The Banach Fixed Point Theorem: selected topics from its hundred-year history

• Published in: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas Vol. 118; no. 4
• Main Authors: Jachymski, Jacek, Jóźwik, Izabela, Terepeta, Małgorzata
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.10.2024
• Subjects: Applications of Mathematics; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Survey; Theoretical; Complete metric space; Applications of the contraction principle; 01A61; 01A60; 47H10; Banach fixed point theorem; 47H09; Semimetric space; Banach space; Contraction; 54H25
• ISSN: 1578-7303; 1579-1505
• DOI: 10.1007/s13398-024-01636-6

On June 24, 1920 Stefan Banach presented his doctoral dissertation titled O operacjach na zbiorach abstrakcyjnych i ich zastosowaniach do równañ całkowych ( On operations on abstract sets and their applications to integral equations ) to the Philosophy Faculty of Jan Kazimierz University in Lvov. He passed his PhD examinations in mathematics, physics and philosophy, and in January 1921 he became a doctor. A year later, he published the results of his doctorate in Fundamenta Mathematicae. Among them there was the theorem known today as the Banach Fixed Point Theorem or the Banach Contraction Principle. It is one of the most famous theorems in mathematics, one of many under the name of Banach. It concerns certain mappings (called contractions) of a complete metric space into itself and it gives the conditions sufficient for the existence and uniqueness of a fixed point of such mapping. In 2022 we had a centenary of publishing this theorem. In the paper, we want to present its most important modifications and generalizations, several contractive conditions, the converse theorems and some applications. It is not possible to provide complete information about what has been written during the last hundred years about the Banach Fixed Point Theorem and we are just trying to touch on some breakthrough moments in the development of the metric fixed point theory. The main purpose of this article is to organize the knowledge on this subject and to elaborate a broad bibliography which all interested persons can refer to.