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.