Advertisement

Contraction Mappings

Banach’s Contraction-Mapping Principle
  • Joel H. Shapiro
Chapter
Part of the Universitext book series (UTX)

Overview

In this chapter we’ll study the best-known of all fixed-point theorems: the Banach Contraction-Mapping Principle, which we’ll apply to Newton’s Method, initial-value problems, and stochastic matrices.Prerequisites. Undergraduate-level real analysis and linear algebra. The basics of metric spaces: continuity and completeness.

References

  1. 5.
    Banach, S.: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fund. Math. 3, 133–181 (1922)zbMATHGoogle Scholar
  2. 17.
    Brin, S., Page, L.: The anatomy of a large-scale hypertextual web search engine. In: Seventh International World-Wide Web Conference (WWW 1998), Brisbane, 14–18 April 1998. Available online at http://infolab.stanford.edu/pub/papers/google.pdf (1998)
  3. 20.
    Bryan, K., Leise, T.: The $25,000,000,000 eigenvector: the linear algebra behind Google. SIAM Rev. 48, 569–581 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 66.
    Lalley, S.: Markov Chains: Basic Theory. Lecture Notes available from http://galton.uchicago.edu/lalley/Courses/312/MarkovChains.pdf (2009)
  5. 70.
    Lindel&f, M.E.: Sur l’application de la méthode des approximations successives aux équations différentiales ordinaires du premier ordre. C. R. Math. Acad. Sci. Paris 114, 454–457 (1894)Google Scholar
  6. 93.
    Perko, L.: Differential Equations and Dynamical Systems, 3rd edn. Springer, New York (2000)zbMATHGoogle Scholar
  7. 101.
    Rudin, W.: Principles of Mathematical Analysis, 3rd edn. McGraw-Hill, New York (1976)zbMATHGoogle Scholar
  8. 117.
    Trefethen, L.N., Bau III, D.: Numerical Linear Algebra. SIAM, Philadelphia (1997)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Joel H. Shapiro
    • 1
  1. 1.Portland State UniversityPortlandUSA

Personalised recommendations