A more robust version of the contraction mapping theorem is given. The first version stated that if f is a contraction on a complete metric space, then it has a unique fixed point. This version states that if some iterate of f is a contract, then f has a unique fixed point.
This is part of a series of lectures on Mathematical Analysis II. Topics covered include continuous and differentiable multi-variable functions on Euclidean space, the chain rule, the implicit function theorem, manifolds, tangent spaces, vector fields, the degree and index of a smooth map, the Euler characteristic, metric spaces, the contraction mapping theorem, existence and uniqueness of solutions to ordinary differential equations, and integral equations.
I speak rather slowly, so you may wish to increase the speed of this video.
These videos were created during the 2017 Spring semester at the UConn CETL Lightboard Room.