Fixed Point Theory: An IntroductionSpringer Science & Business Media, 30 nov. 2001 - 488 pages Approach your problems from the right It isn't that they can't see the solution. It end and begin with the answers. Then, is that they can't see the problem. one day, perhaps you will find the final G. K. Chesterton, The Scandal of Father question. Brown 'The Point of a Pin'. 'The Hermit Clad in Crane Feathers' in R. Van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of mono graphs and textbooks on increasingly specialized topics. However, the 'tree' of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. |
Table des matières
Topological Spaces and Topological Linear Spaces | 1 |
12 COMPACTNESS IN METRIC SPACES MEASURES OF NONCOMPACTNESS | 5 |
13 BAIRE CATEGORY THEOREM | 11 |
14 TOPOLOGICAL SPACES | 12 |
15 LINEAR TOPOLOGICAL SPACES LOCALLY CONVEX SPACES | 19 |
Hilbert Space and Banach Spaces | 26 |
22 HILBERT SPACES | 35 |
23 CONVERGENCE IN x X AND LX | 43 |
65 CONVERGENCE OF ITERATIONS OF NONEXPANSIVE MAPPINGS | 219 |
66 CLASSES OF MAPPINGS RELATED TO NONEXPANSIVE MAPPINGS | 224 |
67 COMPUTATION OF FIXED POINTS FOR CLASSES OF NONEXPANSIVE MAPPINGS | 230 |
MAPPING ON A ROTUND SPACE WITHOUT FIXED POINTS | 231 |
Sequences of Mappings and Fixed Points | 233 |
72 SEQUENCES OF MAPPINGS AND MEASURES OF NONCOMPACTNESS | 242 |
Duality Mappings and Monotone Operators | 245 |
81 DUALITY MAPPINGS | 246 |
24 THE ADJOINT OF AN OPERATOR | 45 |
25 CLASSES OF BANACH SPACES | 46 |
26 MEASURES OF NONCOMPACTNESS IN BANACH SPACES | 63 |
27 CLASSES OF SPECIAL OPERATORS ON BANACH SPACES Let X Y be two Banach spaces and TeLX Y | 65 |
The Contraction Principle | 72 |
3 1 THE PRINCIPLE OF CONTRACTION MAPPING IN COMPLETE METRIC SPACES | 74 |
32 LINEAR OPERATORS AND CONTRACTION MAPPINGS | 78 |
33 SOME GENERALIZATIONS OF THE CONTRACTION MAPPINGS | 79 |
34 HILBERTS PROJECTIVE METRIC AND MAPPINGS OF CONTRACTIVE TYPE | 92 |
35 APPROXIMATE ITERATION | 100 |
36 A CONVERSE OF THE CONTRACTION PRINCIPLE | 103 |
37 SOME APPLICATIONS OF THE CONTRACTION PRINCIPLE | 109 |
Brouwers Fixed Point Theorem | 113 |
42 BROUWERS FIXED POINT THEOREM EQUIVALENT FORMULATIONS | 116 |
43 ROBBINS COMPLEMENTS OF BROUWERS THEOREM | 125 |
44 THE BORSUKULAM THEOREM | 126 |
45 AN ELEMENTARY PROOF OF BROUWERS THEOREM | 132 |
46 SOME EXAMPLES | 139 |
47 SOME APPLICATIONS OF BROUWERS FIXED POINT THEOREM | 140 |
48 THE COMPUTATION OF FIXED POINTS SCARFS THEOREM | 143 |
Schauders Fixed point Theorem and some Generalizations | 152 |
51 THE SCH A UDER FIXED POINT THEOREM | 154 |
52 DARBOS GENERALIZATION OF SCHAUDERS FIXED POINT THEOREM | 159 |
53 KRASNOSELSKllS ROTHES AND ALTMANS THEOREMS | 165 |
SCHAUDERS AND TYCHONOFFS FIXED POINT THEOREM | 168 |
55 SOME APPLICATIONS | 177 |
Fixed Point Theorems for Nonexpansive Mappings and Related Classes of Mappings | 182 |
61 NONEXPANSIVE MAPPINGS | 183 |
62 THE EXTENSION OF NONEXPANSIVE MAPPINGS | 185 |
63 SOME GENERAL PROPERTIESOF NONEXPANSIVE MAPPINGS | 194 |
64 NONEXPANSIVE MAPPINGS ON SOME CLASSES OF BANACH SPACES | 195 |
82 MONOTONE MAPPINGS AND CLASSES OF NONEXPANSIVE MAPPINGS | 254 |
ON REAL BANACH SPACES | 257 |
84 SOME SURJECTIVITY THEOREMS IN COMPLEX BANACH SPACES | 265 |
85 SOME SURJECTIVITY THEOREMS IN LOCALLY CONVEX SPACES | 266 |
86 DUALITY MAPPINGS AND MONOTONICITY FOR SETVALUED MAPPINGS | 271 |
87 SOME APPLICATIONS | 272 |
Families of Mappings and Fixed Points | 276 |
92THE RYLLNARDZEWSKI FIXED POINT THEOREM | 283 |
93 FIXED POINTS FOR FAMILIES OF NONEXPANSIVE MAPPINGS | 287 |
AND FIXED POINTS FOR FAMILIES OF MAPPINGS | 292 |
Fixed Points and SetValued Mappings | 295 |
101 THE POMPEIUHAUSDORFF METRIC | 296 |
102 CONTINUITY FOR SETVALUED MAPPINGS | 299 |
OF SETVALUED MAPPINGS | 301 |
104 SETVALUED CONTRACTION MAPPINGS | 313 |
105 SEQUENCES OF SETVALUED MAPPINGS AND FIXED POINTS | 322 |
Fixed Point Theorems for Mappings on PMSpaces | 325 |
112 CONTRACTION MAPPINGS IN PMSPACES | 328 |
113 PROBABILISTIC MEASURES OF NONCOMPACTNESS | 337 |
114 SEQUENCES OF MAPPINGS AND FIXED POINTS | 341 |
The Topological Degree | 344 |
121 THE TOPOLOGICAL DEGREE IN FINITEDIMENSIONAL SPACES | 345 |
122 THE LERAYSCHAUDER TOPOLOGICAL DEGREE | 360 |
123 LERAYS EXAMPLE | 370 |
124 THE TOPOLOGICAL DEGREE FOR kSET CONTRACTIONS | 372 |
125 THE UNIQUENESS PROBLEM FOR THE TOPOLOGICAL DEGREE | 375 |
126 THE COMPUTATION OF THE TOPOLOGICAL DEGREE | 388 |
127 SOME APPLICATIONS OF THE TOPOLOGICAL DEGREE | 413 |
Bibliography | 419 |
459 | |
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Expressions et termes fréquents
a-set contraction Acad Amer applications assertion Banach space bounded set Brouwer's fixed point Bull C₁ Cauchy sequence class of mappings clearly compact set complete metric space continuous function continuous mapping contraction mapping conv converges convex set defined as follows DEFINITION deg f duality mapping element Example exists F. E. Browder finite finite-dimensional Fixed Point Theorems following properties following result following theorem give Hilbert space implies inequality integer intersection invariant Istrăţescu Lemma Let f linear space locally convex Math measure of noncompactness monotone neighbourhood nonempty Nonexpansive Mappings Nonlinear norm obtain open set Operators Petryshin PM-space point of f primitive set Proc Proof remark satisfies set-valued mapping simplex space and let subset subspace Suppose theorem is proved Theory topological degree topological space uniformly convex space vectors W. A. Kirk x₁ y₁