Nonlinear Functional Analysis

topics. However, only a modest preliminary knowledge is needed. In the first chapter, where we introduce an important topological concept, the so-called topological degree for continuous maps from subsets ofRn into Rn, you need not know anything about functional analysis. Starting with Chapter 2, where infinite dimensions first appear, one should be familiar with the essential step of consider­ ing a sequence or a function of some sort as a point in the corresponding vector space of all such sequences or functions, whenever this abstraction is worthwhile. One should also work out the things which are proved in § 7 and accept certain basic principles of linear functional analysis quoted there for easier references, until they are applied in later chapters. In other words, even the 'completely linear' sections which we have included for your convenience serve only as a vehicle for progress in nonlinearity. Another point that makes the text introductory is the use of an essentially uniform mathematical language and way of thinking, one which is no doubt familiar from elementary lectures in analysis that did not worry much about its connections with algebra and topology. Of course we shall use some elementary topological concepts, which may be new, but in fact only a few remarks here and there pertain to algebraic or differential topological concepts and methods.

1. Topological Degree in Finite Dimensions.- § 1. Uniqueness of the Degree.- 1.1 Notation.- 1.2 From % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qaiaacI % cacuqHPoWvgaqeaiaacMcacaaMf8UaamiDaiaad+gacaaMf8Uabm4q % ayaaraWaaWbaaSqabeaacqGHEisPaaGccaGGOaGaeuyQdCLaaiykaa % aa!4433! $$C(\bar \Omega )\quad to\quad {\bar C^\infty }(\Omega )$$.- 1.3 From Singular to Regular Values.- 1.4 From C?-Maps to Linear Maps.- 1.5 Linear Algebra May Help.- Exercises.- § 2. Construction of the Degree.- 2.1 The Regular Case.- 2.2 From Regular to Singular Values.- 2.3 From % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4qayaara % WaaWbaaSqabeaacaaIYaaaaOGaaiikaiabfM6axjaacMcacaaMf8Ua % amiDaiaad+gacaaMf8Uaae4qaiaabIcacuqHPoWvgaqeaiaacMcaaa % a!437B! $${\bar C^2}(\Omega )\quad to\quad {\text{C(}}\bar \Omega )$$.- Exercises.- § 3. Further Properties of the Degree.- 3.1 Consequences of (d 1)–(d 3).- 3.2 Brouwer’s Fixed Point Theorem.- 3.3 Surjective Maps.- 3.4 The Hedgehog Theorem.- Exercises.- § 4. Borsuk’s Theorem.- 4.1 Borsuk’s Theorem.- 4.2 Some Applications of Borsuk’s Theorem.- Exercises.- § 5. The Product Formula.- 5.1 Preliminaries.- 5.2 The Product Formula.- 5.3 Jordan’s Separation Theorem.- Exercises.- § 6. Concluding Remarks.- 6.1 Degree on Unbounded Sets.- 6.2 Degree in Finite-Dimensional Topological Vector Spaces.- 6.3 A Relation Between the Degrees for Spaces of Different Dimension.- 6.4 Hopf’s Theorem and Generalizations of Borsuk’s Theorem.- 6.5 The Index of an Isolated Solution.- 6.6 Degree and Winding Number.- 6.7 Index of Gradient Maps.- 6.8 Final Remarks.- Exercises.- 2. Topological Degree in Infinite Dimensions.- § 7. Basic Facts About Banach Spaces.- 7.1 Banach’s Fixed Point Theorem.- 7.2 Compactness.- 7.3 Measures of Noncompactness.- 7.4 Compact Subsets of Cx(D).- 7.5 Compact Subsets of Banach Spaces with a Base.- 7.6 Continuous Extensions of Continuous Maps.- 7.7 Differentiability.- 7.8 Remarks.- Exercises.- § 8. Compact Maps.- 8.1 Definitions.- 8.2 Properties of Compact Maps.- 8.3 The Leray-Schauder Degree.- 8.4 Further Properties of the Leray-Schauder Degree.- 8.5 Schauder’s Fixed Point Theorem.- 8.6 Compact Linear Operators.- 8.7 Remarks.- Exercises.- § 9. Set Contractions.- 9.1 Definitions and Examples.- 9.2 Properties of ?-Lipschitz Maps.- 9.3 A Generalization of Schauder’s Theorem.- 9.4 The Degree for ?-Condensing Maps.- 9.5 Further Properties of the Degree.- 9.6 Examples.- 9.7 Linear Set Contractions.- 9.8 Basic Facts from Spectral Theory.- 9.9 Representations of Linear ?-Contractions.- 9.10 Remarks.- Exercises.- § 10. Concluding Remarks.- 10.1 Degree of Maps on Unbounded Sets.- 10.2 Locally Convex Spaces.- 10.3 Degree Theory in Locally Convex Spaces.- 10.4 Degree for Differentiable Maps.- 10.5 Related Concepts.- Exercises.- 3. Monotone and Accretive Operators.- § 11. Monotone Operators on Hilbert Spaces.- 11.1 Monotone Operators on Real Hilbert Spaces.- 11.2 Maximal and Hypermaximal Monotone Operators.- 11.3 The Sum of Hypermaximal Operators.- 11.4 Monotone Operators on Complex Hilbert Spaces.- 11.5 Remarks.- Exercises.- § 12. Monotone Operators on Banach Spaces.- 12.1 Special Banach Spaces.- 12.2 Duality Maps.- 12.3 Monotone Operators.- 12.4 Maximal and Hypermaximal Monotone Operators.- Exercises.- § 13. Accretive Operators.- 13.1 Semi-Inner Products.- 13.2 Accretive Operators.- 13.3 Maximal Accretive and Hyperaccretive Maps.- 13.4 Hyperaccretive Maps and Differential Equations.- 13.5 A Degree for Condensing Perturbations of Accretive Maps.- Exercises.- § 14. Concluding Remarks.- 14.1 Monotonicity.- 14.2 Ordinary Differential Equations in Banach Spaces.- 14.3 Semigroups and Evolution Equations.- Exercises.- 4. Implicit Functions and Problems at Resonance.- § 15. Implicit Functions.- 15.1 Classical Inverse and Implicit Function Theorems.- 15.2 Global Homeomorphisms.- 15.3 An Open Mapping Theorem.- 15.4 Newton’s Method.- 15.5 Scales of Banach Spaces.- 15.6 A ‘Hard’ Implicit Function Theorem.- 15.7 Remarks.- Exercises.- § 16. Problems at Resonance.- 16.1 Applications of Degree Theory.- 16.2 The Lyapunov-Schmidt Method.- 16.3 Examples.- 16.4 Remarks.- Exercises.- 5. Fixed Point Theory.- § 17. Metric Fixed Point Theory.- 17.1 Some Descendants of Banach.- 17.2 When is F a Strict Contraction?.- 17.3 Fixed Points of Nonexpansive Maps.- 17.4 The Browder-Caristi Theorem and Normal Solvability.- Exercises.- § 18. Fixed Point Theorems Involving Compactness.- 18.1 Fixed Points in Open Sets.- 18.2 Fixed Points in Closed Convex Sets.- 18.3 Weakly Inward Maps.- 18.4 Fixed Points of Weakly Inward Maps.- 18.5 The Set of All Fixed Points.- 18.6 Remarks.- Exercises.- 6. Solutions in Cones.- § 19. Cones and Increasing Maps.- 19.1 Cones and Partial Orderings.- 19.2 Positive Linear Functionals.- 19.3 Fixed Points of Increasing Maps.- 19.4 Differentiability with Respect to a Cone.- 19.5 Positive Linear Operators.- 19.6 Order Topologies.- 19.7 Fixed Points of Increasing Maps Once More.- 19.8 Remarks.- Exercises.- § 20. Solutions in Cones.- 20.1 The Fixed Point Index.- 20.2 Fixed Points in Conical Shells.- 20.3 Existence of Several Fixed Points.- 20.4 Weakly Inward Maps.- 20.5 Remarks.- Exercises.- 7. Approximate Solutions.- § 21. Approximation Solvability.- 21.1 Projection Schemes.- 21.2 A-Proper Mappings.- 21.3 Approximation Solvability.- 21.4 Linear A-Proper Maps and Approximation of Isolated Solutions.- 21.5 Remarks.- Exercises.- § 22. A-Proper Maps and Galerkin for Differential Equations.- 22.1 Topological Degrees.- 22.2 Fixed Point Theorems.- 22.3 Galerkin for Differential Equations.- Exercises.- 8. Multis.- § 23. Monotone and Accretive Multis.- 23.1 Definitions.- 23.2 Convex Functionals.- 23.3 Properties of Monotone Multis.- 23.4 Subdifferentials.- 23.5 Dense Single-Valuedness of Monotone Multis.- 23.6 Accretive Multis.- 23.7 Remarks.- Exercises.- § 24. Multis and Compactness.- 24.1 Semicontinuity of Multis.- 24.2 Examples.- 24.3 Continuous Selections.- 24.4 Approximate Selections.- 24.5 Measurable Selections.- 24.6 Degree for ?-Contracting Multis.- 24.7 Fixed Points of Multis.- 24.8 Remarks.- Exercises.- 9. Extremal Problems.- § 25. Convex Analysis.- 25.1 Minima of Convex Functionals.- 25.2 Conjugate Functionals.- 25.3 Second Conjugates.- 25.4 Remarks.- Exercises.- § 26. Extrema Under Constraints.- 26.1 Local Minima of Differentiable Maps.- 26.2 Minima Under Equality Constraints.- 26.3 Examples.- 26.4 More General Constraints.- 26.5 Remarks.- Exercises.- § 27. Critical Points of Functionals.- 27.1 The Minimax Characterization of Eigenvalues.- 27.2 A Variational Method.- 27.3 Category and Genus.- 27.4 Banach Manifolds.- 27.5 Finsler Manifolds.- 27.6 Semigroups Generated by Pseudo-Gradient Fields.- 27.7 Some Consequences of Condition (C).- 27.8 Remarks.- Exercises.- 10. Bifurcation.- § 28. Local Bifurcation.- 28.1 Necessary Conditions.- 28.2 The Odd Multiplicity Case.- 28.3 The Simple Eigenvalue Case.- 28.4 Examples.- 28.5 Bifurcation at Infinity.- 28.6 Banach Algebras May Help.- 28.7 Remarks.- Exercises.- § 29. Global Bifurcation.- 29.1 Global Continua of Solutions.- 29.2 Global Continua in Cones.- 29.3 Secondary Bifurcation.- 29.4 Remarks.- Exercises.- § 30. Further Topics in Bifurcation Theory.- 30.1 Variational Methods.- 30.2 Stability.- 30.3 Hopf Bifurcation and Last Remarks.- Epilogue.- Symbols.