Some of the fundamental results in analysis (inverse function theorem, existence and uniqueness of solutions to ODEs) have slick proofs using the idea of a contraction. So, it seems plausible to me that one might be motivated to study a "contraction space":
I'll define a contraction space as a metric space $(X,d)$ such that there is at least one fixed point of any function $f:X\rightarrow X$ with the property that for all $x,y$ we have $d(f(x),f(y)) \le \frac12 d(x,y)$.
Here's the question: is every contraction space complete?
I feel like the answer should be yes. The only approach I can see for proving this, though, is to find some way to take a Cauchy sequence $(a_n)$ and construct a contraction on the whole space taking each $a_i$ to some $a_j$ with $j > i$, but even for the case of Cauchy sequences in the rationals I don't see an obvious approach.
A related question: suppose we take an arbitrary metric space, and add a fixed point for every contraction the same way we add limits to Cauchy sequences. Is the resulting space then a contraction space?