Banach fixed-point theorem

In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem or Banach–Caccioppoli theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces and provides a constructive method to find those fixed points. It can be understood as an abstract formulation of Picard's method of successive approximations.[1] The theorem is named after Stefan Banach (1892–1945) who first stated it in 1922.[2][3]

  1. ^ Kinderlehrer, David; Stampacchia, Guido (1980). "Variational Inequalities in RN". An Introduction to Variational Inequalities and Their Applications. New York: Academic Press. pp. 7–22. ISBN 0-12-407350-6.
  2. ^ Banach, Stefan (1922). "Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales" (PDF). Fundamenta Mathematicae. 3: 133–181. doi:10.4064/fm-3-1-133-181. Archived (PDF) from the original on 2011-06-07.
  3. ^ Ciesielski, Krzysztof (2007). "On Stefan Banach and some of his results" (PDF). Banach J. Math. Anal. 1 (1): 1–10. doi:10.15352/bjma/1240321550. Archived (PDF) from the original on 2009-05-30.