MathDB
Banach Theorem

Source: Iran 2005

August 27, 2005
functiontopologyalgebra proposedalgebraFixed point

Problem Statement

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.
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