to Nonlinear Equations
  1.5. Accelerated Convergence and Newton's Method
  1.6. The Picard-Lindel6f Theorem
  1.7. The Main Theorem for Iterative Methods for Linear Operator Equations
  1.8. Applications to Systems of Linear Equations
  1.9. Applications to Linear Integral Equations
 CHAPTER 2 The Schauder Fixed-Point Theorem and Compactness
  2.1. Extension Theorem
  2.2. Retracts
  2.3. The Brouwer Fixed-Point Theorem
  2.4. Existence Principle for Systems of Equations
  2.5. Compact Operators
  2.6. The Schauder Fixed-Point Theorem
  2.7. Peano's Theorem
  2.8. Integral Equations with Small Parameters
  2.9. Systems of Integral Equations and Semilinear Differential Equations
  2.10. A General Strategy
  2.11. Existence Principle for Systems of Inequalities
APPLICATIONS OF THE FUNDAMENTAL FIXED-POINT PRINCIPLES
 CHAPTER 3 Ordinary Differential Equations in B-spaces
  3.1. Integration of Vector Functions of One Real Variable t
  3.2. Differentiation of Vector Functions of One Real Variable t
  3.3. Generalized Picard-Lindeirf Theorem
  3.4. Generalized Peano Theorem
  3.5. Gronwall's Lemma
  3.6. Stability of Solutions and Existence of Periodic Solutions
  3.7. Stability Theory and Plane Vector Fields, Electrical Circuits, Limit Cycles
  3.8. Perspectives
 CHAPTER 4 Differential Calculus and the Implicit Function Theorem
  4.1. Formal Differential Calculus
  4.2. The Derivatives of Frrchet and G~teaux
  4.3. Sum Rule, Chain Rule, and Product Rule
  4.4. Partial Derivatives
  4.5. Higher Differentials and Higher Derivatives
  4.6. Generalized Taylor's Theorem
  4.7. The Implicit Function Theorem
  4.8. Applications of the Implicit Function Theorem
  4.9. Attracting and Repelling Fixed Points and Stability
  4.10. Applications to Biological Equilibria
  4.11. The Continuously Differentiable Dependence of the Solutions of Ordinary Differential Equations in B-spaces on the Initial Values and on the Parameters
  4.12. The Generalized Frobenius Theorem and Total Differential Equations
……

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