The fixed-point method presents certain challenges that are important to consider.

The article reviews its origins, as well as the challenges that can arise in its use.Beyond its application in root calculus, the article explores its uses in integro differential and integral equations of the Vito Volterra and Hermann Von Helmholtz type, demonstrating the versatility of the method in various scientific problems.Hahn-Banach generalized this method, proving that every continuous and contractive function has a unique fixed point.His proof is based on Augustin Louis Cauchy's theorem, which states that every Cauchy sequence in a complete metric space is convergent.Because of the high level of abstraction required, this theory is not usually explored in depth in introductory mathematics courses.