Analysis MOC

Contraction map

A contraction map is a function 𝑓 :𝑋 →𝑌 between metric spaces (𝑋,𝑑𝑋) and (𝑌,𝑑𝑌) where for some 0 ≤𝐿 <1

𝑑𝑌(𝑓(𝑥),𝑓(𝑦))≤𝐿𝑑𝑌(𝑥,𝑦)

for all 𝑥,𝑦 ∈𝑋. anal Such a function is herefore a Lipschitz continuous function for which 𝑑𝑌(𝑓(𝑥),𝑓(𝑦)) <𝑑(𝑥,𝑦).

Properties

• Contraction map theorem states contracting endomorphisms of complete metric spaces have a fixed point.

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