Show root using Banach Fixpoint

Question

I'm required to show that: $f(x) = e^x - 4x$ has a root in $(0,1)$ using the Banach Fixpoint theorem.

The fact that $f((0,1)) \neq (0,1)$ confuses me. How do I proceed without knowing that $f$ isn't a map of a set onto itself?

Answer 1

You have to reformulate the requirement $f(x^\star) = e^{x^\star} -4x^\star = 0$ for a root such that the root $x^\star$ is a fixpoint of a function $g(x)$, that is $g(x^\star) = x^\star $. This is the case for $ g(x) = f(x)/4 +x = e^x/4$.

Now you have to show that $g(0,1) \subset (0,1)$ and $|g'(x)|<1$ on $x \in (0,1)$. Then, there is a fixpoint in $(0,1).$