Let $\mathscr{F}:\mathfrak{A}\to\mathfrak{B}$ be a map between two Banach spaces such that for all $\bm{v}\in\mathfrak{A}$,

(6.2)

$$\mathscr{F}(\bm{v})-\mathscr{F}(\bm{0})=\mathscr{L}(\bm{v})+\mathscr{N}(\bm{v}),$$

where the operator $\mathscr{L}:\mathfrak{A}\to\mathfrak{B}$ is linear and the operator $\mathscr{N}:\mathfrak{A}\to\mathfrak{B}$ satisfies $\mathscr{N}(\bm{0})=\bm{0}$. Additionally we assume the following properties:

1. (1)

   (Bounded inverse) $\mathscr{L}:\mathfrak{A}\to\mathfrak{B}$ is an isomorphism and there is some constant $C_{L}>0$ such that

   (6.3)

   $$\|\mathscr{L}^{-1}\|_{op}\leq C_{L},$$

   where $\mathscr{L}^{-1}$ is the inverse of $\mathscr{L}$.

2. (2)

   There exists a constant $C_{N}>0$ and there is some $r_{0}\in(0,\frac{1}{2C_{L}C_{N}})$ satisfying the following:

   1. (a)

      (Controlled nonlinear error) for all $\bm{v}_{1},\bm{v}_{2}\in\overline{B_{r_{0}}(\bm{0})}\subset\mathfrak{A}$,

      (6.4)

      $$\|\mathscr{N}(\bm{v}_{1})-\mathscr{N}(\bm{v}_{2})\|_{\mathfrak{B}}\leq C_{N}\cdot r_{0}\cdot\|\bm{v}_{1}-\bm{v}_{2}\|_{\mathfrak{A}}.$$

   2. (b)

      (Controlled initial error) $\mathscr{F}(\bm{0})$ is effectively controlled as follows,

      (6.5)

      $$\|\mathscr{F}(\bm{0})\|_{\mathfrak{B}}\leq\frac{r_{0}}{4C_{L}}.$$

Then the equation $\mathscr{F}(\bm{x})=\bm{0}$ has a unique solution $\bm{x}\in B_{r_{0}}(\bm{0})$ with the estimate

(6.6)

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