Past exam of the mathematics course of the University of Cambridge / 2002 / ib / Paper 2 / 10E (Analysis II)

The contraction mapping theorem says that a self-map $g$ of a nonempty complete metric space satisfying $d(gx,gy)\le kd(x,y)$ for a constant $0\le k<1$ has exactly one fixed point. Apply it to $g=f^N$ to obtain $p$ with $f^{N}p=p$.

No continuity of $f$ itself is needed: composition givesThus $fp$ is also a fixed point of $f^N$, whose uniqueness forces $fp=p$. Conversely any fixed point of $f$ is fixed by $f^N$, so it must equal $p$. The unique fixed point of the contracting iterate is also the unique fixed point of $f$. The usual nonempty-space hypothesis is necessary; an empty space has no fixed point.

The strict distance inequality implies $d(fx,fy)\le d(x,y)$ for all points, so $f$ is Lipschitz continuous. Henceis a continuous function, since $|g(x)-g(y)|\le d(fx,fy)+d(x,y)\le 2d(x,y)$. On the nonempty compact metric space, $g$ attains its minimum at some $p$.

If $fp\ne p$, the strict distance inequality applied to $fp,p$ givesa contradiction. Thus $fp=p$. If $p,q$ were different fixed points, then $d(p,q)=d(fp,fq)<d(p,q)$, again impossible. There is exactly one fixed point. This proves strict distance decrease on a compact metric space without assuming a uniform contraction constant.

Let $K=\overline{O(x)}$. It is nonempty, closed and bounded in finite-dimensional Euclidean space, hence compact by the Heine-Borel theorem. The strict distance inequality again makes $f$ Lipschitz continuous. If $y\in K$, choose a sequence $y_j\in O(x)$ converging to $y$. Then $f{y}_j\in O(x)$, and continuity gives $f{y}_j\to fy$, so $fy\in K$. This proves $f(K)\subseteq K$.

The restriction to $K$ satisfies strict distance decrease on a compact metric space, so part (b) produces a fixed point there. If a second fixed point existed anywhere in ${\mathbb{R}}^n$, the same strict distance inequality applied to the two points would contradict equality of their distance. Therefore the fixed point is unique in all of ${\mathbb{R}}^n$. This is the bounded-orbit fixed point for a strictly distance-decreasing Euclidean map.