Geometrical Methods of Nonlinear Analysis

Geometrical (in particular, topological) methods in nonlinear analysis were originally invented by Banach, Birkhoff, Kellogg, Schauder, Leray, and others in existence proofs. Since about the fifties, these methods turned out to be essentially the sole approach to a variety of new problems: the investigation of iteration processes and other procedures in numerical analysis, in bifur­ cation problems and branching of solutions, estimates on the number of solutions and criteria for the existence of nonzero solutions, the analysis of the structure of the solution set, etc. These methods have been widely applied to the theory of forced vibrations and auto-oscillations, to various problems in the theory of elasticity and fluid. mechanics, to control theory, theoretical physics, and various parts of mathematics. At present, nonlinear analysis along with its geometrical, topological, analytical, variational, and other methods is developing tremendously thanks to research work in many countries. Totally new ideas have been advanced, difficult problems have been solved, and new applications have been indicated. To enumerate the publications of the last few years one would need dozens of pages. On the other hand, many problems of non­ linear analysis are still far from a solution (problems arising from the internal development of mathematics and, in particular, problems arising in the process of interpreting new problems in the natural sciences). We hope that the English edition of our book will contribute to the further propagation of the ideas of nonlinear analysis.

1. Vector Fields in Finite Dimensional Spaces.- §1. Extensions of Vector Fields.- 1.1 Vector Fields.- 1.2 Extensions of Vector Fields.- 1.3 Sard’s Theorem.- 1.4 Extensions With a Minimal Number of Singular Points.- §2. Homotopic Vector Fields • •.- 2.1 Deformations and Homotopic Fields.- 2.2 Homotopy Criteria.- 2.3 Homotopy Classes.- §3. Rotation of Vector Fields.- 3.1 The Main Properties of the Rotation.- 3.2 Rotation in One-Dimensional Space.- 3.3 Rotation in Two-Dimensional Spaces.- 3.4 Rotation in Dimensional Space (n > 2).- §4. Theorems on Singular Points.- 4.1 Existence of Singular Points.- 4.2 Change of Domain.- 4.3 Algebraic Count of Singular Points.- 4.4 Index of the Singular Point at Infinity.- §5. Hopf’s Theorem.- 5.1 Homotopy Classification.- 5.2 Vector Fields Without Singularities.- §6. Linear Vector Fields.- 6.1 Linear Fields.- 6.2 The Brouwer-Bohl Theorem.- 6.3 Computing the Index of Singular Points for Linearized Fields.- 6.4 Asymptotically Linear Fields.- §7. Product Theorems.- 7.1 A Product Formula for the Index.- 7.2 The Product Formula for Rotations.- 7.3 The Leray-Schauder Lemma.- 7.4 Direct Sum of Vector Fields.- 7.5 A Theorem on Reducible Vector Fields.- 7.6 The Sum Formula for Rotations.- §8. Periodic and Odd Vector Fields.- 8.1 Periodic Mappings.- 8.2 The Congruence Theorem.- 8.3 A Special Case.- 8.4 Odd and Even Vector Fields.- 8.5 A General Theorem.- 8.6 Vector Fields Which are Symmetric With Respect to a Subspace.- §9. Special Coverings of Spheres.- 9.1 The Genus of a Set.- 9.2 Genus With Respect to Periodic Mappings.- 9.3 Deformations Along Great Circles.- 9.4 Proof of Theorem 9.3.- §10. Homogeneous Polynomials.- 10.1 Parity of Rotation.- 10.2 Homogeneous Fields in the Plane.- 10.3 Quadratic Fields.- §11. Smooth Vector Fields.- 11.1 Positive Vector Fields.- 11.2 Analytic Fields in Complex Spaces.- 11.3 Singular Points of Analytic Fields.- §12. Gradient Fields.- 12.1 The Index of a Nonsingular Potential.- 12.2 Quadratic Potentials.- 12.3 Odd and Even Potentials.- 12.4 Homogeneous Potentials.- 12.5 Coercive Potentials.- §13. Periodic and Bounded Solutions of Differential Equations.- 13.1 The Setting of the Problem.- 13.2 The Translation Operator.- 13.3 Guiding Functions.- 13.4 Existence of Periodic Solutions.- 13.5 The Index of Periodic Solutions.- 13.6 Regular Guiding Functions.- 13.7 Examples.- 13.8 Existence of Solutions Which are Bounded on the Whole Axis.- 13.9 General Boundary Value Problems.- §14. Construction of Guiding Functions.- 14.1 Homogeneous Systems.- 14.2 Planar Systems.- 14.3 Equations With Polynomial Right-Hand Side.- 14.4 Periodic and Bounded Solutions.- §15. The Index of a Singular Point of a Planar Vector Field.- 15.1 General Theorems.- 15.2 Simple Singular Rays.- 15.3 Multiple Singular Rays (Noncritical Case).- 15.4 Multiple Singular Rays (Critical Case).- 15.5 A Geometrical Scheme.- 15.6 The Order of Nondegeneracy.- 2. Completely Continuous Vector Fields.- §16. Continuous Fields in Infinite Dimensional Spaces.- 16.1 Continuous Deformations of Fields.- 16.2 Contractions of Spheres.- 16.3 Leray’s Example.- 16.4 Kakutani’s Example.- 16.5 Passing to Smaller Classes of Vector Fields.- §17. Completely Continuous Operators.- 17.1 Definitions.- 17.2 Spaces and Operators.- 17.3 Linear Integral Operators.- 17.4 Spectral Properties of Completely Continuous Linear Operators…...- 17.5 The Frechet Derivative.- 17.6 Taylor’s Formula.- 17.7 The Substitution Operator.- 17.8 The Hammerstein Operator.- 17.9 The General Nonlinear Integral Operator.- 17.10 Asymptotic Derivatives of Integral Operators.- §18. Finite Dimensional Approximations.- 18.1 Projections on Convex Sets.- 18.2 Remarks.- 18.3 Finite Dimensional Approximation of Operators.- 18.4 The Schauder Projections.- 18.5 Extensions of Completely Continuous Operators.- 18.6 Extensions With a Finite Number of Fixed Points.- §19. Homotopy for Completely Continuous Vector Fields.- 19.1 Definitions.- 19.2 Homotopy Criteria.- 19.3 A Remark on the Definition of Homotopy.- 19.4 Uniformly Nonsingular Homotopies.- 19.5 Finite Dimensional Approximations for Compact Deformations.- 19.6 A Special Criterion for Homotopy.- §20. Rotation of Completely Continuous Vector Fields.- 20.1 Rotation of Fields With Finite Dimensional Operators.- 20.2 The Definition in the General Case.- 20.3 Properties of Rotation.- 20.4 Existence of Singular Points.- 20.5 Generalization of Hopfs Theorem.- 20.6 Fields With Zero Rotation.- §21. Linear and Almost Linear Completely Continuous Fields.- 21.1 Rotation of Linear Fields.- 21.2 Asymptotically Linear Fields.- 21.3 One-sided Estimates.- 21.4 Computing the Index of a Regular Fixed Point.- 21.5 Fields With (Blt £2)-Quasilinear Operators.- 21.6 Coercive Differentiable Fields Ill.- 21.7 A Special Case.- 21.8 Odd Fields.- §22. Product of Rotations.- 22.1 The Product Formula for Indices.- 22.2 The Product Formula for Rotations of Vector Fields.- 22.3 Vector Fields on a Direct Sum of Subspaces.- 22.4 Homeomorphisms and Completely Continuous Fields.- 22.5 Homeomorphisms on Boundaries of Domains.- §23. Smooth Completely Continuous Vector Fields.- 23.1 Smale’s Theorem.- 23.2 Positively Oriented Completely Continuous Vector Fields.- 23.3 Completely Continuous Analytic Vector Fields.- §24. Computing the Index of a Singular Point in Critical Cases…..- 24.1 The Setting of the Problem and Notations.- 24.2 The Case of a Simple Critical Point….- 24.3 The General Case.- 24.4 Fields With Analytic Principal Part.- 24.5 The Simple Critical Case.- 24.6 The General Critical Case.- 24.7 The Method of Successive Substitutions.- 24.8 Another Reduction Principle.- 3. Principles of Relatedness.- §25. Invariance Principles for the Rotation.- 25.1 Domains With a Common Core.- 25.2 A Counter-Example.- 25.3 Linear Fields.- 25.4 Some Basic Lemmata.- 25.5 Invariance Principles..- §26. Composition of Operators.- 26.1 The Setting of the Problem.- 26.2 The Main Theorem.- 26.3 Generalizations.- 26.4 Equivalent Conditions.- 26.5 Choice of a New Norm.- 26.6 The Index of a Singular Point for Iterates of Operators.- §27. Transition to Equations in a Subspace.- 27.1 The Setting of the Problem.- 27.2 A Lemma on the Index.- 27.3 The Principle of Relatedness.- §28. Forced Vibrations.- 28.1 A Periodic Boundary Value Problem.- 28.2 Transformations of Related Equations.- 28.3 Integral Equations.- 28.4 Change of Space.- 28.5 The Translation Field and the Basic Theorem.- 28.6 Applications of Guiding Functions.- 28.7 Equations in Banach Spaces.- §29. Boundary Value Problems.- 29.1 Periodic Problems.- 29.2 Construction of Equivalent Integral Equations.- 29.3 Integro-Differential Equations.- 29.4 The Principle of Relatedness for Periodic Problems.- 29.5 A General Boundary Value Problem.- 29.6 A General Principle of Relatedness.- 29.7 Remarks.- §30. The Principle of Relatedness for Elliptic Equations.- 30.1 Linear Equations.- 30.2 Nonlinear Equations.- 30.3 Theorems on the Equality of Rotations.- 30.4 The Principle of Relatedness.- 30.5 Generalizations -.- §31. Vector Fields Involving Iterated Operators.- 31.1 A General Theorem.- 31.2 Passage to a Finite Dimensional Subspace.- 31.3 Passage to Smooth Mappings.- 31.4 Completion of the Proof.- 31.5 Consequences and Generalizations.- 4. Fields With Noncompact Operators.- §32. The Method of Partial Redefinition of Operators.- 32.1 Compactly Supported Operators.- 32.2 Measures of Noncompactness.- 32.3 Fields With Condensing Operators.- 32.4 Computing the Rotation: Examples.- 32.5 Limit Compact Operators.- §33. Fields With Positive Operators.- 33.1 Cones and Partial Orderings.- 33.2 Positive Linear Operators.- 33.3 Nonlinear Positive Operators.- 33.4 Rotation of Fields With Positive Operators.- 33.5 Computing the Rotation.- 33.6 Computing the Index of a Singular Point • • • •.- 33.7 A Perturbation Result.- 33.8 Remarks.- §34. The Method of Partial Inversion.- 34.1 Fields With Invertible Operators.- 34.2 Quasirotation for Partially Invertible Fields.- 34.3 Contractions.- 34.4 An Important Example.- 34.5 A Generalization.- 1 34.6 Rotation mod 2.- 34.7 A Lemma on Linear Fredholm Operators.- 34.8 Nonlinear Fredholm Mappings.- §35. Some Generalizations.- 35.1 Approximation Schemes.- 35.2 Frum-Ketkov’s Theorem.- 35.3 Operators in Topological Vector Spaces.- 35.4 Other Articles.- §36. Multivalued Mappings.- 36.1 Upper Semicontinuous Mappings.- 36.2 Rotation for Fields With Multivalued Operators.- 36.3 Fixed Points.- 36.4 Generalizations.- 36.5 Mappings in Quotient Spaces.- 5. Solvability of Nonlinear Equations.- §37. Invariant Sets and Fixed Points.- 37.1 Generalized Contractions.- 37.2 Schauder’s Principle.- 37.3 A Priori Estimates.- 37.4 Completely Continuous Perturbations of Contractions.- 37.5 Nonexpansive Operators.- 37.6 Construction of Invariant Sets ?.- 37.7 Remarks.- §38. Fixed Points of Monotone Operators.- 38.1 Monotone Operators.- 38.2 Monotonically Limit Compact Operators.- 38.3 An Example.- §39. Dissipative Operators.- 39.1 Dissipative Equations.- 39.2 Browder’s Principle.- 39.3 Existence of Periodic Solutions.- 39.4 Fixed Points for Dissipative Operators.- 39.5 Existence of a A-Centre.- §40. Almost Linear Equations.- 40.1 Equations With Completely Continuous Operators.- 40.2 Equations With Smooth Operators.- 40.3 A Lemma §47. Nonzero Solutions of Parametrized Equations.- 47.1 The Setting of the Problem.- 47.2 A Lemma on Nonhomotopic Fields.- 47.3 Continuous Branches of Solutions.- 47.4 Operators With a Monotone Minorant.- 47.5 The Principle of Topological Closure ’...- 47.6 Eigenvectors of Concave Operators.- §48. Connectivity Principles.- 48.1 Normally Solvable Equations.- 48.2 Strongly Smoothable Operators.- 48.3 Equations With Nonexpansive Operators.- 48.4 Equations With Differentiable Operators.- 48.5 The Structure of the Solution Funnel.- 48.6 Remarks.- 7. Construction of Solutions.- §49. The Method of Successive Approximation.- 49.1 Convergence.- 49.2 Equations With Concave Operators.- 49.3 Contractions for Comparable Elements.- 49.4 Equations With Invertible Positive Operators.- §50. Approximating Equations.- 50.1 The Setting of the Problem.- 50.2 The Method of Galerkin-Petrov.- 50.3 The Ljapunov-Cesari Method.- 50.4 Tonelli’s Method.- 50.5 Numerical Quadrature Methods.- §51. Error Estimates.- 51.1 The Setting of the Problem...- 51.2 A General Scheme.- 51.3 Remarks.- §52. The Index of a Stable Solution.- 52.1 Successive Iterations and Stability...- 52.2 A Criterion for Lack of Asymptotic Stability.- 52.3 Continuous Deformations of Systems With Isolated Equilibrium State….- 52.4 The Index of a Ljapunov-Stable Equilibrium State.- 52.5 Balls With Handles.- 52.6 Remarks.- 8. Small Perturbations of Nonlinear Equations.- §53. Perturbations and Existence Theorems.- 53.1 The Principle of Nonzero Rotation.- 53.2 Weakly Coupled Equations.- 53.3 An Application of the Principles on Relatedness and Invariance of Rotation.- 53.4 On a Class of Boundary Value Problems.- §54. Perturbations of Isolated Solutions.- 54.1 Solutions With Nonzero Index.- 54.2 The General Implicit Function Theorem.- 54.3 The Bifurcation Equation.- 54.4 (n, m)-Rotation.- 54.5 Systems of Scalar Equations With a Surplus Number of Unknowns…..- 54.6 Parametrized Equations.- 54.7 Equations With Completely Continuous Operators.- §55. Functionalizing the Parameter.- 55.1 A General Scheme.- 55.2 The Index of a Cycle.- 55.3 Computing the Index of a Cycle.- 55.4 The Theorem on Relatedness.- 55.5 Autonomous Equations With Time-Lag.- 55.6 Remarks.- §56. The Principle of Changing Index.- 56.1 Bifurcation Points.- 56.2 Necessary Conditions.- 56.3 Sufficient Criteria.- 56.4 Continuous Branches of Nonzero Solutions.- 56.5 Leading Nonlinearities.- 56.6 Critical Loads.- 56.7 Solutions of Large Norm.- §57. Stability of Critical Values.- 57.1 Eigenvectors of Gradient Operators.- 57.2 Even Functionals.- 57.3 Small Perturbations.- 57.4 Some Lemmata on Weakly Continuous Functionals.- 57.5 Sets of Finite Genus.- 57.6 Stability of Critical Values.- 57.7 Bifurcation Points for Equations With Gradient Operators.- References.- List of Symbols.