Let $\phi : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ be a map and let $x ^ { * }$ be a fixed point of $\phi$. Let $A \in \mathbf { M } _ { 2 } ( \mathbb { R } )$ be a matrix satisfying $\rho ( A ) < 1$, and let $M > 0$ be a real number. We assume that $\phi$ satisfies $$\forall x \in \mathbb { R } ^ { 2 } , \quad \left\| \phi ( x ) - \phi \left( x ^ { * } \right) - A \left( x - x ^ { * } \right) \right\| \leqslant M \left\| x - x ^ { * } \right\| ^ { 2 } .$$ Show that there exists $\varepsilon > 0$ such that for every $x _ { 0 } \in \mathbb { R } ^ { 2 }$ satisfying $\left\| x _ { 0 } - x ^ { * } \right\| < \varepsilon$, the sequence $\left( x _ { n } \right) _ { n \geqslant 0 }$ defined by $x _ { n + 1 } = \phi \left( x _ { n } \right)$ (for $n \geqslant 0$ ) converges to $x ^ { * }$ when $n \rightarrow + \infty$.
Let $\phi : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ be a map and let $x ^ { * }$ be a fixed point of $\phi$. Let $A \in \mathbf { M } _ { 2 } ( \mathbb { R } )$ be a matrix satisfying $\rho ( A ) < 1$, and let $M > 0$ be a real number. We assume that $\phi$ satisfies
$$\forall x \in \mathbb { R } ^ { 2 } , \quad \left\| \phi ( x ) - \phi \left( x ^ { * } \right) - A \left( x - x ^ { * } \right) \right\| \leqslant M \left\| x - x ^ { * } \right\| ^ { 2 } .$$
Show that there exists $\varepsilon > 0$ such that for every $x _ { 0 } \in \mathbb { R } ^ { 2 }$ satisfying $\left\| x _ { 0 } - x ^ { * } \right\| < \varepsilon$, the sequence $\left( x _ { n } \right) _ { n \geqslant 0 }$ defined by $x _ { n + 1 } = \phi \left( x _ { n } \right)$ (for $n \geqslant 0$ ) converges to $x ^ { * }$ when $n \rightarrow + \infty$.