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Is there a fixed-point theorem regarding mappings of the form $T:2^S\to 2^S$, i.e. mappings that map subsets of $S$ to other subsets in $S$:

$$ T:S\supseteq A \mapsto T(A)\subseteq S $$

Where $S$ is a normed space.

In the particular case which I study, $S$ can be assumed to be convex and compact and for every $A\subseteq S$, it holds that $T(A)$ is convex whenever $T(A)$ is nonempty.

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Is there some particular structure on your $S$ that allows you to speak about "compact" and "convex" subsets? –  Henning Makholm Feb 8 '12 at 16:24
@HenningMakholm: Assume that $S\subset \Re^n$ with the standard Euclidean norm. But of course (if possible) it would be nice if we could just assume that $S$ is a topological vector space, otherwise the structure of an Euclidean space suffices for my needs. –  Pantelis Sopasakis Feb 8 '12 at 16:36

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