Skip to main content
Log in

A fixed point theorem including the last theorem of Poincaré

  • Published:
Mathematical Programming Submit manuscript

Abstract

Poincaré's last theorem is the most famous among those theorems which are not subsumed by the Lefschetz fixed point theorem. A fixed point theorem is proved directly and constructively which in a special case reduces to the last theorem of Poincaré.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. G.D. Birkhoff, “Proof of Poincaré's geometric theorem”,Transactions of the American Mathematical Society 14 (1913) 14–22.

    Google Scholar 

  2. G.D. Birkhoff, “An extension of Poincaré's last geometric theorem”,Acta Mathematica 47 (1925) 298–311.

    Google Scholar 

  3. G.D. Birkhoff,Dynamical systems (Am. Math. Soc., Providence, R.I., 1927).

    Google Scholar 

  4. F.E. Browder, “On continuity of fixed points under deformations of continuous mappings”,Summa Brasil. Math. 4 (1960).

  5. D.I.A. Cohen, “On the Sperner Lemma”,Journal of Combinatorial Theory 2 (1967) 585–587.

    Google Scholar 

  6. B.C. Eaves, “Homotopies for the computation of fixed points”,Mathematical Programming 3 (1972) 1–22.

    Google Scholar 

  7. B.C. Eaves and R. Saigal, “Homotopies for the computation of fixed points on unbounded regions”,Mathematical Programming 3 (1972) 225–237.

    Google Scholar 

  8. M.L. Fisher and F.J. Gould, “An algorithm for the nonlinear complementarity problem”,Mathematical Programming 6 (1974) 281–300.

    Google Scholar 

  9. C.B. Garcia, “A hybrid algorithm for the computation of fixed points”,Management Science, to appear.

  10. T. Hansen and H. Scarf, “On the applications of a recent combinatorial algorithm”, Cowles Foundation Discussion Paper No. 272 (1969).

  11. H.W. Kuhn, “Simplicial approximation of fixed points”,Proceedings of the National Academy of Science 61 (1968) 1238–1242.

    Google Scholar 

  12. L.A. Pars,A treatise on dynamical systems (Heinemann, London, 1965).

    Google Scholar 

  13. S. Lefschetz,Introduction to topology (Princeton Press, Princeton, 1949).

    Google Scholar 

  14. C.E. Lemke and J.T. Howson, “Equilibrium points of bimatrix games”,SIAM Review 12 (1964) 413–423.

    Google Scholar 

  15. C.E. Lemke, “Bimatrix equilibrium points and mathematical programming”,Management Science 11 (1965) 681–689.

    Google Scholar 

  16. O.H. Merrill, “Applications and extensions of an algorithm that computes fixed points of certain non-empty convex upper semicontinuous point to set mappings”, Tech. Rept. No. 71-7, University of Michigan (1971).

  17. H. Poincaré. “Sur un théorème de gdometrie”,Rendiconti del Circolo Matematico di Palermo 33 (1912) 375–407.

    Google Scholar 

  18. H. Scarf, “The approximation of fixed points of a continuous mapping”,SIAM Review 15 (1967) 1328–1343.

    Google Scholar 

  19. L. Shapley, “A note on the Lemke-Howson algorithm”, Rept. R-1538-PR, The RAND Corporation (1974).

  20. D.R. Smart,Fixed point theorems (Cambridge Press, Cambridge, 1974).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Garcia, C.B. A fixed point theorem including the last theorem of Poincaré. Mathematical Programming 9, 227–239 (1975). https://doi.org/10.1007/BF01681345

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01681345

Keywords

Navigation