Comptes Rendus
no. 9
Mathematical Analysis/Mathematical Economics
An extension of Brouwer's fixed point theorem allowing discontinuities

Presented by: Pierre-Louis Lions

Philippe Bich1
1 CEREMADE, Université de Paris Dauphine, place du Maréchal de Lattre de Tassigny, 75775 Paris cedex 16, France
Comptes Rendus. Mathématique, Volume 338 (2004) no. 9, pp. 673-678.

In this article, we extend Brouwer's fixed point theorem – which states that every continuous mapping f:B→B (a closed ball of ℝ n ) must have a fixed point – by allowing discontinuities of f, and we apply this extension to equilibrium theory in Economics.

Nous étendons dans cet article le théorème du point fixe de Brouwer – qui dit que toute fonction f continue de B dans B (une boule fermée de ℝ n ) admet un point fixe – en autorisant un certain type de discontinuité de f sur un ensemble éventuellement infini, et appliquons cette extension à la théorie de l'équilibre général en économie.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2004.03.001

Philippe Bich 1

1 CEREMADE, Université de Paris Dauphine, place du Maréchal de Lattre de Tassigny, 75775 Paris cedex 16, France 
@article{CRMATH_2004__338_9_673_0,
     author = {Philippe Bich},
     title = {An extension of {Brouwer's} fixed point theorem allowing discontinuities},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {673--678},
     publisher = {Elsevier},
     volume = {338},
     number = {9},
     year = {2004},
     doi = {10.1016/j.crma.2004.03.001},
     language = {en},
}
TY  - JOUR
AU  - Philippe Bich
TI  - An extension of Brouwer's fixed point theorem allowing discontinuities
JO  - Comptes Rendus. Mathématique
PY  - 2004
SP  - 673
EP  - 678
VL  - 338
IS  - 9
PB  - Elsevier
DO  - 10.1016/j.crma.2004.03.001
LA  - en
ID  - CRMATH_2004__338_9_673_0
ER  - 
%0 Journal Article
%A Philippe Bich
%T An extension of Brouwer's fixed point theorem allowing discontinuities
%J Comptes Rendus. Mathématique
%D 2004
%P 673-678
%V 338
%N 9
%I Elsevier
%R 10.1016/j.crma.2004.03.001
%G en
%F CRMATH_2004__338_9_673_0
Philippe Bich. An extension of Brouwer's fixed point theorem allowing discontinuities. Comptes Rendus. Mathématique, Volume 338 (2004) no. 9, pp. 673-678. doi : 10.1016/j.crma.2004.03.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.03.001/

[1] P. Bich, Some fixed point theorems allowing discontinuities, in press

[2] L.E.J. Brouwer U ¨ber Abbildung von Mannigfaltigkeiten, Math. Ann., Volume 71 (1912), pp. 97-115

[3] G. Debreu Theory of Value, Wiley, New York, 1959

[4] D. Duffie; W. Shaffer Equilibrium in incomplete markets, I. A basic model of generic existence, J. Math. Econom., Volume 14 (1985) no. 3, pp. 285-300

[5] M.W. Hirsch Differentiel Topology, Graduate Texts in Math., vol. 33, Springer-Verlag, New York, 1994

[6] M.W. Hirsch; M. Magill; A. Mas-Colell A geometric approach to a class of equilibrium existence theorems, J. Math. Econom., Volume 19 (1990) no. 1–2, pp. 95-106

[7] M. Magill; W. Shafer Incomplete markets (W. Hildenbrand; H. Sonnenschein, eds.), Handbook of Mathematical Economics, vol. 4, 1991, pp. 167-194

[8] J.W. Milnor; J.D. Stasheff Characteristic Classes, Ann. Math. Stud., vol. 76, Princeton University Press, NJ, 1974

Cited by Sources:

Comments - Policy