Does every real- analytic function defined in the neighborhood of$ x= 0$ must have a fixed point?

Question

let $f$ be a solution for an ODE and it is a real analytic function in defined in neighborhood of x=0 over $\mathbb{R}$.

Question: Does every real- analytic function defined in the neighborhood of$ x= 0$ must have a fixed point?

Answer 1

It doesn't. In fact, it has to only if it is lipschitzian with k<1, which is, for analytic functions: $|f'(x)|≤ k <1$

Answer 2

No. Consider f(x) = exp(x) on ( - 1 , 1).