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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?

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  • $\begingroup$ followed links, eventually found your publications; maybe you should type in a few of your favorite results of the type you mean, at least by name. I never know what to say when people ask for names of things, never sure what is the purpose of knowing a name. For early students, i think it may be to search on Google, but... $\endgroup$
    – Will Jagy
    Commented Sep 21, 2014 at 21:15
  • $\begingroup$ The reason I'm asking is that I'm writing a booklet on fixed point theory, and I'm in search for a good section title;). When writing papers, I never needed a single name to refer to such theorems. $\endgroup$
    – mbork
    Commented Sep 21, 2014 at 21:17
  • $\begingroup$ Makes sense.... $\endgroup$
    – Will Jagy
    Commented Sep 21, 2014 at 21:24
  • $\begingroup$ Is the map usually required surjective? $\endgroup$
    – Will Jagy
    Commented Sep 22, 2014 at 0:28
  • $\begingroup$ No. (I'll try to include some examples of such theorems when I have a bit of spare time, maybe today in the afternoon.) $\endgroup$
    – mbork
    Commented Sep 22, 2014 at 10:58

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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.

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    $\begingroup$ If $ f:X\rightarrow Y $ then it makes sense to write $ f:X\rightarrow X\cup Y $ as well. In that sense every function is a superfunction. $\endgroup$
    – Bjørn Kjos-Hanssen
    Commented Sep 22, 2014 at 3:33
  • $\begingroup$ Interesting. But I would use your @Bjørn notation (one just above) only on special occasions. $\endgroup$
    – Włodzimierz Holsztyński
    Commented Sep 22, 2014 at 4:01
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Ecumenical fixed point theorems.

Surjective, prophetic, heraldic, clairvoyant, omnipresent, gestalt, symbiotic

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  • $\begingroup$ :-) -- very good! $\endgroup$
    – Włodzimierz Holsztyński
    Commented Sep 21, 2014 at 23:09
  • $\begingroup$ ..., encompassing, generous, overwhelming, possessive, sybaritic, ... (in the alphabetic order). $\endgroup$
    – Włodzimierz Holsztyński
    Commented Sep 22, 2014 at 2:41
  • $\begingroup$ @WłodzimierzHolsztyński, all good suggestions. You seem to know this area, are the maps under discussion usually surjective? $\endgroup$
    – Will Jagy
    Commented Sep 22, 2014 at 2:43
  • $\begingroup$ Perhaps, @Will Jagy, someone like Andrzej Granas would know (I am not well read, sorry). $\endgroup$
    – Włodzimierz Holsztyński
    Commented Sep 22, 2014 at 3:00
  • $\begingroup$ @WłodzimierzHolsztyński, alright. That strikes me as a possible name or part of a phrase, a likely feature of many such things, either as a hypothesis or early conclusion on the way to proof. $\endgroup$
    – Will Jagy
    Commented Sep 22, 2014 at 3:28

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