How to use the Contraction Mapping Theorem to prove the following result: Let $X$ and $Y$ be Banach spaces, let $a>0$, and let $B_a=B_a\left(z_0\right)=\left\{z \in X:\left\|z-z_0\right\| \leq a\right\}$. Suppose that $F$ is a $C^1$ map of $B_a$ into $Y$, with $F^{\prime}\left(z_0\right)$ invertible, and satisfying, for some $0<\theta<1$, $$ \left\|F^{\prime}\left(z_0\right)^{-1} F\left(z_0\right)\right\| \leq(1-\theta) a, $$ and $$ \left\|F^{\prime}\left(z_0\right)^{-1}\right\|\left\|F^{\prime}(z)-F^{\prime}\left(z_0\right)\right\| \leq \theta \quad \text { for all } z \in B_a . $$ Then there is a unique solution in $B_a$ of $F(z)=0$.

How to use the contraction mapping theorem to prove the following result: Let $X$ and $Y$ be Banach spaces, let $a>0$, and let $$B_a=B_a\left(z_0\right)=\left\{z \in X:\left\|z-z_0\right\| \leq a\right\}.$$ Suppose that $F$ is a $C^1$ map of $B_a$ into $Y$, with $F^{\prime}\left(z_0\right)$ invertible, and satisfying, for some $0<\theta<1$, $$ \left\|F^{\prime}\left(z_0\right)^{-1} F\left(z_0\right)\right\| \leq(1-\theta) a, $$ and $$ \left\|F^{\prime}\left(z_0\right)^{-1}\right\|\left\|F^{\prime}(z)-F^{\prime}\left(z_0\right)\right\| \leq \theta \quad \text { for all } z \in B_a . $$ Then there is a unique solution in $B_a$ of $F(z)=0$.