An introduction to nonlinear boundary value problems, Volume 109 (Mathematics in Science and Engineering)

✍ Scribed by Lakshmikantham, Stephen R. Bernfeld

• Publisher: Academic Press
• Year: 1974
• Tongue: English
• Leaves: 399
• Edition: 1
• Category: Library

✦ Table of Contents

Front Cover
An Introduction to Nonlinear Boundary Value Problems
Copyright Page
Contents
Preface
Acknowledgments
Chapter 1. Methods Involving Differential Inequalities
1.0. Introduction
1.1. Existencein the small
1.2. Upper and Lower Solutions
1.3. The Modified Function
1.4. Nagumo’s Condition
1.5. Existence in the Large
1.6. Lyapunov–Like Functions
1.7. Existence on Infinite Intervals
1.8. Super- and Subfunctions
1.9. Properties of Subfunctions
1.10. Perron’s Method
1.11. Modified Vector Function
1.12. Nagumo’s Condition (Continued)
1.13. Existence in the Large for Systems
1.14. Further Results for Systems
1.15. Notes and comments
Chapter 2. Shooting Type Methods
2.0. Introduction
2.1. Uniqueness Implies Existence
2.2. General Linear Boundary Conditions
2.3. Weaker Uniqueness Conditions
2.4. Nonlinear Boundary Conditions
2.5. Angular Function Technique
2.6. Fundamental Lemmas
2.7. Existence
2.8. Uniqueness
2.9. Estimation of Number of Solutions
2.10. Existence of Infinite Number of Solutions
2.11. Nonlinear Boundary Conditions
2.12. Notes and Comments
Chapter 3. Topological Methods
3.0. Introduction
3.1. Solution Funnels
3.2. Application to Second-Order Equations
3.3. Wazewski Retract Method
3.4. Generalized Differential Equations
3.5. Dependence of Solutions on Boundary Data
3.6. Notes and Comments
Chapter 4. Functional Analytic Methods
4.0. Introduction
4.1. Linear Problems for Linear Systems
4.2. Linear Problems for Nonlinear Systems
4.3. Interpolation Problems
4.4. Further Nonlinear Problems
4.5. Generalized Spaces
4.6. Integral Equations
4.7. Application to Existence and Uniqueness
4.8. Method of A Priori Estimates
4.9. Bounds for Solutions in Admissible Subspaces
4.10. Leray Schauder’s Alternative
4.11. Application of Leray-Schauder’s Alternative
4.12. Periodic Boundary Conditions
4.13. Set-Valued Mappings and Functional Equations
4.14. General Linear Problems
4.15. General Results for Set-Valued Mappings
4.16. Set-Valued Differential Equations
4.17. Notes and Comments
Chapter 5. Extensions to Functional Differential Equations
5.0. Introduction
5.1. Existence in the Small
5.2. Existence in the Large
5.3. Shooting Method
5.4. Nonhomogeneous Linear Boundary Conditions
5.5. Linear Problems
5.6. Nonlinear Problems
5.7. Degenerate Cases
5.8. Notes and Comments
Chapter 6. Selected Topics
6.0. Introduction
6.1. Newton's Method
6.2. The Goodman-Lance Method
6.3. The Method of Quasilinearization
6.4. Nonlinear Eigenvalue Problems
6.5. n-Parameter Families and Interpolation Problems
6.6. Notes and Comments
Bibliography
Additional Bibliography
Index