Originally Posted by

**Mollier** Hi,

I had my exam about a week ago, and here's one problem from it.

For which values of $\displaystyle a$ can we use the fixed-point iteration $\displaystyle x_{n+1}=g(x_n)$ to solve the following equation,

$\displaystyle x + a = e^{-x}$ ?

I know the fixed-point iteration will converge if $\displaystyle |g'(x)|<1$ for all $\displaystyle x$. I get,

$\displaystyle x = e^{-x}-a = g(x)$, and so $\displaystyle |g'(x)|=|e^{-x}|<1$. From this I conclude that $\displaystyle x>0$.

I now define two function,$\displaystyle f(x)=x+a$ and $\displaystyle g(x)=e^{-x}$. When $\displaystyle x>0$, a solution exists when $\displaystyle f(x)=g(x)$, that is when $\displaystyle a<1$.

Something smells here, but I don't know what. Hints are very welcome!