Title Genericity in nonlinear analysis / by Simeon Reich, Alexander J. Zaslavski

Published New York : Springer, 2013

Table of Contents
1.	Introduction					1
1.1.	Hyperbolic Spaces					1
1.2.	Successive Approximations					2
1.3.	Contractive Mappings					3
1.4.	Infinite Products					5
1.5.	Contractive Set-Valued Mappings					7
1.6.	Nonexpansive Set-Valued Mappings					9
1.7.	Porosity					10
1.8.	Examples					12
2.	Fixed Point Results and Convergence of Powers of Operators					15
2.1.	Convergence of Iterates for a Class of Nonlinear Mappings					15
2.2.	Convergence of Iterates of Typical Nonexpansive Mappings					23
2.3.	A Stability Result in Fixed Point Theory					29
2.4.	Well-Posed Null and Fixed Point Problems					34
2.5.	Mappings in a Finite-Dimensional Euclidean Space					37
2.6.	Approximate Fixed Points					42
2.7.	Generic Existence of Small Invariant Sets					47
2.8.	Many Nonexpansive Mappings Are Strict Contractions					51
2.9.	Krasnosel'skii-Mann Iterations of Nonexpansive Operators					55
2.10.	Power Convergence of Order-Preserving Mappings					63
2.11.	Positive Eigenvalues and Eigenvectors					72
2.12.	Proof of Theorem 2.48					75
2.13.	Auxiliary Results for Theorems 2.49--2.51					78
2.14.	Proofs of Theorems 2.49 and 2.50					83
2.15.	Proof of Theorem 2.51					85
2.16.	Convergence of Inexact Orbits for a Class of Operators					87
2.17.	Proofs of Theorem 2.65 and Corollary 2.66					89
2.18.	Proof of Theorem 2.67					92
2.19.	Proof of Theorem 2.68					93
2.20.	Proof of Theorem 2.69					96
2.21.	Inexact Orbits of Nonexpansive Operators					97
2.22.	Convergence to Attracting Sets					100
2.23.	Nonconvergence to Attracting Sets					103
2.24.	Convergence and Nonconvergence to Fixed Points					106
2.25.	Convergence to Compact Sets					110
2.26.	An Example of Nonconvergence to Compact Sets					113
3.	Contractive Mappings					119
3.1.	Many Nonexpansive Mappings Are Contractive					119
3.2.	Attractive Sets					121
3.3.	Attractive Subsets of Unbounded Spaces					124
3.4.	A Contractive Mapping with no Strictly Contractive Powers					129
3.5.	A Power Convergent Mapping with no Contractive Powers					132
3.6.	A Mapping with Nonuniformly Convergent Powers					134
3.7.	Two Results in Metric Fixed Point Theory					136
3.8.	A Result on Rakotch Contractions					144
3.9.	Asymptotic Contractions					149
3.10.	Uniform Convergence of Iterates					153
3.11.	Well-Posedness of Fixed Point Problems					157
3.12.	A Class of Mappings of Contractive Type					159
3.13.	A Fixed Point Theorem for Matkowski Contractions					166
3.14.	Jachymski-Schroder-Stein Contractions					170
3.15.	Two Results on Jachymski-Schroder-Stein Contractions					175
4.	Dynamical Systems with Convex Lyapunov Functions					181
4.1.	Minimization of Convex Functionals					181
4.2.	Proof of Proposition 4.3					183
4.3.	Proofs of Theorems 4.1 and 4.2					185
4.4.	Examples					188
4.5.	Normal Mappings					190
4.6.	Existence of a Normal A ε Ac					192
4.7.	Auxiliary Results					193
4.8.	Proof of Theorem 4.12					194
4.9.	Proof of Theorem 4.13					195
4.10.	Proof of Theorem 4.14					196
4.11.	Normality and Porosity					197
4.12.	Proof of Theorem 4.18					198
4.13.	Proof of Theorem 4.19					200
4.14.	Convex Functions Possessing a Sharp Minimum					202
5.	Relatively Nonexpansive Operators with Respect to Bregman Distances					205
5.1.	Power Convergence of Operators in Banach Spaces					205
5.2.	Power Convergence for a Class of Continuous Mappings					206
5.3.	Preliminary Lemmata for Theorems 5.1--5.6					209
5.4.	Proofs of Theorems 5.1--5.6					213
5.5.	A Class of Uniformly Continuous Mappings					217
5.6.	An Auxiliary Result					218
5.7.	Proofs of Theorems 5.11 and 5.12					219
5.8.	Mappings with a Uniformly Continuous Bregman Function					222
5.9.	Proofs of Theorems 5.15 and 5.16					223
5.10.	Generic Power Convergence to a Retraction					226
5.11.	Two Lemmata					228
5.12.	Convergence of Powers of Uniformly Continuous Mappings					230
5.13.	Convergence to a Retraction					231
5.14.	Auxiliary Results					231
5.15.	Proof of Theorem 5.21					233
5.16.	Proofs of Theorems 5.22 and 5.23					235
5.17.	Convergence of Powers for a Class of Continuous Operators					241
5.18.	Proofs of Theorems 5.32--5.34					242
6.	Infinite Products					247
6.1.	Nonexpansive and Uniformly Continuous Operators					247
6.2.	Asymptotic Behavior					249
6.3.	Nonexpansive Retractions					250
6.4.	Preliminary Results					252
6.5.	Proofs of Theorems 6.1 and 6.2					255
6.6.	Proofs of Theorems 6.3 and 6.4					257
6.7.	Proofs of Theorems 6.5, 6.6 and 6.7					259
6.8.	Hyperbolic Spaces					263
6.9.	Infinite Products of Order-Preserving Mappings					263
6.10.	Existence of a Unique Fixed Point					265
6.11.	Asymptotic Behavior					269
6.12.	Preliminary Lemmata for Theorems 6.16--6.20					271
6.13.	Proofs of Theorems 6.16 and 6.17					276
6.14.	Proofs of Theorems 6.18 and 6.19					277
6.15.	Proof of Theorem 6.20					280
6.16.	Infinite Products of Positive Linear Operators					282
6.17.	Proof of Theorem 6.24					287
6.18.	Proof of Theorem 6.26					290
6.19.	Proof of Theorem 6.27					295
6.20.	Homogeneous Order-Preserving Mappings					301
6.21.	Preliminary Lemmata for Theorems 6.41--6.43					305
6.22.	Proofs of Theorems 6.41 and 6.42					314
6.23.	Proof of Theorem 6.43					315
6.24.	Infinite Products of Affine Operators					321
6.25.	A Generic Fixed Point Theorem for Affine Mappings					323
6.26.	A Weak Ergodic Theorem for Affine Mappings					327
6.27.	Affine Mappings with a Common Fixed Point					329
6.28.	Proofs of Theorems 6.64, 6.65 and 6.66					330
6.29.	Weak Convergence					334
6.30.	Proofs of Theorems 6.67 and 6.68					335
6.31.	Affine Mappings with a Common Set of Fixed Points					336
6.32.	Infinite Products of Resolvents of Accretive Operators					339
6.33.	Auxiliary Results					343
6.34.	Proof of Theorem 6.71					345
6.35.	Proof of Theorem 6.72					348
7.	Best Approximation					353
7.1.	Well-Posedness and Porosity					353
7.2.	Auxiliary Results					357
7.3.	Proofs of Theorems 7.3--7.5					360
7.4.	Generalized Best Approximation Problems					363
7.5.	Theorems 7.8--7.11					365
7.6.	A Basic Lemma					367
7.7.	Proofs of Theorems 7.8--7.11					372
7.8.	A Porosity Result in Best Approximation Theory					374
7.9.	Two Lemmata					375
7.10.	Proof of Theorem 7.13					379
7.11.	Porous Sets and Generalized Best Approximation Problems					380
7.12.	A Basic Lemma					383
7.13.	Proofs of Theorems 7.16--7.18					389
8.	Descent Methods					397
8.1.	Discrete Descent Methods for a Convex Objective Function					397
8.2.	An Auxiliary Result					401
8.3.	Proof of Theorem 8.2					403
8.4.	A Basic Lemma					406
8.5.	Proofs of Theorems 8.3 and 8.4					409
8.6.	Methods for a Nonconvex Objective Function					412
8.7.	An Auxiliary Result					416
8.8.	Proof of Theorem 8.8					417
8.9.	A Basic Lemma for Theorems 8.9 and 8.10					419
8.10.	Proofs of Theorems 8.9 and 8.10					421
8.11.	Continuous Descent Methods					424
8.12.	Proof of Theorem 8.14					427
8.13.	Proof of Theorem 8.15					428
8.14.	Regular Vector-Fields					432
8.15.	Proofs of Propositions 8.16 and 8.17					434
8.16.	An Auxiliary Result					436
8.17.	Proof of Theorem 8.18					437
8.18.	Proof of Theorem 8.19					438
8.19.	Proof of Theorem 8.20					439
8.20.	Proof of Theorem 8.21					440
8.21.	Most Continuous Descent Methods Converge					441
8.22.	Proof of Proposition 8.24					442
8.23.	Proof of Theorem 8.25					443
9.	Set-Valued Mappings					449
9.1.	Contractive Mappings					449
9.2.	Star-Shaped Spaces					451
9.3.	Convergence of Iterates of Set-Valued Mappings					453
9.4.	Existence of Fixed Points					457
9.5.	An Auxiliary Result and the Proof of Proposition 9.16					459
9.6.	Proof of Theorem 9.14					460
9.7.	Proof of Theorem 9.15					461
9.8.	An Extension of Theorem 9.15					462
9.9.	Generic Existence of Fixed Points					464
9.10.	Topological Structure of the Fixed Point Set					470
9.11.	Approximation of Fixed Points					473
9.12.	Approximating Fixed Points in Caristi's Theorem					479
10.	Minimal Configurations in the Aubry-Mather Theory					481
10.1.	Preliminaries					481
10.2.	Spaces of Functions					484
10.3.	The Main Results					486
10.4.	Preliminary Results for Assertion 1 of Theorem 10.9					487
10.5.	Preliminary Results for Assertion 2 of Theorem 10.9					493
10.6.	Proof of Proposition 10.11					502
	References					513
	Index					519

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Description 1 online resource Series Developments in mathematics ; 34 Developments in mathematics ; 34. Contents Introduction -- Fixed Point Results and Convergence of Powers of Operators -- Contractive Mappings -- Dynamical Systems with Convex Lyapunov Functions -- Relatively Nonexpansive Operators with Respect to Bregman Distances -- Infinite Products -- Best Approximation -- Descent Methods -- Set-Valued Mappings -- Minimal Configurations in the Aubry-Mather Theory Summary This book presents an extensive collection of state-of-the-art results and references in nonlinear functional analysis demonstrating how the generic approach proves to be very useful in solving many interesting and important problems. Nonlinear analysis plays an ever-increasing role in theoretical and applied mathematics, as well as in many other areas of science such as engineering, statistics, computer science, economics, finance, and medicine. The text may be used as supplementary material for graduate courses in nonlinear functional analysis, optimization theory and approximation theory, a Analysis calculus optimalisatie optimization wiskunde mathematics functionaalanalyse functional analysis optimalisatiemethoden optimization methods Mathematics (General) Wiskunde (algemeen) Bibliography Includes bibliographical references and index Notes Print version record Subject Nonlinear functional analysis. MATHEMATICS -- Calculus. MATHEMATICS -- Mathematical Analysis. Nonlinear functional analysis Form Electronic book Author Zaslavski, Alexander J., author. ISBN 9781461495338 1461495334

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