Fixed-point-like theorems on subspaces
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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 ≤ 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 Copyright © 2004 Hindawi Publishing Corporation Fixed Point Theory and Applications 2004:3 (2004) 159–171 2000 Mathematics Subject Classification: 47H04, 47H10, 47H11 URL: http://dx.doi.org/10.1155/S1687182004406056
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Fixed-point-like theorems on subspaces
and the appendix recalls the main properties of the Grassmannian manifold, used in this paper. 2. Statement of the results 2.1. Preliminaries. A correspondence Φ from a set X to a set Y is a map from X to the set of all the subsets of Y , and the graph of Φ, denoted G(Φ), is defined by G(Φ) = {(x, y) ∈ X × Y | y ∈ Φ(x)}. A mapping ϕ : X → Y is said to be a selection of Φ if ϕ(x) ∈ Φ(x) for all x ∈ X. If A is a subset of X, we let Φ(A) = x∈A Φ(x), and the restriction of Φ to A, denoted Φ|A , is the correspondence from A to Y defined by Φ|A (x) = Φ(x) if x ∈ A. If X and Y are topological spaces, the correspondence Φ is said to be lower semicontinuous (l.s.c.) (resp., upper semicontinuous (u.s.c.)) if for every open set U ⊂ Y , the set {x ∈ X | Φ
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