Fixed Point TheoryThe aim of this monograph is to give a unified account of the classical topics in fixed point theory that lie on the border-line of topology and non linear functional analysis, emphasizing developments related to the Leray Schauder theory. Using for the most part geometric methods, our study cen ters around formulating those general principles of the theory that provide the foundation for many of the modern results in diverse areas of mathe matics. The main text is self-contained for readers with a modest knowledge of topology and functional analysis; the necessary background material is collected in an appendix, or developed as needed. Only the last chapter pre supposes some familiarity with more advanced parts of algebraic topology. The "Miscellaneous Results and Examples", given in the form of exer cises, form an integral part of the book and describe further applications and extensions of the theory. Most of these additional results can be established by the methods developed in the book, and no proof in the main text relies on any of them; more demanding problems are marked by an asterisk. The "Notes and Comments" at the end of paragraphs contain references to the literature and give some further information about the results in the text. |
Contents
| 1 | |
| 9 | |
2 OrderTheoretic Results | 25 |
Theorem of Borsuk and Topological Transversality | 40 |
Some Simple Consequences | 94 |
Notes and Comments | 104 |
6 Fixed Points for Compact Maps in Normed Linear Spaces | 112 |
7 Further Results and Applications | 142 |
The LefschetzHopf Theory 14 Singular Homology | 369 |
Invariance of Homology under Barycentric Subdivision | 378 |
Excision | 384 |
Axiomatization | 386 |
Comparison of Homologies Künneth Theorem | 391 |
Homology and Topological Degree | 397 |
Miscellaneous Results and Examples | 402 |
Notes and Comments | 409 |
Homology and Fixed Points | 197 |
9 The LefschetzHopf Theorem and Brouwer Degree | 223 |
Theorem of BorsukHirsch | 236 |
Maps of Even and of OddDimensional Spheres | 237 |
Degree and Homotopy Theorem of Hopf | 239 |
Vector Fields on Spheres | 241 |
Miscellaneous Results and Examples | 243 |
Notes and Comments | 245 |
LeraySchauder Degree and Fixed Point Index 10 Topological Degree in R | 249 |
PL Maps of Polyhedra | 250 |
Polyhedral Domains in R Degree for Generic Maps | 251 |
Local Constancy and Homotopy Invariance | 254 |
Degree for Continuous Maps | 258 |
Some Properties of Degree | 260 |
Extension to Arbitrary Open Sets | 262 |
Axiomatics | 263 |
The Main Theorem on the Brouwer Degree in R | 266 |
Extension of the Antipodal Theorem | 268 |
Miscellaneous Results and Examples | 270 |
Notes and Comments | 274 |
11 Absolute Neighborhood Retracts | 279 |
ARs and ANRS | 280 |
Local Properties | 281 |
Pasting ANRS Together | 283 |
Theorem of Hanner | 285 |
Homotopy Properties | 287 |
Generalized LeraySchauder Principle in ANRS | 289 |
Miscellaneous Results and Examples | 292 |
Notes and Comments | 300 |
12 Fixed Point Index in ANRS | 305 |
Axioms for the Index | 308 |
The LeraySchauder Index in Normed Linear Spaces | 309 |
Commutativity of the Index | 312 |
Fixed Point Index for Compact Maps in ANRS | 315 |
The LeraySchauder Continuation Principle in ANRS | 317 |
Simple Consequences and Index Calculations | 321 |
Local Index of an Isolated Fixed Point | 326 |
Miscellaneous Results and Examples | 329 |
Notes and Comments | 333 |
EMMANUEL SPERNER 19051980 | 334 |
13 Further Results and Applications | 338 |
Application of the Index to Nonlinear PDEs | 344 |
The LeraySchauder Degree | 348 |
Extensions of the Borsuk and BorsukUlam Theorems | 352 |
The LeraySchauder Index in Locally Convex Spaces | 354 |
Miscellaneous Results and Applications | 357 |
Notes and Comments | 364 |
15 Lefschetz Theory for Maps of ANRs | 413 |
Generalized Lefschetz Number | 418 |
Lefschetz Maps and Lefschetz Spaces | 420 |
Lefschetz Theorem for Compact Maps of ANRS | 423 |
Asymptotic Fixed Point Theorems for ANRS | 425 |
Basic Classes of Locally Compact Maps | 426 |
Asymptotic LefschetzType Results in ANRS | 429 |
Periodicity Index of a Map Periodic Points | 431 |
Miscellaneous Results and Examples | 434 |
Notes and Comments | 437 |
16 The Hopf Index Theorem | 441 |
Homology of Polyhedra with Attached Cones | 444 |
The Hopf Index Theorem in Polyhedral Domains | 447 |
The Hopf Index Theorem in Arbitrary ANRs | 448 |
The LefschetzHopf Fixed Point Index for ANRS | 450 |
Some Consequences of the Index | 451 |
Miscellaneous Results and Examples | 456 |
Notes and Comments | 458 |
17 Further Results and Applications | 463 |
Fixed Points for SelfMaps of Arbitrary Compacta | 465 |
Forming New Lefschetz Spaces from Old by Domination | 467 |
Fixed Points in Linear Topological Spaces | 469 |
Fixed Points in NEScompact Spaces | 471 |
General Asymptotic Fixed Point Results | 474 |
7 Domination of ANRS by Polytopes | 475 |
Miscellaneous Results and Examples | 483 |
Notes and Comments | 488 |
Selected Topics 18 FiniteCodimensional Čech Cohomology | 491 |
Preliminaries | 492 |
Continuous Functors | 500 |
The Čech Cohomology Groups HnX | 506 |
L Ab | 511 |
Cohomology Theory on L | 513 |
Miscellaneous Results and Examples | 521 |
Notes and Comments | 523 |
19 Vietoris Fractions and Coincidence Theory | 531 |
Category of Fractions | 532 |
Vietoris Maps and Fractions | 534 |
Induced Homomorphisms and the Lefschetz Number | 536 |
Coincidence Spaces | 537 |
Some General Coincidence Theorems | 539 |
20 Further Results and Supplements | 551 |
Preliminaries | 588 |
| 620 | |
List of Standard Symbols | 668 |
| 678 | |
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Common terms and phrases
abelian group acyclic map Assume Banach space Borsuk bounded called Čech Čech cohomology chain closed commutative diagram compact fields compact map compact metric compact space continuous map convex set defined DEFINITION degree denote endomorphism exists extends finite finite-dimensional Fix(ƒ fixed point free fixed point index fixed point space fixed point theorem function functor given Hn(X homology groups homomorphism implies inclusion induced integer invariant isomorphism Lefschetz map Lefschetz number Lefschetz space LEMMA Leray Leray-Schauder Let f let ƒ linear subspace linear topological space locally convex space map f map ƒ Math metric space morphisms natural transformation nonempty normed linear space open set open subset pair polyhedron properties Prove retract satisfying Schauder Show simplex simplicial singular homology topological space U₁ vector space Vietoris