Let $\mathscr{F}:\mathfrak{A}\to\mathfrak{B}$ be a $C^{1}$-map between two Banach spaces such that $\mathscr{F}(x)-\mathscr{F}(0)=\mathscr{L}(x)+\mathscr{N}(x)$, where the operator $\mathscr{L}:\mathfrak{A}\to\mathfrak{B}$ is linear and $\mathscr{N}(0)=0$. Assume that

1. (1)

   $\mathscr{L}$ is an isomorphism with $\|\mathscr{L}^{-1}\|\leq C_{1}$,

2. (2)

   there are constants $r>0$ and $C_{2}>0$ with $r<\frac{1}{3C_{1}C_{2}}$ such that

   1. (a)

      $\|\mathscr{N}(x)-\mathscr{N}(y)\|_{\mathfrak{B}}\leq C_{2}\cdot(\|x\|_{\mathfrak{A}}+\|y\|_{\mathfrak{A}})\cdot\|x-y\|_{\mathfrak{A}}$ for all $x,y\in B_{r}(0)\subset{\mathfrak{A}}$,

   2. (b)

      $\|\mathscr{F}(0)\|_{\mathfrak{B}}\leq\frac{r}{2C_{1}}$,

then there exists a unique solution to $\mathscr{F}(x)=0$ in $\mathfrak{A}$ such that

(9.92)

$$\|x\|_{\mathfrak{A}}\leq 2C_{1}\cdot\|\mathscr{F}(0)\|_{\mathfrak{B}}.$$