Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Equilibria of set-valued maps
on nonconvex domains

Authors: H. Ben-El-Mechaiekh and W. Kryszewski
Journal: Trans. Amer. Math. Soc. 349 (1997), 4159-4179
MSC (1991): Primary 47H10, 47H04; Secondary 54C55
MathSciNet review: 1401763
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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 $\Phi :K\longrightarrow 2^E$ is an upper hemicontinuous set-valued map with nonempty closed convex values satisfying the tangency condition

\begin{equation*}\Phi (x)\cap T_K^r(x)\neq \emptyset \text { for all }x\in K, \end{equation*}

then there exists $x_0\in K$ such that $0\in \Phi (x_0).$ Here, $T_K^r(x)$ denotes a new concept of retraction tangent cone to $K$ at $x$ suited for compact neighborhood retracts. When $K$ is locally convex at $x,T_K^r(x)$ 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.

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Additional Information

H. Ben-El-Mechaiekh
Affiliation: Instytut Matematyki, Uniwersitet Mikolaja Kopernika, ul. Chopina 12/18, 87-100 Toruń, Poland
Address at time of publication: Department of Mathematics, Sultan Qaboos Universisty, P.O. Box 50, Al-Khod, Oman

W. Kryszewski
Affiliation: Instytut Matematyki, Uniwersitet Mikolaja Kopernika, ul. Chopina 12/18, 87-100 Toruń, Poland
Email: and wkrysz@plunlo51.bitnet

Keywords: Equilibria, nonconvex set-valued maps, compact neighborhood retracts, normal and tangent retraction cones and inwardness, $L-$retracts
Received by editor(s): October 17, 1994
Received by editor(s) in revised form: March 18, 1996
Additional Notes: Research supported by the Natural Sciences and Engineering Research Council of Canada under grant OGP0042422
Article copyright: © Copyright 1997 American Mathematical Society