Give an example of contraction function that has no fixed points.

Question

So we have a function $f: \mathbb{R} \rightarrow \mathbb{R}$ which is a contraction mapping. Can we find any example of $f$ that has no fixed points? We take into account standard Euclidean distance.

Answer 1

We have the Banach fixed-point theorem, which says you cannot.

The intuition: if you are in a metric space which is complete (meaning all Cauchy sequences have a limit), then a contraction applied to an arbitrary starting point will define a Cauchy sequence through its iterations, which in turn defines a limit. With some work, you can show that the limit of this sequence is a fixed point of the function. With a bit more work, you show that actually any such sequence gives you the same fixed point, regardless of the starting point, so you even get uniqueness!

Good point from the comments: In this context, a contraction is understood as a function $f$ such that $$ d(f(x), f(y)) \leq c d(x, y),$$ uniformly, for some constant $c <1$. Having $$ d(f(x), f(y)) < d(x, y),$$ is not sufficient, as one can concoct counterexamples!