Equilibria of set-valued maps on nonconvex domains

Abstract

Abstract. We present new theorems on the existence of equilibria (or zeros) of convex as well as nonconvex set-valued maps defined on compact neighborhood retracts of normed spaces. The maps are subject to tangency conditions expressed in terms of new concepts of normal and tangent cones to such sets. Among other things, we show that if K is a compact neighborhood retract with nontrivial Euler characteristic in a Banach space E , and Φ : K −→ 2 E is an upper hemicontinuous set-valued map with nonempty closed convex values satisfying the tangency conditionthen there exists x 0 ∈ K such that 0 ∈ Φ(x 0 ). Here, T r K (x) denotes a new concept of retraction tangent cone to K at x suited for compact neighborhood retracts. When K is locally convex at x, T r K (x) coincides with the usual tangent cone of convex analysis.Special attention is given to neighborhood retracts having "lipschitzian behavior", called L−retracts below. This class of sets is very broad; it contains compact homeomorphically convex subsets of Banach spaces, epi-Lipschitz subsets of Banach spaces, as well as proximate retracts. Our results thus generalize classical theorems for convex domains, as well as recent results for nonconvex sets.

Differential inclusions with constraints in Banach spaces

“…t ∈ I} is weakly compact in L 1 (I, E). 5 Applying it we infer that (r • u n ) converges weakly in L 1 (I, E) to some v ∈ L 1 (I, E) (over a subsequence). By (11), for a.e.…”

Generalized Poincaré-Hopf Theorem for Compact Nonsmooth Regions

“…Jongen et al [18] also provide related uniqueness results, but they derive these from the generalization of Morse theory. Ben-El-Mechaiekh and Kryszewski [1], Cornet [7], and Cornet and Czarnecki [8,9] use the axiomatic index (degree) theory (see Ortega and Rheinboldt [24], Chapter 6) to prove the existence of zeros of vector valued functions and correspondences, but they do not investigate uniqueness issues. Dierker [10], Mas-Colell [20], Varian [28], and Hildenbrand and Kirman [15] use the Poincaré-Hopf theorem to prove uniqueness results for general equilibrium economies with boundary conditions that restrict the equilibrium to be in the interior of the region (i.e., a zero of the vector field), and Eraslan and McLennan [11] use the axiomatic index theory to prove uniqueness in a bargaining game.…”

“…We next recall the notion of a normal cone which will be used in our analysis (see Clarke [5] or Cornet [7]): Definition 2.1. Let M be given by (1).…”

“…Generalized index theorem. We consider a region M given by (1) and present an extension of the Poincaré-Hopf theorem that applies for a generalized notion of critical points of a function over M. Figure 1).…”

“…Definition 3.2. Let M be a region given by (1). Let U be an open set containing M and F U → n be a function that is continuously differentiable at x ∈ Cr F M , where x is complementary and nondegenerate.…”