Nonlinearity and Functional Analysis is a collection of lectures that aim to present a systematic description of fundamental nonlinear results and their applicability to a variety of concrete problems taken from various fields of mathematical analysis. For decades, great mathematical interest has focused on problems associated with linear operators and the extension of the well-known results of linear algebra to an infinite-dimensional context. This interest has been crowned with deep insights, and the substantial theory that has been developed has had a profound influence throughout the mathematical sciences.
This volume comprises six chapters and begins by presenting some background material, such as differential-geometric sources, sources in mathematical physics, and sources from the calculus of variations, before delving into the subject of nonlinear operators. The following chapters then discuss local analysis of a single mapping and parameter dependent perturbation phenomena before going into analysis in the large. The final chapters conclude the collection with a discussion of global theories for general nonlinear operators and critical point theory for gradient mappings.
This book will be of interest to practitioners in the fields of mathematics and physics, and to those with interest in conventional linear functional analysis and ordinary and partial differential equations.
1. Cover image - p. Cover 2. Title page - p. Cover 3. Table of Contents - p. Cover 4. Inside Front Cover - p. Cover 5. Copyright page - p. ifc1 6. Dedication - p. ifc1 7. Dedication - p. v 8. Preface - p. vi 9. Notation and Terminology - p. xv 10. Suggestions for the Reader - p. xvii 11. Part I: Preliminaries - p. xix 12. Part I: Preliminaries - p. xix 13. Chapter 1: Background Material - p. 2 14. Chapter 2: Nonlinear Operators - p. 63 15. Part II: Local Analysis - p. 108 16. Part II: Local Analysis - p. 108 17. Chapter 3: Local Analysis of a Single Mapping - p. 110 18. Chapter 4: Parameter Dependent Perturbation Phenomena - p. 148 19. Part III: Analysis in the Large - p. 213 20. Part III: Analysis in the Large - p. 213 21. Chapter 5: Global Theories for General Nonlinear Operators - p. 216 22. 5.1 Linearization - p. 217 23. 5.2 Finite-Dimensional Approximations - p. 231 24. 5.3 Homotopy, the Degree of Mappings, and Its Generalizations - p. 243 25. 5.4 Homotopy and Mapping Properties of Nonlinear Operators - p. 265 26. 5.5 Applications to Nonlinear Boundary Value Problems - p. 283 27. Notes - p. 295 28. Chapter 6: Critical Point Theory for Gradient Mappings - p. 298 29. 6.1 Minimization Problems - p. 299 30. 6.2 Specific Minimization Problems from Geometry and Physics - p. 312 31. 6.3 Isoperimetric Problems - p. 324 32. 6.4 Isoperimetric Problems in Geometry and Physics - p. 336 33. 6.5 Critical Point Theory of Marston Morse in Hilbert Space - p. 353 34. 6.6 The Critical Point Theory of Ljusternik and Schnirelmann - p. 366 35. 6.7 Applications of the General Critical Point Theories - p. 374 36. Notes - p. 387 37. Appendix A: On Differentiable Manifolds - p. 390 38. Appendix B: On the Hodge—Kodaira Decomposition for Differential Forms - p. 395 39. References - p. 398 40. Index - p. 408 41. Pure and Applied Mathematics - p. 408
Nonlinearity and Functional Analysis is a collection of lectures that aim to present a systematic description of fundamental nonlinear results and their applicability to a variety of concrete problems taken from various fields of mathematical analysis. For decades, great mathematical interest has focused on problems associated with linear operators and the extension of the well-known results of linear algebra to an infinite-dimensional context. This interest has been crowned with deep insights, and the substantial theory that has been developed has had a profound influence throughout the mathematical sciences.
This volume comprises six chapters and begins by presenting some background material, such as differential-geometric sources, sources in mathematical physics, and sources from the calculus of variations, before delving into the subject of nonlinear operators. The following chapters then discuss local analysis of a single mapping and parameter dependent perturbation phenomena before going into analysis in the large. The final chapters conclude the collection with a discussion of global theories for general nonlinear operators and critical point theory for gradient mappings.
This book will be of interest to practitioners in the fields of mathematics and physics, and to those with interest in conventional linear functional analysis and ordinary and partial differential equations.
1. Cover image - p. Cover 2. Title page - p. Cover 3. Table of Contents - p. Cover 4. Inside Front Cover - p. Cover 5. Copyright page - p. ifc1 6. Dedication - p. ifc1 7. Dedication - p. v 8. Preface - p. vi 9. Notation and Terminology - p. xv 10. Suggestions for the Reader - p. xvii 11. Part I: Preliminaries - p. xix 12. Part I: Preliminaries - p. xix 13. Chapter 1: Background Material - p. 2 14. Chapter 2: Nonlinear Operators - p. 63 15. Part II: Local Analysis - p. 108 16. Part II: Local Analysis - p. 108 17. Chapter 3: Local Analysis of a Single Mapping - p. 110 18. Chapter 4: Parameter Dependent Perturbation Phenomena - p. 148 19. Part III: Analysis in the Large - p. 213 20. Part III: Analysis in the Large - p. 213 21. Chapter 5: Global Theories for General Nonlinear Operators - p. 216 22. 5.1 Linearization - p. 217 23. 5.2 Finite-Dimensional Approximations - p. 231 24. 5.3 Homotopy, the Degree of Mappings, and Its Generalizations - p. 243 25. 5.4 Homotopy and Mapping Properties of Nonlinear Operators - p. 265 26. 5.5 Applications to Nonlinear Boundary Value Problems - p. 283 27. Notes - p. 295 28. Chapter 6: Critical Point Theory for Gradient Mappings - p. 298 29. 6.1 Minimization Problems - p. 299 30. 6.2 Specific Minimization Problems from Geometry and Physics - p. 312 31. 6.3 Isoperimetric Problems - p. 324 32. 6.4 Isoperimetric Problems in Geometry and Physics - p. 336 33. 6.5 Critical Point Theory of Marston Morse in Hilbert Space - p. 353 34. 6.6 The Critical Point Theory of Ljusternik and Schnirelmann - p. 366 35. 6.7 Applications of the General Critical Point Theories - p. 374 36. Notes - p. 387 37. Appendix A: On Differentiable Manifolds - p. 390 38. Appendix B: On the Hodge—Kodaira Decomposition for Differential Forms - p. 395 39. References - p. 398 40. Index - p. 408 41. Pure and Applied Mathematics - p. 408
