QUESTION. It is possible to say, using historical references, which mathematician came up with the idea of using Banach's Contraction Principle to prove the inverse function theorem? If not, what is the earliest known source where such proof appeared?
I did a search on the Mathematics history books specialized in the subject, but I did not find any clues. For example:
Krantz, Steven G.; Parks, Harold R. The implicit function theorem. History, theory, and applications. Birkhäuser Boston, Inc., Boston, MA, 2002.
Dontchev, Asen L.; Rockafellar, R. Tyrrell Implicit functions and solution mappings. A view from variational analysis. Springer Monographs in Mathematics. Springer, Dordrecht, 2009