Conditions for nonexpansive mappings to have fixed points

Question

This is exercise 4.30 of Foundations of Functional Analysis, Ponnusamy. This is the whole problem:

Find a condition under which nonexpansive mappings have fixed points.

I have absolutely no idea where to start. Any help/insight would be appreciated.

Answer 1

Your starting point is to look at the conditions for fixed-point theorems, and specifically the Banach fixed-point theorem.

You might also think about why the map

$f(x)=x+\frac 1 2 \text{ : }0\le x <\frac 1 2\\f(x)=x-\frac 1 2\text{ : }\frac 1 2 \le x <1$

From $[0,1)$ to itself does not have a fixed point.