The proof most commonly of the Inverse Function Theorem seen in textbooks of relies on the Banach fixed-point theorem.
QUESTION. It is possible to say, using historical references, which mathematician was the idea of using Banach's Contration Principle to prove the inverse function theorem ? If not, in which reference of the century XIX or century XX first appeared this proof?
I did a search on the Mathematics history books specialized in the subject. 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
But I did not get (at least my noticing) any track.