Why do we study multi-valued(set valued) mappings? I am working on a problem that has to do with multi-valued mappings. Precisely, Iterative methods for Fixed points for multivalued mappings.  However, I have no clear motivation for studying such mappings, especially with regards to Hausdorff metric.
Why do we need to study multivalued mappings? What are the major/possible areas of applications? 
 A: I can't say anything about the applications to Hausdorff metrics, but there's one case where a multi-valued mapping comes up in a big way, and for a good reason:
assigning, to a nonzero point of the plane, an "angle", with points on the positive x-axis getting an angle of $0$, the positive $y$-axis getting angle $\pi/2$, etc. 
The problem with trying to do this nicely is that you get up all the way to nearly $2\pi$, and your assignments are continuous...and then at $2\pi$, you reach the $+x$-axis, and you're back to 0, so things aren't continuous. Also, we sometimes like to talk about angles greater than $2\pi$. Both these problems are addressed by saying that the value assigned to a point isn't a single angle, but a whole family of them, where any two differ by a multiple of $2 \pi$.
Why not just say "We're going to assign, to each nonzero point of the plane, a point on the unit circle?", since that's what's going on under the hood, so to speak. Well, one answer is "we're really good at doing arithmetic with real numbers, so it's pretty convenient to be able to treat angles as reals, or as something as close to reals as possible while still being consistent."
