Fixed Point Theorem

An accurate map of California is spread out flat on a table in Evans Hall, in Berkeley. Prove that there is exactly one point on the map lying directly over the point it represents.

Define a sequence of positive numbers as follows. Let $x_0 > 0$ be any positive number, and let $x_{n+1} = (1+x_n)^{-1}$. Prove that this sequence converges, and find its limit.

Let $f : \mathbb {R} \to \mathbb {R}$ be monotonically increasing (perhaps discontinuous). Suppose $0 < f(0)$ and $f(100) < 100$. Prove $f(x) = x$ for some $x$.

Let $K$ be a nonempty compact set in a metric space with distance function $d$. Suppose that $\varphi : K \to K$ satisfies

for all $x \neq y$ in $K$. Show there exists precisely one point $x \in K$ such that $x = \varphi (x)$.

Let $K$ be a continuous function on the unit square $0 \leq x, y \leq 1$ satisfying $\left|K(x,y)\right| < 1$ for all $x$ and $y$. Show that there is a continuous function $f(x)$ on $[0,1]$ such that we have

Can there be more than one such function $f$?

Let $g$ be a continuous real valued function on $[0,1]$. Prove that there exists a continuous real valued function $f$ on $[0,1]$ satisfying the equation

Show there is a unique continuous real valued function $f : [0,1] \to \mathbb {R}$ such that

Let $(M,d)$ be a nonempty complete metric space. Let $S$ map $M$ into $M$, and write $S^2$ for $S \circ S$; that is, $S^2(x) = S(S(x))$. Suppose that $S^2$ is a strict contraction; that is, there is a constant $\lambda < 1$ such that for all points $x, y \in M$, $d(S^2(x), S^2(y)) \leq \lambda d(x,y)$. Show that $S$ has a unique fixed point in $M$.