Optima and equilibria: an introduction to nonlinear analysis
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| Main Author: | |
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| Format: | Book |
| Language: | German French |
| Published: |
Berlin ; Heidelberg ; New York ; Barcelona ; Budapest ; Hong Kon
Springer
1998
|
| Edition: | 2. ed. |
| Series: | Graduate texts in mathematics
140 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaurverz. S. 421 - 424. - Orig.-Ausg. gesondert u.d.T.: L'analyse non linéaire et ses motivations économiques und Exercices d'analyse non linéaire |
| Physical Description: | XVII, 429 S. graph. Darst. |
| ISBN: | 3540649832 |
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| 100 | 1 | |a Aubin, Jean-Pierre |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Optima and equilibria |b an introduction to nonlinear analysis |c Jean-Pierre Aubin. Transl. from the French by Stephen Wilson |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Berlin ; Heidelberg ; New York ; Barcelona ; Budapest ; Hong Kon |b Springer |c 1998 | |
| 300 | |a XVII, 429 S. |b graph. Darst. | ||
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Record in the Search Index
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| adam_text | Table of Contents
Part I Nonlinear Analysis: Theory
1 Minimisation Problems: General Theorems 9
1.1 Introduction 9
1.2 Definitions 9
1.3 Epigraph 10
1.4 Lower Sections 11
1.5 Lower Semi continuous Functions 11
1.6 Lower Semi compact Functions 13
1.7 Approximate Minimisation of Lower Semi continuous Func¬
tions on a Complete Space 15
1.8 Application to Fixed point Theorems 17
2 Convex Functions and Proximation, Projection and Separa¬
tion Theorems 21
2.1 Introduction 21
2.2 Definitions 21
2.3 Examples of Convex Functions 24
2.4 Continuous Convex Functions 25
2.5 The Proximation Theorem 27
2.6 Separation Theorems 31
3 Conjugate Functions and Convex Minimisation Problems . 35
3.1 Introduction 35
3.2 Characterisation of Convex Lower Semi continuous Functions 37
3.3 Fenchel s Theorem 39
3.4 Properties of Conjugate Functions 43
3.5 Support Functions 48
3.6 The Cramer Transform 52
4 Subdifferentials of Convex Functions 57
4.1 Introduction 57
4.2 Definitions 61
4.3 Subdifferentiability of Convex Continuous Functions 64
xii Table of Contents
4.4 Subdifferentiability of Convex Lower Semi continuous Func¬
tions 66
4.5 Subdifferential Calculus 67
4.6 Tangent and Normal Cones 70
5 Marginal Properties of Solutions of Convex Minimisation
Problems 75
5.1 Introduction 75
5.2 Fermat s Rule 76
5.3 Minimisation Problems with Constraints 80
5.4 Principle of Price Decentralisation 82
5.5 Regularisation and Penalisation 84
6 Generalised Gradients of Locally Lipschitz Functions .... 87
6.1 Introduction 87
6.2 Definitions 87
6.3 Elementary Properties 91
6.4 Generalised Gradients 95
6.5 Normal and Tangent Cones to a Subset 97
6.6 Fermat s Rule for Minimisation Problems with Constraints . 99
7 Two person Games. Fundamental Concepts and Examples 101
7.1 Introduction 101
7.2 Decision Rules and Consistent Pairs of Strategies 102
7.3 Brouwer s Fixed point Theorem (1910) 104
7.4 The Need to Convexify: Mixed Strategies 105
7.5 Games in Normal (Strategic) Form 106
7.6 Pareto Optima 108
7.7 Conservative Strategies 110
7.8 Some Finite Games 112
7.9 Cournot s Duopoly 116
8 Two person Zero sum Games:
Theorems of Von Neumann and Ky Fan 125
8.1 Introduction 125
8.2 Value and Saddle Points of a Game 125
8.3 Existence of Conservative Strategies 130
8.4 Continuous Partitions of Unity 135
8.5 Optimal Decision Rules 137
9 Solution of Nonlinear Equations and Inclusions 143
9.1 Introduction 143
9.2 Upper Hemi continuous Set valued Maps 144
9.3 The Debreu Gale NikaMo Theorem 148
9.4 The Tangential Condition 149
Table of Contents xiii
9.5 The Fundamental Theorem for the Existence of Zeros of a
Set valued Map 150
9.6 The Viability Theorem 152
9.7 Fixed point Theorems 154
9.8 Equilibrium of a Dynamical Economy 155
9.9 Variational Inequalities 157
9.10 The Leray^Schauder Theorem 159
9.11 Quasi variational Inequalities 160
9.12 Shapley s Generalisation of the Three Poles Lemma 162
10 Introduction to the Theory of Economic Equilibrium .... 167
10.1 Introduction 167
10.2 Exchange Economies 168
10.3 The Walrasian Mechanism 169
10.4 Another Mechanism for Price Decentralisation 173
10.5 Collective Budgetary Rule 174
11 The Von Neumann Growth Model 179
11.1 Introduction 179
11.2 The Von Neumann Model 179
11.3 The Perron Frobenius Theorem 184
11.4 Surjectivity of the M matrices 187
12 n person Games 189
12.1 Introduction 189
12.2 Non cooperative Behaviour 189
12.3 n person Games in Normal (Strategic) Form 190
12.4 Non cooperative Games with Constraints (Metagames) . . . 192
12.5 Pareto Optima 193
12.6 Behaviour of Players in Coalitions 196
12.7 Cooperative Games Without Side Payments 197
12.8 Evolutionary Games 205
13 Cooperative Games and Fuzzy Games 209
13.1 Introduction 209
13.2 Coalitions, Fuzzy Coalitions and Generalised Coalitions of n
Players 209
13.3 Action Games and Equilibrium Coalitions 214
13.4 Games with Side Payments 216
13.5 Core and Shapley Value of Standard Games 224
xiv Table of Contents
Part II Nonlinear Analysis: Examples
14 Exercises 235
14.1 Exercises for Chapter 1 Minimisation Problems: General
Theorems 235
14.2 Exercises for Chapter 2 Convex Functions and Proximation,
Projection and Separation Theorems 240
14.3 Exercises for Chapter 3 Conjugate Functions and Convex
Minimisation Problems 245
14.4 Exercises for Chapter 4 Subdifferentials of Convex Functions 254
14.5 Exercises for Chapter 5 Marginal Properties of Solutions of
Convex Minimisation Problems 261
14.6 Exercises for Chapter 6 Generalised Gradients of Locally
Lipschitz Functions 268
14.7 Exercises for Chapter 8 Two person Zero sum Games: The¬
orems of Von Neumann and Ky Fan 275
14.8 Exercises for Chapter 9 Solution of Nonlinear Equations and
Inclusions 280
14.9 Exercises for Chapter 10 Introduction to the Theory of Eco¬
nomic Equilibrium 285
14.10 Exercises for Chapter 11 The Von Neumann Growth Model 290
14.11 Exercises for Chapter 12 n person Games 290
14.12 Exercises for Chapter 13 Cooperative Games and Fuzzy
Games 297
15 Statements of Problems 301
15.1 Problem 1 Set valued Maps with a Closed Graph 301
15.2 Problem 2 Upper Semi continuous Set valued Maps .... 301
15.3 Problem 3 Image of a Set valued Map 302
15.4 Problem 4 Inverse Image of a Set valued Map 302
15.5 Problem 5 Polars of a Set valued Map 303
15.6 Problem 6 Marginal Functions 303
15.7 Problem 7 Generic Continuity of a Set valued Map with a
Closed Graph 304
15.8 Problem 8 Approximate Selection of an Upper Semi continuous
Set valued Map 304
15.9 Problem 9 Continuous Selection of a Lower Semi continuous
Set valued Map 305
15.10 Problem 10 Interior of the Image of a Convex Closed Cone 305
15.11 Problem 11 Discrete Dynamical Systems 308
15.12 Problem 12 Fixed Points of Contractive Set valued Maps . 310
15.13 Problem 13 Approximate Variational Principle 311
15.14 Problem 14 Open Image Theorem 311
15.15 Problem 15 Asymptotic Centres 313
15.16 Problem 16 Fixed Points of Non expansive Mappings . . . 314
Table of Contents xv
15.17 Problem 17 Orthogonal Projectors onto Convex Closed Cones 315
15.18 Problem 18 Gamma convex functions 316
15.19 Problem 19 Proper Mappings 317
15.20 Problem 20 Fenchel s Theorem for the Functions L(x, Ax) 319
15.21 Problem 21 Conjugate Functions of x — L(x, Ax) 320
15.22 Problem 22 Hamiltonians and Partial Conjugates 321
15.23 Problem 23 Lack of Convexity and Fenchel s Theorem for
Pareto Optima 322
15.24 Problem 24 Duality in Linear Programming 323
15.25 Problem 25 Lagrangian of a Convex Minimisation Problem 324
15.26 Problem 26 Variational Principles for Convex Lagrangians 325
15.27 Problem 27 Variational Principles for Convex Hamiltonians 326
15.28 Problem 28 Approximation to Fermat s Rule 327
15.29 Problem 29 Transposes of Convex Processes 327
15.30 Problem 30 Cones with a Compact Base 329
15.31 Problem 31 Regularity of Tangent Cones 329
15.32 Problem 32 Tangent Cones to an Intersection 330
15.33 Problem 33 Derivatives of Set valued Maps with Convex
Graphs 331
15.34 Problem 34 Epiderivatives of Convex Functions 332
15.35 Problem 35 Subdifferentials of Marginal Functions 333
15.36 Problem 36 Values of a Game Associated with a Covering . 333
15.37 Problem 37 Minimax Theorems with Weak Compactness
Assumptions 334
15.38 Problem 38 Minimax Theorems for Finite Topologies ... 335
15.39 Problem 39 Ky Fan s Inequality 336
15.40 Problem 40 Ky Fan s Inequality for Monotone Functions . 337
15.41 Problem 41 Generalisation of the Gale Nikaido Debreu The¬
orem 338
15.42 Problem 42 Equilibrium of Coercive Set valued Maps . . . 339
15.43 Problem 43 Eigenvectors of Set valued Maps 339
15.44 Problem 44 Positive Eigenvectors of Positive Set valued Maps 340
15.45 Problem 45 Some Variational Principles 341
15.46 Problem 46 Generalised Variational Inequalities 341
15.47 Problem 47 Monotone Set valued Maps 343
15.48 Problem 48 Walrasian Equilibrium for Set valued Demand
Maps 344
16 Solutions to Problems 347
16.1 Problem 1 Solution. Set valued Maps with a Closed Graph 347
16.2 Problem 2 Solution. Upper Semi continuous set valued Maps 347
16.3 Problem 3 Solution. Image of a Set valued Map 348
16.4 Problem 4 Solution. Inverse Image of a Set valued Map . . 348
16.5 Problem 5 Solution. Polars of a Set valued Map 350
16.6 Problem 6 Solution. Marginal Functions 350
xvi Table of Contents
16.7 Problem 7 Solution. Generic Continuity of a Set valued Map
with a Closed Graph 351
16.8 Problem 8 Solution. Approximate Selection of an Upper
Semi continuous Set valued Map 351
16.9 Problem 9 Solution. Continuous Selection of a Lower Semi
continuous Set valued Map 352
16.10 Problem 10 Solution. Interior of the Image of a Convex
Closed Cone 352
16.11 Problem 11 Solution. Discrete Dynamical Systems 356
16.12 Problem 12 Solution. Fixed Points of Contractive Set valued
Maps 358
16.13 Problem 13 Solution. Approximate Variational Principle . 359
16.14 Problem 14 Solution. Open Image Theorem 360
16.15 Problem 15 Solution. Asymptotic Centres 362
16.16 Problem 16 Solution. Fixed Points of Non expansive Map¬
pings 363
16.17 Problem 17 Solution. Orthogonal Projectors onto Convex
Closed Cones 365
16.18 Problem 18 Solution. Gamma convex Functions 366
16.19 Problem 19 Solution. Proper Mappings 367
16.20 Problem 20 Solution. Fenchel s Theorem for the Functions
L(x,Ax) 368
16.21 Problem 21 Solution. Conjugate Functions of x — L(x, Ax) 369
16.22 Problem 22 Solution. Hamiltonians and Partial Conjugates 369
16.23 Problem 23 Solution. Lack of Convexity and Fenchel s The¬
orem for Pareto Optima 370
16.24 Problem 24 Solution. Duality in Linear Programming . . . 372
16.25 Problem 25 Solution. Lagrangian of a Convex Minimisation
Problem 373
16.26 Problem 26 Solution. Variational Principles for Convex La
grangians 374
16.27 Problem 27 Solution. Variational Principles for Convex
Hamiltonians 374
16.28 Problem 28 Solution. Approximation to Fermat s Rule . . 375
16.29 Problem 29 Solution. Transposes of Convex Processes . . . 376
16.30 Problem 30 Solution. Cones with a Compact Base 377
16.31 Problem 31 Solution. Regularity of Tangent Cones 378
16.32 Problem 32 Solution. Tangent Cones to an Intersection . . 379
16.33 Problem 33 Solution. Derivatives of Set valued Maps with
Convex Graphs 381
16.34 Problem 34 Solution. Epiderivatives of Convex Functions . 382
16.35 Problem 35 Solution. Subdifferentials of Marginal Functions 383
16.36 Problem 36 Solution. Values of a Game Associated with a
Covering 384
Table of Contents xvii
16.37 Problem 37 Solution. Minimax Theorems with Weak Com¬
pactness Assumptions 385
16.38 Problem 38 Solution. Minimax Theorems for Finite Topolo¬
gies 386
16.39 Problem 39 Solution. Ky Fan s Inequality 387
16.40 Problem 40 Solution. Ky Fan s Inequality for Monotone
Functions 388
16.41 Problem 41 Solution. Generalisations of the Gale Nikai do
Debreu Theorem 389
16.42 Problem 42 Solution. Equilibrium of Coercive Set valued
Maps 390
16.43 Problem 43 Solution. Eigenvectors of Set valued Maps . . . 391
16.44 Problem 44 Solution. Positive Eigenvectors of Positive Set
valued Maps 391
16.45 Problem 45 Solution. Some Variational Principles 391
16.46 Problem 46 Solution. Generalised Variational Inequalities . 393
16.47 Problem 47 Solution. Monotone Set valued Maps 395
16.48 Problem 48 Solution. Walrasian Equilibrium for Set valued
Demand Maps 397
Appendix
17 Compendium of Results 401
17.1 Nontrivial, Convex, Lower Semi continuous Functions .... 401
17.2 Convex Functions 403
17.3 Conjugate Functions 404
17.4 Separation Theorems and Support Functions 405
17.5 Subdifferentiability 408
17.6 Tangent and Normal Cones 409
17.7 Optimisation 411
17.8 Two Person Games 413
17.9 Set valued Maps and the Existence of Zeros and Fixed Points 415
References 421
Index 425
|
| any_adam_object | 1 |
| author | Aubin, Jean-Pierre |
| author_facet | Aubin, Jean-Pierre |
| author_role | aut |
| author_sort | Aubin, Jean-Pierre |
| author_variant | j p a jpa |
| building | Verbundindex |
| bvnumber | BV012097159 |
| callnumber-first | Q - Science |
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| callnumber-raw | QA427.A9313 1998 |
| callnumber-search | QA427.A9313 1998 |
| callnumber-sort | QA 3427 A9313 41998 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 860 SK 870 |
| ctrlnum | (OCoLC)39695501 (DE-599)BVBBV012097159 |
| dewey-full | 519.3 519.321 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.3 519.3 21 |
| dewey-search | 519.3 519.3 21 |
| dewey-sort | 3519.3 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| genre | 1\p (DE-588)4151278-9 Einführung gnd-content |
| genre_facet | Einführung |
| id | DE-604.BV012097159 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:21:40Z |
| institution | BVB |
| isbn | 3540649832 |
| language | German French |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008192576 |
| oclc_num | 39695501 |
| open_access_boolean | |
| owner | DE-92 DE-20 DE-91G DE-BY-TUM DE-706 DE-521 DE-634 DE-188 DE-19 DE-BY-UBM |
| owner_facet | DE-92 DE-20 DE-91G DE-BY-TUM DE-706 DE-521 DE-634 DE-188 DE-19 DE-BY-UBM |
| physical | XVII, 429 S. graph. Darst. |
| publishDate | 1998 |
| publishDateSearch | 1998 |
| publishDateSort | 1998 |
| publisher | Springer |
| record_format | marc |
| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| spelling | Aubin, Jean-Pierre Verfasser aut Optima and equilibria an introduction to nonlinear analysis Jean-Pierre Aubin. Transl. from the French by Stephen Wilson 2. ed. Berlin ; Heidelberg ; New York ; Barcelona ; Budapest ; Hong Kon Springer 1998 XVII, 429 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 140 Literaurverz. S. 421 - 424. - Orig.-Ausg. gesondert u.d.T.: L'analyse non linéaire et ses motivations économiques und Exercices d'analyse non linéaire Análisis matemático Convexe functies gtt Economie gtt Economía matemática Niet-lineaire analyse gtt Speltheorie gtt Wirtschaft Nonlinear theories Mathematical analysis Economics, Mathematical Gleichgewichtstheorie (DE-588)4071876-1 gnd rswk-swf Mathematisches Modell (DE-588)4114528-8 gnd rswk-swf Nichtlineare Analysis (DE-588)4177490-5 gnd rswk-swf Gleichgewicht (DE-588)4121372-5 gnd rswk-swf Spieltheorie (DE-588)4056243-8 gnd rswk-swf Optimierung (DE-588)4043664-0 gnd rswk-swf 1\p (DE-588)4151278-9 Einführung gnd-content Spieltheorie (DE-588)4056243-8 s Gleichgewichtstheorie (DE-588)4071876-1 s Nichtlineare Analysis (DE-588)4177490-5 s DE-604 Gleichgewicht (DE-588)4121372-5 s Mathematisches Modell (DE-588)4114528-8 s 2\p DE-604 Optimierung (DE-588)4043664-0 s 3\p DE-604 Graduate texts in mathematics 140 (DE-604)BV000000067 140 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008192576&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Aubin, Jean-Pierre Optima and equilibria an introduction to nonlinear analysis Graduate texts in mathematics Análisis matemático Convexe functies gtt Economie gtt Economía matemática Niet-lineaire analyse gtt Speltheorie gtt Wirtschaft Nonlinear theories Mathematical analysis Economics, Mathematical Gleichgewichtstheorie (DE-588)4071876-1 gnd Mathematisches Modell (DE-588)4114528-8 gnd Nichtlineare Analysis (DE-588)4177490-5 gnd Gleichgewicht (DE-588)4121372-5 gnd Spieltheorie (DE-588)4056243-8 gnd Optimierung (DE-588)4043664-0 gnd |
| subject_GND | (DE-588)4071876-1 (DE-588)4114528-8 (DE-588)4177490-5 (DE-588)4121372-5 (DE-588)4056243-8 (DE-588)4043664-0 (DE-588)4151278-9 |
| title | Optima and equilibria an introduction to nonlinear analysis |
| title_auth | Optima and equilibria an introduction to nonlinear analysis |
| title_exact_search | Optima and equilibria an introduction to nonlinear analysis |
| title_full | Optima and equilibria an introduction to nonlinear analysis Jean-Pierre Aubin. Transl. from the French by Stephen Wilson |
| title_fullStr | Optima and equilibria an introduction to nonlinear analysis Jean-Pierre Aubin. Transl. from the French by Stephen Wilson |
| title_full_unstemmed | Optima and equilibria an introduction to nonlinear analysis Jean-Pierre Aubin. Transl. from the French by Stephen Wilson |
| title_short | Optima and equilibria |
| title_sort | optima and equilibria an introduction to nonlinear analysis |
| title_sub | an introduction to nonlinear analysis |
| topic | Análisis matemático Convexe functies gtt Economie gtt Economía matemática Niet-lineaire analyse gtt Speltheorie gtt Wirtschaft Nonlinear theories Mathematical analysis Economics, Mathematical Gleichgewichtstheorie (DE-588)4071876-1 gnd Mathematisches Modell (DE-588)4114528-8 gnd Nichtlineare Analysis (DE-588)4177490-5 gnd Gleichgewicht (DE-588)4121372-5 gnd Spieltheorie (DE-588)4056243-8 gnd Optimierung (DE-588)4043664-0 gnd |
| topic_facet | Análisis matemático Convexe functies Economie Economía matemática Niet-lineaire analyse Speltheorie Wirtschaft Nonlinear theories Mathematical analysis Economics, Mathematical Gleichgewichtstheorie Mathematisches Modell Nichtlineare Analysis Gleichgewicht Spieltheorie Optimierung Einführung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008192576&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000067 |
| work_keys_str_mv | AT aubinjeanpierre optimaandequilibriaanintroductiontononlinearanalysis |