Math 6420 • PDE

Let \((X,d)\) be a complete metric space and \(f\) a contraction on \(X\) with contraction
constant \(\alpha<1\). Let \(x_f\) be the unique fixed point of \(f\). Fix \(x_0\in X\) and
consider the sequence of points defined by \(x_n=f(x_{n-1})\) for \(n\in\mathbb{N}\).
\begin{enumerate}
\item[(a)]
Prove that
\[
  d(x_n,x_f)\le \frac{\alpha^n}{1-\alpha}d(x_0,x_1).
\]
Hint: consider \(d(x_m,x_n)\) for \(m>n\).

\item[(b)]
Let \(S=[1,\infty)\) and \(f:S\to S\) be defined by
\(f(x)=\frac{1}{2}(x+2/x)\). Prove that \(f\) is a contraction mapping on \(S\) and compute
a contraction constant. Determine all fixed points of \(f\) on \(S\). Using part (a),
explain why the sequence \((x_n)\) might be useful.
\end{enumerate}