I'm stuck at item (b) and would appreciate some help.
Let $E$ be a Banach space, $r>0$ and $F:B_{r}(0) \subset E \rightarrow E$ such that $$ F(0) = 0 \quad,\quad \mathrm{Lip}(F-\mathrm{id}_{E}) = \lambda < 1 $$ where $\mathrm{Lip}$ denotes the Lipschitz constant. Show that:
a) F is injective
b) $B_{s}(0) \subset \mathrm{Im}(F)$ where $s = (1-\lambda)r$ and there is an inverse $G$ of $F$ defined in $B_{s}(0)$ such that $\mathrm{Lip}(G-\mathrm{id}_{E}) = \frac{\lambda}{1-\lambda}$.
My solution of item (a):
Let $x,y \in E$ such that $F(x)=F(y)$ and $x \neq y$. If $d$ denotes the metric in $E$ induced from it's norm, then $$ d(F(x)-x,F(y)-y) = d(x,y) \leq \lambda d(x,y) < d(x,y) $$ and $d(x,y) < d(x,y)$ cannot be true.