Skip to main content

The fixed point index and fixed point theorems

Part of the Lecture Notes in Mathematics book series (LNMCIME,volume 1537)

Keywords

  • Banach Space
  • Periodic Solution
  • Open Subset
  • Open Neighborhood
  • Fixed Point Theorem

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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   69.99
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. R. Arens and J. Eells, On embedding uniform and topological spaces, Pacific J. Math 6 (1956), 397–403.

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. R.H. Bing, The elusive fixed point property, American Math. Monthly, February, (1969), 119–132.

    Google Scholar 

  3. K. Borsuk, Theory of Retracts, Polish Sci. Publ., Warsaw, 1967.

    MATH  Google Scholar 

  4. F.E. Browder, On a generalization of the Schauder fixed point theorem, Duke Math. J. 26 (1959), 291–304.

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. _____, Another generalization of the Schauder fixed point theorem, Duke Math. J. 32 (1965), 399–406.

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. _____, A further generalization of the Schauder fixed point theorem, Duke Math. J. 32 (1965), 575–578.

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. _____, On the fixed point index for continuous mappings of locally connected spaces, Summa Brasil. Math. 4 (1960), 253–293.

    MathSciNet  Google Scholar 

  8. _____, Local and global properties of nonlinear mappings in Banach spaces, Instituto Naz. di Alta Mat. Symposia Math. 2 (1968), 13–35.

    MathSciNet  Google Scholar 

  9. _____, Asymptotic fixed point theorems, Math. Ann. 185 (1970), 38–60.

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. R.F. Brown, The Lefschetz Fixed Point Theorem, Scott Foresman Co., Glenview, Illinois, 1971.

    MATH  Google Scholar 

  11. G. Darbo, Punti uniti in trasformazioni a condiminio non compatto, Rend. Sem. Math. Univ. Padova 24 (1955), 84–92.

    MathSciNet  MATH  Google Scholar 

  12. A. Deleanu, Théorie des points fixes sur les retractes de voisinage des espaces convexoides, Bull. Soc. Math. France 89 (1959), 235–243.

    MathSciNet  MATH  Google Scholar 

  13. A. Dold, Fixed point index and fixed point theorems for euclidean neighborhood retracts, Topology 4 (1965), 1–8.

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. _____, Lectures on Algebraic Topology, vol. 200, Springer-Verlag, New York, 1972.

    CrossRef  MATH  Google Scholar 

  15. J. Dugundji, An extension of Tietze's theorem, Pacific J. Math. 1 (1951), 353–367.

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. J. Dugundji and A. Granas, Fixed Point Theory, vol. 1, Polska Akademia Nauk, Monografie Matematyczne, Tom 61, Warsaw, 1982.

    Google Scholar 

  17. J. Eells and G. Fournier, La théorie des points fixes des applications à itérée condensante, Bull Soc. Math. France, Mémoire 45 (1976), 91–120.

    MathSciNet  MATH  Google Scholar 

  18. C. Fenske and H.-O. Peitgen, Fixed points of zero index in asymptotic fixed point theory, Pacific J. Math 66 (1976), 391–410.

    CrossRef  MathSciNet  MATH  Google Scholar 

  19. G. Fournier, Généralisations du théoreme de Lefschetz pour des espaces non-compacts, II. Bull. Acad. Pol. Sci. 23 (1975), 701–706.

    MathSciNet  MATH  Google Scholar 

  20. R.L. Frum-Ketkov, Mappings into a Banach space sphere, Dokl. Akad. Nauk. SSSR 175 (1967), 1229–1231=Soviet Math. Dokl. 8 (1967), 1004–1007.

    MathSciNet  MATH  Google Scholar 

  21. R.B. Grafton, A periodicity theorem for autonomous functional differential equations, J. Differential Equations 6 (1969), 87–109.

    CrossRef  MathSciNet  MATH  Google Scholar 

  22. A. Granas, Some theorems in fixed point theory, Bull Acad. Polon. Sci. 17 (1969), 131–137.

    MathSciNet  MATH  Google Scholar 

  23. _____, The Leray-Schauder index and the fixed point theory for arbitrary ANR's, Bull. Soc. Math. France 100 (1972), 209–228.

    MathSciNet  MATH  Google Scholar 

  24. J. Hale, Asymptotic Behaviour of Dissipative Systems, Mathematical Surveys and Monographs, number 25, Amer. Math. Soc., Providence, Rhode Island.

    Google Scholar 

  25. O. Hanner, Some theorems on absolute neighborhood retracts, Arkiv for Matematik 1 (1951), 389–408.

    CrossRef  MathSciNet  MATH  Google Scholar 

  26. _____, Retraction and extension of mappings of metric and non-metric spaces, Arkiv for Matematik 2 (1952–1954), 315–360.

    CrossRef  MathSciNet  MATH  Google Scholar 

  27. G.S. Jones, The existence of periodic solutions of f′(x)=−αf(x−1)[1+f(x)], J. Math Anal. Appl. 4 (1962), 440–469.

    CrossRef  MathSciNet  MATH  Google Scholar 

  28. _____, Periodic motions in Banach space and applications to functional differential equations, Contributions to Differential Equations 3 (1964), 75–106.

    MathSciNet  Google Scholar 

  29. M.A. Krasnosel'skii and P.P. Zabreiko, Iterations of operators and the fixed point index, Soviet Math. Doklady 12 (1971), 294–298.

    MathSciNet  Google Scholar 

  30. C. Kuratowski, Sur les espaces complets, Fund. Math. 15 (1930), 301–309.

    MATH  Google Scholar 

  31. J. Leray and J. Schauder, Topologie et équations fonctionnelles, Ann. Sci. Ecole Norm. Sup. 51 (1934), 45–78.

    MathSciNet  MATH  Google Scholar 

  32. J. Leray, Sur les équations et les transformations, J. Math. Pures Appl. 24 (1945), 201–248.

    MATH  Google Scholar 

  33. _____, La theorie des points fixes et ses applications en analyse, Proc. International Congress Math., Cambridge, vol. 2, (1950), pp. 202–208.

    MathSciNet  Google Scholar 

  34. _____, Theorie des points fixes, indice total et nombre de Lefschetz, Bull. Soc. Math. France 87 (1959), 221–233.

    MathSciNet  MATH  Google Scholar 

  35. J. Mallet-Paret and R.D. Nussbaum, Global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation, Annali di Mat. Pura Appl. 145 (1986), 33–128.

    CrossRef  MathSciNet  MATH  Google Scholar 

  36. J. Mallet-Paret and R.D. Nussbaum, Boundary layer phenomena for differential-delay equations with state dependent time lags, I, to appear in Arch. Rat. Mech. and Analysis

    Google Scholar 

  37. L. Nirenberg, Topics in Nonlinear Functional Analysis, N.Y.U. Courant Institute Lecture Notes, New York, 1974.

    MATH  Google Scholar 

  38. R.D. Nussbaum, The fixed point index and asymptotic fixed point theorems for k-set-contractions, Bull. Amer. Math. Soc. 75 (1969), 490–495.

    CrossRef  MathSciNet  MATH  Google Scholar 

  39. _____, The radius of the essential spectrum, Duke Math J. 38 (1970), 473–478.

    CrossRef  MathSciNet  MATH  Google Scholar 

  40. _____, The fixed point index for local condensing maps, Ann. Mat. Pura Appl. 89 (1971), 217–258.

    CrossRef  MathSciNet  MATH  Google Scholar 

  41. _____, A geometric approach to the fixed point index, Pacific J. Math. 39 (1971), 751–766.

    CrossRef  MathSciNet  MATH  Google Scholar 

  42. _____, Asymptotic fixed point theory for local condensing maps, Math. Ann. 191 (1971), 181–195.

    CrossRef  MathSciNet  MATH  Google Scholar 

  43. _____, Some asymptotic fixed point theorems, Transactions Amer. Math. Soc. 171 (1972), 349–375.

    CrossRef  MathSciNet  MATH  Google Scholar 

  44. _____, Periodic solutions of some nonlinear autonomous functional differential equations, Ann. Mat. Pura Appl. 101 (1974), 263–306.

    CrossRef  MathSciNet  MATH  Google Scholar 

  45. _____, Periodic solutions of some nonlinear autonomous functional differential equations, II, J. Differential Equations 14 (1973), 360–394.

    CrossRef  MathSciNet  MATH  Google Scholar 

  46. ____, A global bifurcation theorem with applications to functional differential equations, J. Functional Analysis 19 (1975), 319–338.

    CrossRef  MathSciNet  MATH  Google Scholar 

  47. _____, The range of periods of periodic solutions of x′(t)=−αf(x(t−1)), J. Math. Anal. Appl. 58 (1977), 280–292.

    CrossRef  MathSciNet  MATH  Google Scholar 

  48. _____, Periodic solutions of nonlinear, autonomous functional differential equations, in Functional Differential Equations and Approximation of Fixed Points, H.-O. Peitgen and H.-O. Walther, eds., Springer Verlag Lecture Notes in Mathematics, vol. 730, Sringer-Verlag, New York, 1979, pp. 283–325.

    CrossRef  Google Scholar 

  49. _____, Generalizing the fixed point index, Math. Ann. 228 (1977), 259–278.

    CrossRef  MathSciNet  MATH  Google Scholar 

  50. _____, A periodicity threshold theorem for some nonlinear integral equations, SIAM J. Math. Anal. 9 (1978), 356–376.

    CrossRef  MathSciNet  MATH  Google Scholar 

  51. _____, Periodic solutions of some nonlinear integral equations, in Dynamical Systems, A. Bednarek and L. Cesari, eds., Academic Press, New York, 1977, pp. 221–251.

    Google Scholar 

  52. _____, The Fixed Point Index and Some Applications, volume 94 in the collection, "Séminaire de Mathematiques Supérieures", Les Presses de l'Université Montreal, C.P. 6128, succ. "A", Montreal, Québec, Canada H3C3J7, 1985.

    Google Scholar 

  53. _____, Eigenvectors of nonlinear positive operators and the linear Krein-Rutman theorem, in Fixed Point Theory, E. Fadell and G. Fournier, eds, Springer Lecture Notes in Mathematics, vol. 886, Spring-Verlag, New York, 1981, pp. 309–331.

    CrossRef  Google Scholar 

  54. H.-O. Peitgen, On the Lefschetz number for iterates of continuous mappings, Proc. Amer. Math. Soc. 54 (1976), 441–444.

    CrossRef  MathSciNet  MATH  Google Scholar 

  55. P. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis 7 (1971), 487–513.

    CrossRef  MathSciNet  MATH  Google Scholar 

  56. _____, Some aspects of nonlinear eigenvalue problems, Rocky Mountain Math. J. 3 (1973), 161–202.

    CrossRef  MathSciNet  MATH  Google Scholar 

  57. B.N. Sadovskii, Limit compact and condensing operators, Russian Math. Surveys 27 (1972), 85–155.

    CrossRef  MathSciNet  Google Scholar 

  58. H. Steinlein, Ein Satz über den Leray-Schauderschen Abbildungsgrad, Math. Z. 120 (1972), 176–208.

    CrossRef  MathSciNet  MATH  Google Scholar 

  59. _____, Über die verallgemeinerten Fixpunktindizes von Iterierten verdichtender Abbildungen, Manuscripta Math. 8 (1972), 251–266.

    CrossRef  MathSciNet  MATH  Google Scholar 

  60. R.B. Thompson, A unified approach to local and global fixed point indices, Advances in Math. 3 (1969), 1–72.

    CrossRef  MathSciNet  MATH  Google Scholar 

  61. A.J. Tromba, The beer barrel theorem, in Functional Differential Equations and Approximation of Fixed Points, H.-O. Peitgen and H.-O. Walther, eds., Springer Verlag Lecture Notes in Mathematics, vol. 730, pp. 484–488.

    Google Scholar 

  62. _____, A general asymptotic fixed point theorem, J. für die Reine u. Angew. Math. 332 (1982), 118–125.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1993 Springer-Verlag

About this chapter

Cite this chapter

Nussbaum, R.D. (1993). The fixed point index and fixed point theorems. In: Furi, M., Zecca, P. (eds) Topological Methods for Ordinary Differential Equations. Lecture Notes in Mathematics, vol 1537. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0085077

Download citation

  • DOI: https://doi.org/10.1007/BFb0085077

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-56461-4

  • Online ISBN: 978-3-540-47563-7

  • eBook Packages: Springer Book Archive