Contractive Fixed Point Theorm for Vector Value Functions

Question

Consider the following question:

I tried to look on the function $g(x)=f(x)+[x_1,x_2]$ , so if I could prove that $g(x)\in S$ for $x\in S$, then by the contractive fixed point theorm there exists some vector $a$ such that $g(a)=a$ and therefore $f(a)=0$. I prove it in the following ways:

1. I tried to prove that $||g(x)||<=1$ for $||x||<=1$ but I couldn't do it, the calculations got too complex.
2. I tried to prove that $|g(b)-g(a)|<=K|(b-a)|$ for $a,b\in S$ and $0<K<1$ but I couldn't find any such $K$.
3. I tried to use the mean value theorm - since for every $a,b \in S$ there exists some vector $c\in S$ such that $g(b)-g(a)=(\nabla g)(c)(b-a)$ then if I can prove that $||(\nabla g)(c)||<1$ for $c\in S$ then I can find a $K$ for section 2.

Am I going in the right direction? Can I get any hints?