Suppose that $f:E\to F$(between Banach spaces), is of the form $$f(x)=f(0)+D(x)+N(x).$$ Here $D$ is a linear term, whose kernel is of finite dimension, and admits a right inverse $G$, i.e. $D(G)(\cdot)=Id_F$. The nonlinear term $N(x)$ satisfies that $$\|G(N(x))-G(N(y))\|\leq C(\|x\|+\|y\|)\|x-y\|$$
for some constant $C>0$ and $x,y$ in a small neighborhood $B_{\epsilon(C)}(0)$.
Q: How to show that there is a unique zero point $x_0$ of $f$, i.e. $f(x_0)=0$, in $B_\epsilon(0)\cap G(F)$, with the initial condition $\|G(f(0))\|\leq\epsilon/2$?