What do you call a fixed point theorem for a mapping from a subset of a space to the whole space? There are a number of fixed point theorems in which we have a map from some subset of a (metric, topological, ...) space to the whole space.  (Usually, there is some condition regarding the behavior of the function on the boundary of the set in question; an example would be the various nonlinear alternatives, see Granas & Dugundji's monograph.)
Is there any name such theorems share in common?
 A: I used to be (and still am) interested in the lovely topic f.p.p. but don't know any publications which would provide an answer to your question. Thus you have prompted me to come up with respective terminology.
Let me be systematic. First of all we need a name for functions from a set to its superset. Let's call them superfunctions:

A function $\ f:X\rightarrow Y\ $ is called a superfunction $\ \ \Leftarrow:\Rightarrow\ \ X\subseteq Y.\ $ A superfunction $\ f:X\rightarrow Y\ $ is true (resp. proper) $\ \ \Leftarrow:\Rightarrow\ \ X\subseteq f(X)\ \ $(resp. $\ X\subset f(X)$).

In practice, once may often use term superfunction to mean true (or proper) superfuntions only. One should say so explicitly then.
This is the keyword: superfunction. Now one can formulate all kind of s.f.p.p. (super fixed point property) for the needed classes of superfunctions with respect to the respective applications. Thus there will be a whole family of related notions.
A: Ecumenical fixed point theorems.
Surjective, prophetic, heraldic, clairvoyant, omnipresent, gestalt, symbiotic
