The greatest mathematicians, such as Archimedes, Newton, and Gauss, always united theory and applications in equal measure. Felix Klein There exists the remarkable possibility that one can master a subject mathemati cally, without really understanding its essence. Albert Einstein Don't give us numbers: give us insight! A contemporary natural scientist to a mathematician Numerous questions in physics, chemistry, biology, and economics lead to nonlinear problems; for example, deformation of rods, plates, and shells; behavior of plastic materials; surface waves of fluids; flows around objects in fluids or gases; shock waves in gases; movement of viscous fluids; equilibrium forms of rotating fluids in astrophysics; determination of the shape of the earth through gravitational measu- ments; behavior of magnetic fields of astrophysical objects; melting processes; chemical reactions; heat radiation; processes in nuclear reactors; nonlinear oscillation in physics, chemistry, and biology; 2 Introduction existence and stability of periodic and quasiperiodic orbits in celestial mechanics; stability of physical, chemical, biological, ecological, and economic processes; diffusion processes in physics, chemistry, and biology; processes with entropy production, and self-organization of systems in physics, chemistry, and biology; study of the electrical potential variation in the heart through measure ments on the body surface to prevent heart attacks; determining material constants or material laws (e. g.

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InhaltsangabeFundamental Fixed-Point Principles.- 1 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-Lindelof 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.- 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.- 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-Lindelöf 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.- 4 Differential Calculus and the Implicit Function Theorem.- §4.1. Formal Differential Calculus.- §4.2. The Derivatives of Fréchet and Gâteaux.- §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.- §4.13. Diffeomorphisms and the Local Inverse Mapping Theorem.- §4.14. Proper Maps and the Global Inverse Mapping Theorem.- §4.15. The Suijective Implicit Function Theorem.- §4.16. Nonlinear Systems of Equations, Subimmersions, and the Rank Theorem.- §4.17. A Look at Manifolds.- §4.18. Submersions and a Look at the Sard-Smale Theorem.- §4.19. The Parametrized Sard Theorem and Constructive Fixed-Point Theory.- 5 Newton's Method.- §5.1. A Theorem on Local Convergence.- §5.2. The Kantorovi? Semi-Local Convergence Theorem.- 6 Continuation with Respect to a Parameter.- §6.1. The Continuation Method for Linear Operators.- §6.2. B-spaces of Hölder Continuous Functions.- §6.3. Applications to Linear Partial Differential Equations.- §6.4. Functional-Analytic Interpretation of the Existence Theorem and its Generalizations.- §6.5. Applications to Semi-linear Differential Equations.- §6.6. The Implicit Function Theorem and the Continuation Method.- §6.7. Ordinary Differential Equations in B-spaces and the Continuation Method.- §6.8. The Leray-Schauder Principle.- §6.9. Applications to Quasi-linear Elliptic Differential Equations.- 7 Positive Operators.- §7.1. Ordered B-spaces.- §7.2. Monotone Increasing Operators.- §7.3. The Abstract Gronwall Lemma and its Applications to Integral Inequalities.- §7.4. Supersolutions, Subsolutions, Iterative Methods, and Stability.- §7.5. Applications.- §7.6. Minorant Methods and Positive Eigensolutions.- §7.7. Applications.- §7.8. The Krein-Rutman Theorem and its Applications.- §7.9. Asymptotic Linear Operators.- §7.10. Main Theorem for Operators of Monotone Type.- §7.11. Application to a Heat Conduction Problem.- §7.12. Existence of Three Solutions.- §7.13.