Lipschitz continuous set-valued map Let $X$ and $Y$ be metric spaces and $F:X\to2^Y$ be a set-valued map. Suppose that $2^Y$ is endowed with the Hausdorff metric. I wonder about sufficient conditions on $F$ that ensure this map is Lipschitz continuous with a given constant $c$. I think there may be several situations, so I'd be interested in a paper/book chapter that treats this problem from different perspectives.
For example, I am particularly interested in the following situation. Suppose that $U$ is another metric space, $G:X\to 2^U$ is a set-valued map and $h:X\times U\to Y$ is an ordinary map. Which conditions $G$ and $h$ need to satisfy to ensure Lipschitz continuity of $F$ given by
$$
  F(x) = h(x,G(x)) = \bigcup_{u\in G(x)}\{h(x,u)\}.
$$
Please feel free to retag.
 A: Since the Hausdorff metric does not distinguish between a set and its closure, we can take the quotient of $2^Y$ by the equivalence relation "$A\sim B$ if $\overline{A}=\overline{B}$", reducing the problem to the study of the space of closed subsets. This is usually called a hyperspace, though hyperspaces come in different flavors: e.g., in normed spaces one may want to consider only convex subsets, in other contexts people look at hyperspaces of continua. Much of the research in the area was traditionally  on topological structure of hyperspaces, the Lipschitz structure got attention relatively recently. The introduction of Bi-Lipschitz embedding of compacta hyperspaces by Tyson gives pointers to the literature.
A Lipschitz map between metric spaces induces a Lipschitz map between their hyperspaces. And a non-Lipschitz map between metric spaces induces a hyperspace map that is not Lipschitz (consider one-point subsets).
In your case I would introduce $\tilde G:X\to 2^{X\times U}$ defined by $G(x) = \{x\}\times G(x)$. Observe that $\tilde G$ is Lipschitz continuous if and only if $G$ is. 
The map $F$ is the composition of $\tilde G$ with the hyperspace map induced by $h$. If both $\tilde G$ and $h$ are Lipschitz continuous, conclusion follows. Otherwise, there is nothing to say in this generality: although the composition of non-Lipschitz maps may happen to be Lipschitz by a lucky coincidence, there is no useful general statement that ensures this. 
