The Contraction Mapping Theorem and the Implicit and Inverse Function Theorems

denote the open ball of radius a centred on the origin in IR. If the function ~g : Ba → IR d obeys there is a constant G < 1 such that ‖~g(~x)− ~g(~y)‖ ≤ G ‖~x− ~y‖ for all ~x, ~y ∈ Ba (H1) ‖~g(~0)‖ < (1−G)a (H2) then the equation ~x = ~g(~x) has exactly one solution. Discussion of hypothesis (H1): Hypothesis (H1) is responsible for the word “Contraction” in the name of the theorem. Because G < 1 (and it is crucial that G is strictly smaller than 1) the distance between the images ~g(~x) and ~g(~y) of ~x and ~y is strictly smaller than the original distance between ~x and ~y. Thus the function g contracts distances. Note that, when the dimension d = 1 and the function g is C, |g(x)− g(y)| = ∣