Past exam of the fixed point methods for nonlinear partial differential equations course of the University of Oxford / 2008 / 1

Here “contractive” means strictly distance-decreasing: $\boxed{d(Tx,Ty)<d(x,y)\text{ whenever }x\ne y}$. “Strongly contractive” means a contraction mapping with one uniform constant $q\in [0,1)$ satisfying $\boxed{d(Tx,Ty)\le qd(x,y)\text{ for all }x,y\in M}$.

If distinct $p,q$ were both fixed points, strict decrease would give $d(p,q)=d(Tp,Tq)<d(p,q)$, a contradiction. Thus there is at most one fixed point. This proves uniqueness if a fixed point exists; the pointwise strict inequality alone does not assert existence.

The contraction mapping theorem states that a self-map $T$ of a nonempty complete metric space with a uniform contraction constant $q<1$ has a unique fixed point $p$. For every starting point, the iteration $x_{n+1}=T{x}_n$ converges to $p$, with $d(x_n,p)\le q^{n}d(x_1,x_0)/(1-q)$.

For the cosine fixed-point iteration, any real $x_0$ gives $x_1\in [-1,1]$. This closed interval is complete, and cosine maps it into $[\cos 1,1]\subseteq [-1,1]$. The mean value theorem giveswith $\sin (1)<1$. Apply the contraction mapping theorem to the tail starting at $x_1$. Therefore $\boxed{x_n\to p,p=\cos p}$ for every real initial value. Numerically $p\approx 0.739085$. The global real-line derivative bound is only one; entering the invariant interval supplies the strict uniform bound needed by the theorem.

The Brouwer fixed-point theorem says that every continuous map $T:{\overline{B}}_R(0)\to {\overline{B}}_R(0)$ has a fixed point. To use the supplied outward vector field zero theorem, first extend the construction to all of ${\mathbb{R}}^n$. LetThe radial retraction onto a closed ball $P_R$ is continuous, so $g$ is a continuous vector field on the required whole space. On $|x|=R$, the Cauchy-Schwarz inequality givesThe supplied theorem gives $g(x_0)=0$ with $|x_0|\le R$. There $P_R(x_0)=x_0$, hence $\boxed{T(x_0)=x_0}$. This explicit extension avoids presuming that the original map $T$ was defined outside the ball.