Equilibria of set-valued maps on nonconvex domains
Hichem Ben-El-Mechaiekh, Wojciech Kryszewski · Transactions of the American Mathematical Society · 1997
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 K is a compact neighborhood retract with nontrivial Euler characteristic in a Banach space E E , and Φ : K ⟶ 2 E \Phi :K\longrightarrow 2^E is an upper hemicontinuous set-valued map with nonempty closed convex values satisfying the tangency condition Φ ( x ) ∩ T K r ( x ) ≠ ∅ for all x ∈ K , \begin{equation*} \Phi (x)\cap T_K^r(x) eq \emptyset \text { for all }x\in K, \end{equation*} then there exists x 0 ∈ K x_0\in K such that 0 ∈ Φ ( x 0 ) . 0\in \Phi (x_0). Here, T K r ( x ) T_K^r(x) denotes a new concept of retraction tangent cone to K K at x x suited for compact neighborhood retracts. When K K is locally convex at