MathDB

6

Part of 2005 Iran MO (3rd Round)

Problems(1)

Banach Theorem

Source: Iran 2005

8/27/2005
Suppose A⊆RmA\subseteq \mathbb R^m is closed and non-empty. Let f:A→Af:A\to A is a lipchitz function with constant less than 1. (ie there exist c<1c<1 that ∣f(x)−f(y)∣<c∣x−y∣, ∀x,y∈A)|f(x)-f(y)|<c|x-y|,\ \forall x,y \in A). Prove that there exists a unique point x∈Ax\in A such that f(x)=xf(x)=x.
functiontopologyalgebra proposedalgebraFixed point