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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