Chapter 7: Contraction Mappings and Applications

Fixed-point theorems are among the most powerful tools in mathematical analysis. They are found in nearly every area of pure and applied mathematics. In Corollary 5.9.16, we saw a very simple example of a fixed-point theorem, namely, if a map f : [a, b] → [a, b] is continuous, then it has a fixed point. This result generalizes to higher dimensions in what is called the Brouwer fixed-point theorem, which states that a continuous map on the closed unit ball D(0, 1) ⊂ 𝔽n into itself has a fixed point. The Brouwer fixed-point theorem can be generalized further to infinite dimensions by the Leray-Schauder fixed-point theorem, which is important in both functional analysis and partial differential equations.