This information is indicative and can be subject to change.
Equilibrium, fixed-points and computations
Teacher: Philippe Bich
E-mail: bich@univ-paris1.fr
ECTS : 2.5
Evaluation: written Exam or project.
Previsional Place and time: ROOM S2 of MSE, 15H30-17H, 12 sessions from January to march.
Prerequisites: Logic and Set Theory. Analysis in finite dimensional spaces (compact subsets, open subsets, closed subsets, metric, sequences, continuity, …), convexity.
Aim of the course: During the last 30 years, fixed-point theory has entailed important progresses in Economic Theory, Finance, Game Theory, Decision Theory, Network theory… This course covers basic fixed-point methods that interact with these fields. We shall cover Three important questions: Existence, Uniqueness (or number of equilibria) and computation (of equilibria or of fixed-points), using several kind of methods.
Syllabus:
1) Introduction; the fixed-point property. Deformation retract, homeomorphism.
2) Topological degree (definition, properties, applications).
3) Brouwer fixed-point theorem; existence of zero for inward vector fields.
4) Sperner Lemma; proof of Brouwer; computation.
5) Multivalued functions, Kakutani’s Theorem, existence of a maximal element.
6) Schauder’s theorem and the infinite dimensional case.
7) Ordered fixed-point theorems.
8) Banach Fixed-point theorem and some applications.
9) Examples, applications (depending on time)
References: Lecture notes will be given. See also: “Fixed-point theory” Granas-Dugundji.
Examen de l'année 2022:
| exam-point-fixe-2022.pdf |
