Fixed-point theorem

About: Fixed-point theorem is a research topic. Over the lifetime, 17501 publications have been published within this topic receiving 279091 citations.

Convex Analysis and Monotone Operator Theory in Hilbert Spaces

Abstract: This book provides a largely self-contained account of the main results of convex analysis and optimization in Hilbert space. A concise exposition of related constructive fixed point theory is presented, that allows for a wide range of algorithms to construct solutions to problems in optimization, equilibrium theory, monotone inclusions, variational inequalities, best approximation theory, and convex feasibility. The book is accessible to a broad audience, and reaches out in particular to applied scientists and engineers, to whom these tools have become indispensable.

Invariance theory, the heat equation, and the Atiyah-Singer index theorem

Abstract: Pseudo-Differential Operators Introduction Fourier Transform and Sobolev Spaces Pseudo-Differential Operators on Rm Pseudo-Differential Operators on Manifolds Index of Fredholm Operators Elliptic Complexes Spectral Theory The Heat Equation Local Index Formula Variational Formulas Lefschetz Fixed Point Theorems The Zeta Function The Eta Function Characteristic Classes Introduction Characteristic Classes of Complex Bundles Characteristic Classes of Real Bundles Complex Projective Space Invariance Theory The Gauss-Bonnet Theorem Invariance Theory and Pontrjagin Classes Gauss-Bonnet for Manifolds with Boundary Boundary Characteristic Classes Singer's Question The Index Theorem Introduction Clifford Modules Hirzebruch Signature Formula Spinors The Spin Complex The Riemann-Roch Theorem K-Theory The Atiyah-Singer Index Theorem The Regularity at s = 0 of the Eta Function Lefschetz Fixed Point Formulas Index Theorem for Manifolds with Boundary The Eta Invariant of Locally Flat Bundles Spectral Geometry Introduction Operators of Laplace Type Isospectral Manifolds Non-Minimal Operators Operators of Dirac Type Manifolds with Boundary Other Asymptotic Formulas The Eta Invariant of Spherical Space Forms A Guide to the Literature Acknowledgment Introduction Bibliography Notation

Topics in metric fixed point theory

Abstract: Introduction 1. Preliminaries 2. Banach's contraction principle 3. Nonexpansive mappings: introduction 4. The basic fixed point theorems for nonexpansive mappings 5. Scaling the convexity of the unit ball 6. The modulus of convexity and normal structure 7. Normal structure and smoothness 8. Conditions involving compactness 9. Sequential approximation techniques 10. Weak sequential approximations 11. Properties of fixed point sets and minimal sets 12. Special properties of Hilbert space 13. Applications to accretivity 14. Nonstandard methods 15. Set-valued mappings 16. Uniformly Lipschitzian mappings 17. Rotative mappings 18. The theorems of Brouwer and Schauder 19. Lipschitzian mappings 20. Minimal displacement 21. The retraction problem References.

Asymptotic methods in statistical decision theory

Abstract: 1 Experiments-Decision Spaces.- 1 Introduction.- 2 Vector Lattices-L-Spaces-Transitions.- 3 Experiments-Decision Procedures.- 4 A Basic Density Theorem.- 5 Building Experiments from Other Ones.- 6 Representations-Markov Kernels.- 2 Some Results from Decision Theory: Deficiencies.- 1 Introduction.- 2 Characterization of the Spaces of Risk Functions: Minimax Theorem.- 3 Deficiencies Distances.- 4 The Form of Bayes Risks-Choquet Lattices.- 3 Likelihood Ratios and Conical Measures.- 1 Introduction.- 2 Homogeneous Functions of Measures.- 3 Deficiencies for Binary Experiments: Isometries.- 4 Weak Convergence of Experiments.- 5 Boundedly Complete Experiments.- 6 Convolutions: Hellinger Transforms.- 7 The Blackwell-Sherman-Stein Theorem.- 4 Some Basic Inequalities.- 1 Introduction.- 2 Hellinger Distances: L1-Norm.- 3 Approximation Properties for Likelihood Ratios.- 4 Inequalities for Conditional Distributions.- 5 Sufficiency and Insufficiency.- 1 Introduction.- 2 Projections and Conditional Expectations.- 3 Equivalent Definitions for Sufficiency.- 4 Insufficiency.- 5 Estimating Conditional Distributions.- 6 Domination, Compactness, Contiguity.- 1 Introduction.- 2 Definitions and Elementary Relations.- 3 Contiguity.- 4 Strong Compactness and a Result of D. Lindae.- 7 Some Limit Theorems.- 1 Introduction.- 2 Convergence in Distribution or in Probability.- 3 Distinguished Sequences of Statistics.- 4 Lower-Semicontinuity for Spaces of Risk Functions.- 5 A Result on Asymptotic Admissibility.- 8 Invariance Properties.- 1 Introduction.- 2 The Markov-Kakutani Fixed Point Theorem.- 3 A Lifting Theorem and Some Applications.- 4 Automatic Invariance of Limits.- 5 Invariant Exponential Families.- 6 The Hunt-Stein Theorem and Related Results.- 9 Infinitely Divisible, Gaussian, and Poisson Experiments.- 1 Introduction.- 2 Infinite Divisibility.- 3 Gaussian Experiments.- 4 Poisson Experiments.- 5 A Central Limit Theorem.- 10 Asymptotically Gaussian Experiments: Local Theory.- 1 Introduction.- 2 Convergence to a Gaussian Shift Experiment.- 3 A Framework which Arises in Many Applications.- 4 Weak Convergence of Distributions.- 5 An Application of a Martingale Limit Theorem.- 6 Asymptotic Admissibility and Minimaxity.- 11 Asymptotic Normality-Global.- 1 Introduction.- 2 Preliminary Explanations.- 3 Construction of Centering Variables.- 4 Definitions Relative to Quadratic Approximations.- 5 Asymptotic Properties of the Centerings $$\hat{Z}$$.- 6 The Asymptotically Gaussian Case.- 7 Some Particular Cases.- 8 Reduction to the Gaussian Case by Small Distortions.- 9 The Standard Tests and Confidence Sets.- 10 Minimum ?2 and Relatives.- 12 Posterior Distributions and Bayes Solutions.- 1 Introduction.- 2 Inequalities on Conditional Distributions.- 3 Asymptotic behavior of Bayes Procedures.- 4 Approximately Gaussian Posterior Distributions.- 13 An Approximation Theorem for Certain Sequential Experiments.- 1 Introduction.- 2 Notations and Assumptions.- 3 Basic Auxiliary Lemmas.- 4 Reduction Theorems.- 5 Remarks on Possible Applications.- 14 Approximation by Exponential Families.- 1 Introduction.- 2 A Lemma on Approximate Sufficiency.- 3 Homogeneous Experiments of Finite Rank.- 4 Approximation by Experiments of Finite Rank.- 5 Construction of Distinguished Sequences of Estimates.- 15 Sums of Independent Random Variables.- 1 Introduction.- 2 Concentration Inequalities.- 3 Compactness and Shift-Compactness.- 4 Poisson Exponentials and Approximation Theorems.- 5 Limit Theorems and Related Results.- 6 Sums of Independent Stochastic Processes.- 16 Independent Observations.- 1 Introduction.- 2 Limiting Distributions for Likelihood Ratios.- 3 Conditions for Asymptotic Normality.- 4 Tests and Distances.- 5 Estimates for Finite Dimensional Parameter Spaces.- 6 The Risk of Formal Bayes Procedures.- 7 Empirical Measures and Cumulatives.- 8 Empirical Measures on Vapnik-?ervonenkis Classes.- 17 Independent Identically Distributed Observations.- 1 Introduction.- 2 Hilbert Spaces Around a Point.- 3 A Special Role for $$\sqrt{n}$$: Differentiability in Quadratic Mean.- 4 Asymptotic Normality for Rates Other than $$\sqrt{n}$$.- 5 Existence of Consistent Estimates.- 6 Estimates Converging at the $$\sqrt{n}$$-Rate.- 7 The Behavior of Posterior Distributions.- 8 Maximum Likelihood.- 9 Some Cases where the Number of Observations Is Random.- Appendix: Results from Classical Analysis.- 1 The Language of Set Theory.- 2 Topological Spaces.- 3 Uniform Spaces.- 4 Metric Spaces.- 5 Spaces of Functions.- 6 Vector Spaces.- 7 Vector Lattices.- 8 Vector Lattices Arising from Experiments.- 9 Lattices of Numerical Functions.- 10 Extensions of Positive Linear Functions.- 11 Smooth Linear Functionals.- 12 Derivatives and Tangents.