Skip to main content
SpringerLink
Log in

An \(\bar \varepsilon \) - Perturbation of Brouwer’s definition of degree

  • Conference paper
  • First Online:
Fixed Point Theory

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 886))

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 54.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.95
Price excludes VAT (USA)
  • 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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. ALEXANDER, J.C. and YORKE, J.A.: The homotopy continuation method: Numerically implementable topological prodedures, Trans. AMS, 242 (1978), 271–284.

    Article  MathSciNet  MATH  Google Scholar 

  2. ALEXANDROFF, P. and HOPF, H.: Topologie, berichtigter Reprint, Berlin, Heidelberg, New York, Springer-Verlag (1974), Die Grundlehren der mathematischen Wissenschaften, Bd. 45.

    Book  MATH  Google Scholar 

  3. ALLGOWER, E.L. and GEORG, K.: Simplicial and continuation methods for approximating fixed points and solutions to systems of equations, SIAM Review, 22 (1980), 28–85.

    Article  MathSciNet  MATH  Google Scholar 

  4. ALLGOWER, E.L. and GEORG, K.: Homotopy methods for approximating several solutions to nonlinear systems of equations, in: Numerical Solution of Highly Nonlinear Problems, W. Forster (Ed), North-Holland Publishing Company-Amsterdam, New York, Oxford (1980), 253–270.

    Google Scholar 

  5. AMANN, H.: Lectures on some fixed point theorems, Monografias de Matemática, Instituto de matemática pura e aplicada, Rio de Janeiro (1974).

    Google Scholar 

  6. AMANN, H. and WEISS, S.A.: On the uniqueness of the topological degree, Math. Z. 130 (1973), 39–54.

    Article  MathSciNet  MATH  Google Scholar 

  7. BROUWER, L.E.J.: Beweis der Invarianz der Dimensionszahl, Math. Ann. 70 (1911), 161–165.

    Article  MathSciNet  MATH  Google Scholar 

  8. BROUWER, L.E.J.: Über Abbildung von Mannigfaltigkeiten, Math. Ann. 71 (1912), 97–115.

    Article  MATH  Google Scholar 

  9. CELLINA, A. and LASOTA, A.: A new approach to the definition of topological degree for multi-valued mappings, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 47 (1969), 434–440.

    MathSciNet  MATH  Google Scholar 

  10. CHOW, S.N., MALLET-PARET, J. and YORKE, J.A.: Finding zeros of maps: Homotopy methods that are constructive with probability one, Math. Comp. 32 (1978), 887–899.

    Article  MathSciNet  MATH  Google Scholar 

  11. CRONIN, J.: Fixed points and topological degree in nonlinear analysis, Providence, R.I., Amer. Math. Soc. (1964).

    MATH  Google Scholar 

  12. DEIMLING, K.: Nichtlineare Gleichungen und Abbildungsgrade, Berlin, Heidelberg, New York, Springer-Verlag (1974).

    Book  MATH  Google Scholar 

  13. DOLD, A.: Lectures on algebraic topology, Berlin, Heilderberg, New York, Springer-Verlag (1972), Die Grundlehren der mathematischen Wissenschaften, Bd. 200.

    Book  MATH  Google Scholar 

  14. DE RHAM, G.: Variétés différentiables, Paris, Hermann (1955).

    MATH  Google Scholar 

  15. EAVES, B.C.: A short course in solving equations with PL-homotopies, SIAM-AMS Proceedings 9 (1976), 73–143.

    MathSciNet  MATH  Google Scholar 

  16. EAVES, B.C. and SCARF, H.: The solution of systems of piecewise linear equations, Math. of Op. Res. 1 (1976), 1–27.

    Article  MathSciNet  MATH  Google Scholar 

  17. EILENBERG, S. and STEENROD, N.: Foundations of algebraic topology, Princeton, Univ. Press (1952).

    Book  MATH  Google Scholar 

  18. EISENBUD, D. and LEVINE, H. I.: The topological degree of a finite C∞-map germ, in: Structural Stability, the Theory of Catastrophes, and Applications in the Sciences, Springer-Verlag (1976), Lectures Notes in Mathematics 535, 90–98.

    Google Scholar 

  19. EISENACK, G. and FENSKE, C.C.: Fixpunkttheorie, Mannheim, Wien, Zürich, Bibliographisches Institut (1978).

    MATH  Google Scholar 

  20. EPSTEIN, D.B.A.: The degree of a map, Proc. London Math. Soc. 16 (1966), 369–383.

    Article  MathSciNet  MATH  Google Scholar 

  21. FENSKE, C.C.: Analytische Theorie des Abbildungsgrades in Banachräumen, Math. Nachr. 48 (1971), 279–290. *** DIRECT SUPPORT *** A00J4360 00011

    Article  MathSciNet  MATH  Google Scholar 

  22. FÜHRER, L.: Ein elementarer analytischer Beweis zur Eindeutigkeit des Abbidungsgrades im Rn, Math. Nachr. 54 (1972), 259–267.

    Article  MathSciNet  MATH  Google Scholar 

  23. GARCIA, C.B.: Computation of solutions to nonlinear equations under homotopy invariance, Math. of Op. Res. 2 (1977), 25–29.

    Article  MathSciNet  MATH  Google Scholar 

  24. GRANAS, A.: Sur la notion du degré topologique pour une cartaine classe de transformations multivalentes dans les espaces de Banach, Bull. Acad. Polon. Sci. Sér. Sci. Mat. Astronom. Phys. 7 (1959), 191–194.

    MathSciNet  MATH  Google Scholar 

  25. GREUB, W., HALPERIN, S. and VANSTONE, R.: Connections, Curvature, and Cohomology, vol I, Academic Press, New York, London (1972).

    MATH  Google Scholar 

  26. GUILLEMIN, V. and POLLACK, A.: Differential topology, Prentice Hall, Inc. New Jersey (1974).

    MATH  Google Scholar 

  27. HEINZ, E.: An elementary analytic theory of degree of mappings in n-dimensional spaces, J. Math. Mech. 8 (1959), 231–247.

    MathSciNet  MATH  Google Scholar 

  28. HIRSCH, M.W.: Differential topology. Berlin, Heidelberg, New York, Springer-Verlag (1976), Graduate Texts in Mathematics 33.

    MATH  Google Scholar 

  29. Hocking, J.G. and Young, G.S.: Topology, Addison-Wesley Publishing Company, Inc. (1961).

    Google Scholar 

  30. HUDSON, J.F.P.: Piecewise linear topology, Benjamin, New York, (1969).

    MATH  Google Scholar 

  31. JÜRGENS, H., PEITGEN, H.O. and SAUPE, D.: Topological perturbations in the numerical study of nonlinear eigenvalue and bifurcation problems, to appear in: Proc. Symposium on Analysis and Computation of Fixed Points, Madison, May 7–8, 1979, S.M. Robinson (Ed.), Academic Press, New York.

    Google Scholar 

  32. LERAY, J. and SCHAUDER, J.P.: Topologie et équations fonctionnelles, Ann. Ecole Norm. Sup. (3) 51 (1934), 45–78.

    MathSciNet  MATH  Google Scholar 

  33. LLOYD, N.G.: Degree theory, Cambridge University Press (1978).

    Google Scholar 

  34. MA, T.W.: Topological degree for set-valued compact vector fields in locally convex spaces, Dissertationes Math. 92 (1972).

    Google Scholar 

  35. MILNOR, J.: Topology from the differentiable viewpoint, 2nd printing, Charlottesville, The University of Virginia Press (1969).

    MATH  Google Scholar 

  36. NAGUMO, M.: A theory of degree of mapping based on infinitesimal analysis, Amer. J. Math. 73 (1951), 485–496.

    Article  MathSciNet  MATH  Google Scholar 

  37. NIREMBERG, L.: Topics in nonlinear functional analysis, Courant Institute of Mathematical Sciences, New York University (1974).

    Google Scholar 

  38. NUSSBAUM, R.D.: On the uniqueness of the topological degree for k-set contractions, Math. Z. 137 (1974), 1–6.

    Article  MathSciNet  MATH  Google Scholar 

  39. PEITGEN, H.O. and PRÜFER, M.: The Leray-Schauder continuation method is a constructive element in the numerical study of nonlinear eigenvalue and bifurcation problems, in: Functional Differential Equations and Approximation of Fixed Points, H.O. Peitgen and H.O. Walther (Eds), Springer-Verlag (1979), Lectures Notes in Mathematics 730, 326–409.

    Google Scholar 

  40. PEITGEN, H.O., SAUPE, D. and SCHMITT, K.: Nonlinear elliptic boundary value problems versus finite difference approximations: numerically irrelevant solutions, to appear in: J. Reine Angew. Math.

    Google Scholar 

  41. PONTRYAGIN, L.S.: Smooth manifolds and their applications in homotopy theory, AMS Translations, Ser. 2, II (1959), 1–114.

    MathSciNet  MATH  Google Scholar 

  42. PRÜFER, M.: Calculating global bifurcation, in: Continuation Methods, H.J. Wacker (Ed), Academic Press, New York, San Francisco, London (1978), 187–213.

    Google Scholar 

  43. PRÜFER, M.: Simpliziale Topologie und globale Verzweigung, Dissertation, University of Bonn (1978).

    Google Scholar 

  44. ROURKE, C.P. and SANDERSON, B.J.: Introduction to piecewise-linear topology, Berlin, Heidelberg, New York, Springer-Verlag (1972).

    Book  MATH  Google Scholar 

  45. SCHWARTZ, J.: Nonlinear functional analysis, Gordon and Breach, New York, (1969).

    MATH  Google Scholar 

  46. SIEGBERG, H.W.: Brouwer degree: history and numerical computation, in: Numerical Solution of Highly Nonlinear Problems, W. Forster (Ed.), North-Holland Publishing Company Amsterdam, New York, Oxford (1980), 389–411.

    Google Scholar 

  47. SPANIER, E.: Algebraic topology, New York, Mc Graw-Hill, (1966).

    MATH  Google Scholar 

  48. SPIVAK, M.: A comprehensive introduction to differential geometry, vol I, Boston: Publish or Perish Inc. (1970).

    MATH  Google Scholar 

  49. TODD, M.J.: The computation of fixed points, Berlin, Heidelberg, New York, Springer-Verlag (1976), Lectures Notes in Economics and Mathematical Systems 124.

    Book  MATH  Google Scholar 

  50. TODD, M.J.: Orientation in complementary pivoting, Math. of Op. Res. 1 (1976), 54–66.

    Article  MathSciNet  MATH  Google Scholar 

  51. ZEIDLER, E.: Existenz, Eindeutigkeit, Eigenschaften und Anwendungen des Abbildungsgrades in Rn, in: Theory of Nonlinear Operators, Proceedings of a summer school held in Oct. 1972 at Neuendorf, Akademie-Verlag, Berlin (1974), 259–311. *** DIRECT SUPPORT *** A00J4360 00012

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fachbereich Mathematik Forschungsschwerpunkt “Dynamische Systeme”, Universität Bremen, Bibliothekstrasse, Postfach 330 440, 2800, Bremen 33, West Germany

    Heinz-Otto Peitgen & Hans-Willi Siegberg

Authors
  1. Heinz-Otto Peitgen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hans-Willi Siegberg
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Edward Fadell Gilles Fournier

Rights and permissions

Reprints and permissions

Copyright information

© 1981 Springer-Verlag

About this paper

Cite this paper

Peitgen, HO., Siegberg, HW. (1981). An \(\bar \varepsilon \) - Perturbation of Brouwer’s definition of degree. In: Fadell, E., Fournier, G. (eds) Fixed Point Theory. Lecture Notes in Mathematics, vol 886. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0092192

Download citation

  • DOI: https://doi.org/10.1007/BFb0092192

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-11152-8

  • Online ISBN: 978-3-540-38600-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation