I was working on the area of fixed point iteration. I couldn't find any historical account regarding the subject. Who was the first to use the this method?
There are several "fixpoint iteration methods" and several fixed point theorems underlying them. One of the earliest uses was "Picard's iteration method" for proving existence of solutions of ODE. It is based on the "Banach fixed point theorem", though Banach was not born yet when Picard discovered it.
Edit. Thanks to those who recalled Newton's method, which was discovered in a special case in Babylon thousands years ago. There are indeed very many "fixpoint methods".