 Solution

On $(0,1)$ the map is positive. Its maximum occurs at $x=1/\sqrt{3}$ and equals $2\lambda /(3\sqrt{3})$. Thus if $\lambda <3\sqrt{3}/2$, then $f((0,1))\subset (0,1)$. Since $f$ is odd, also $f((-1,0))\subset (-1,0)$. Induction givesThe case $x_0=0$ is constant. These intervals are positively invariant sets for the discrete dynamical system.