Fixed-point-like Theorems on Subspaces

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 ≤ dimV , 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 EilenbergMontgomery’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 MasColell [8] and Borsuk-Ulam’s theorem. The proof of the main result is given in Section 3