Lipschitz not contraction with a unique fixed point

Question

Define the scalar function $ g:\mathbb{R}\longrightarrow \mathbb{R}$ by $$ g(x):= \dfrac{x^2}{1+|x| +x^{2}}. $$ It is clear that $g$ is Lipschitzian and has a unique fixed point $x=0$ (one can prove that easily by assuming the converse). My question is to prove that $g$ is not a strict contraction. There is any hint or any idea? Thank you in advance!

Answer 1

Since $\sup_{x\in\mathbb{R}} |g'(x)|$ is a Lipschitz constant for $g$,then this graph looks like $g$ is a strict contraction with $\alpha\approx0.532$.