Exercise with a fixed point iteration

Question

I have a problem with this exercise.

I have to prove that the fixed point iteration $x_{k}=e^{-2-\frac{x_{k}}{2}}$ converges for any $x_{0} \in \mathbb{R}$. I want to use the Banach fixed-point theorem but I have problems proving that the function is a contraction in $\mathbb{R}$.

If we have $f(x)=e^{-2-\frac{x_{k}}{2}}$ this example proves that it is not a contraction in $\mathbb{R}$:

$|f(-10)-f(0)|\approx 20 \nleqslant c*|-10-0|$ for any $c \in [0,1)$.

Am I thinking anything wrong?

Is there any other way to prove the convergence of this fixed point iteration?

Answer 1

It is easy to prove from $f(x)$ being positive but strictly decreasing over all $\mathbb R$ that

• any negative number maps to a positive number in one iteration
• any nonnegative number maps to a number in $[0,e^{-2}]$ in one iteration
• any number in $[0,e^{-2}]$ maps to a number in $[e^{-2-e^{-2}/2},e^{-2}]\subset[0,e^{-2}]$, so $f$ is a contraction mapping over $[0,e^{-2}]$

Thus two iterations suffice to contract $\mathbb R$ to an interval where the Banach fixed-point theorem applies, so the fixed-point iteration is globally convergent.