Math 6420 • PDE

Suppose that \(T\) is a transformation of a complete metric space
\(X\), and \(T^n\) (for a certain positive integer \(n\)) is a contraction with fixed point \(a\).
Show that
\begin{enumerate}
\item[(1)] \(a\) is a fixed point of \(T\);
\item[(2)] \(T\) has only one fixed point;
\item[(3)] the sequence of successive iterations \(u_0\), \(u_1=Tu_0\), \(u_2=Tu_1,\ldots\)
tends to \(a\) for any initial element \(u_0\in X\).
\end{enumerate}