2.II.10E

(a) Let $f$ be a map of a complete metric space $(X, d)$ into itself, and suppose that there exists some $k$ in $(0,1)$, and some positive integer $N$, such that $d\left(f^{N}(x), f^{N}(y)\right) \leqslant$ $k d(x, y)$ for all distinct $x$ and $y$ in $X$, where $f^{m}$ is the $m$ th iterate of $f$. Show that $f$ has a unique fixed point in $X$.

(b) Let $f$ be a map of a compact metric space $(X, d)$ into itself such that $d(f(x), f(y))<d(x, y)$ for all distinct $x$ and $y$ in $X$. By considering the function $d(f(x), x)$, or otherwise, show that $f$ has a unique fixed point in $X$.

(c) Suppose that $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ satisfies $|f(x)-f(y)|<|x-y|$ for every distinct $x$ and $y$ in $\mathbb{R}^{n}$. Suppose that for some $x$, the orbit $O(x)=\left\{x, f(x), f^{2}(x), \ldots\right\}$ is bounded. Show that $f$ maps the closure of $O(x)$ into itself, and deduce that $f$ has a unique fixed point in $\mathbb{R}^{n}$.

[The Contraction Mapping Theorem may be used without proof providing that it is correctly stated.]

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