Skip to main content

On numerical approximation of fixed points in C[0,1]

Part of the Lecture Notes in Mathematics book series (LNM,volume 730)

Abstract

The present paper discusses the connection between fixed points in function spaces and finite dimensional approximations. Utilizing algebraic properties of the space of continuous functions C[0,1] we obtain

  1. (i)

    a simple criterion for testing consistency of an approximation.

    Theorem: Consistency is equivalent to TN(a)=a for all N and for all generators {a}={a1, a2, ..., an} of the algebra,

  2. (ii)

    an understanding of stability from an algebraic point of view,

  3. (iii)

    a simple proof for the equivalence theorem.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   54.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   69.95
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Allgower, E.L.; Jeppson, M.M., The Approximation of Solutions of Mildly Nonlinear Elliptic Boundary Value Problems having several Solutions, Springer Lecture Notes 333, Berlin-Heidelberg-New York 1973.

    Google Scholar 

  2. Wilmuth, R.J., The Computation of Fixed Points, Ph.D. Thesis, Stanford University 1973.

    Google Scholar 

  3. Allgower, E.L., Application of a Fixed Point Search Algorithm to Nonlinear Boundary Value Problems having several Solutions, p. 87–p. 111, in reference 4.

    Google Scholar 

  4. Karamardian, S. (Ed.), Fixed Points Algorithms and Applications, Academic Press, New York 1977.

    MATH  Google Scholar 

  5. Courant, R.; Friedrichs, K.; Lewy, H., Über die partiellen Differentialgleichungen der mathematischen Physik, Math. Ann. 100, 32–74 (1928).

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. Lax, P.D.; Richtmyer, R.D., Survey of the Stability of Linear Finite Difference Equations, Comm. Pure Appl. Math. 9, 267–293 (1956).

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. Dahlquist, G., Convergence and Stability in the Numerical Integration of Ordinary Differential Equations, Math. Scand. 4, 33–53 (1956).

    MathSciNet  MATH  Google Scholar 

  8. Henrici, P., Discrete Variable Methods in Ordinary Differential Equations, Wiley, New York 1962.

    MATH  Google Scholar 

  9. Henrici, P., Error Propagation for Difference Methods, Wiley, New York 1963.

    MATH  Google Scholar 

  10. Stummel, F., Discrete Convergence of Mappings, p. 285–310, in: Topics in Numerical Analysis, Academic Press, London 1973.

    Google Scholar 

  11. Petryshyn, W.V., Nonlinear Equations Involving Noncompact Operators, p. 206–233, in: Nonlinear Functional Analysis, American Mathematical Society, Providence, R.I. 1970.

    CrossRef  Google Scholar 

  12. Schauder, J., Der Fixpunktsatz in Funktionalräumen, Studia Math. 2, 171–180 (1930).

    MATH  Google Scholar 

  13. Lusternik, L.A.; Sobolev, V.J., Elements of Functional Analysis, Wiley, New York 1961.

    MATH  Google Scholar 

  14. Simmons, G.F., Introduction to Topology and Modern Analysis, McGraw Hill, New York 1963.

    MATH  Google Scholar 

  15. Stone, H.S., Discrete Mathematical Structures and their Applications, Science Research Associates, Chicago 1973.

    MATH  Google Scholar 

  16. Gautschi, W., Numerical Integration of Ordinary Differential Equations Based on Trigonometric Polynomials, Numer. Math. 3, 381–397 (1961).

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. Lambert, J.D., Computational Methods in Ordinary Differential Equations, Wiley, New York 1973.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1979 Springer-Verlag

About this paper

Cite this paper

Forster, W. (1979). On numerical approximation of fixed points in C[0,1]. In: Peitgen, HO., Walther, HO. (eds) Functional Differential Equations and Approximation of Fixed Points. Lecture Notes in Mathematics, vol 730. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0064314

Download citation

  • DOI: https://doi.org/10.1007/BFb0064314

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-09518-7

  • Online ISBN: 978-3-540-35129-0

  • eBook Packages: Springer Book Archive