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Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: FIXED-POINT-LIKE THEOREMS ON SUBSPACES | FIXED-POINT-LIKE THEOREMS ON SUBSPACES PHILIPPE BICH AND BERNARD CORNET Received 8 June 2004 We prove a fixed-point-like theorem for multivalued mappings defined on the finite Cartesian product of Grassmannian manifolds and convex sets. Our result generalizes two different kinds of theorems the fixed-point-like theorem by Hirsch et al. 1990 or Husseini et al. 1990 and the fixed-point theorem by Gale and Mas-Colell 1975 which generalizes Kakutani s theorem 1941 . 1. Introduction In this paper we prove a fixed-point-like theorem for multivalued mappings defined on the finite Cartesian product of Grassmannian manifolds and convex sets. Let k be an integer and let V be a Euclidean space such that 0 k dim V then the k-Grassmannian manifold of V denoted Gk V is the set of all the k-dimensional subspaces of V. The set Gk V is a smooth compact manifold but in general it does not satisfy properties such as convexity or acyclicity and its Euler characteristic may be null. This prevents the use of classical fixed-point theorems as Brouwer s 2 Kakutani s 14 or Eilenberg-Montgomery s theorem 7 . Our main result generalizes two different kinds of theorems the fixed-point-like theorem by Hirsch et al. 11 or Husseini et al. 13 and the fixed-point theorem by Gale and Mas-Colell 8 which generalizes Kakutani s theorem 14 . As in 11 13 we will mainly use techniques from degree theory. As a consequence of our main result we first deduce the standard fixed-point theorems when the variable is in a convex domain such as Brouwer and Kakutani s theorem and second Borsuk-Ulam s theorem. The main result of this paper is directly motivated by the existence problem of equilibria in economic models with incomplete markets in 1 it is used to extend the classical existence result by Duffie and Shafer 6 to the nontransitive case. The paper is organized as follows. The main result is stated in Section 2 together with some direct consequences of it namely the results by Hirsch et al. 11 Gale and .
