Positivity

, Volume 6, Issue 3, pp 243–260

Euler Characteristic and Fixed-Point Theorems

  • B. Cornet
Article

Abstract

We propose a geometric definition of the Euler characteristic χ(M) for the class of compact epi-Lipschitzian sets M⊂Rn and we provide existence theorems of (generalized) equilibria for set-valued mappings F when the domain M of F is neither assumed to be convex, nor smooth but has a nonzero Euler characteristic.

Euler characteristic Fixed-point Normal and tangent cones 

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References

  1. 1.
    Ben-El-Mechaiekh, H. and Kryszewski, W.: Equilibria of set-valued maps on nonconvex domains, Trans. Amer. Math. Soc. 349 (1997), 4159-4179.Google Scholar
  2. 2.
    Bergman, G. and Halpern, B.: A fixed-point theorem for inward and outward Maps, Trans. Amer. Math. Soc. 130 (1968), 353-358.Google Scholar
  3. 3.
    Bonnisseau, J.-M. and Cornet B.: Existence of marginal cost pricing equilibria in an economy with several nonconvex firms, Econometrica 58 (1990), 661-682.Google Scholar
  4. 4.
    Bonnisseau, J.-M. and Cornet B.: Existence of marginal cost pricing equilibria: the nonsmooth case, International Economic Review 31 (1990), 685-708.Google Scholar
  5. 5.
    Bonnisseau, J.-M. and Cornet, B.: Fixed-point theorem and Morse's lemma for Lipschitzian functions, J. Math. Anal. Appl. 151 (1990), 532-549.Google Scholar
  6. 6.
    Bonnisseau, J.-M. and Cornet, B.: Equilibrium theory with increasing returns: the existence problem, in General Equilibrium Theory and Applications,W. Barnett et al. (eds.), International Symposia in Economic Theory and Econometrics, Cambridge University Press, 1991, 65-82.Google Scholar
  7. 7.
    Browder, F.: The fixed point theory of multivalued mappings in topological vector spaces, Math. Ann. 177 (1968), 283-301.Google Scholar
  8. 8.
    Cellina, A. and Lasota, A.: A new approach to the definition of topological degree for multivalued mappings, Rendiconti dell'Academia Nazionale dei Lincei Sc. fis. mat. e nat. XLVII (1969), 434-440.Google Scholar
  9. 9.
    Cellina, A.: A theorem on the approximation of compact multivalued mappings, Rend. dell' Academ. Naz. dei Lincei Sc. fis. mat. e nat. 47 (1969), 429-433.Google Scholar
  10. 10.
    Clarke, F.: Generalized gradients and applications, Transactions of the American Mathematical Society 205 (1975), 247-262.Google Scholar
  11. 11.
    Clarke, F.: Optimization and Nonsmooth Analysis, Wiley, New-York, 1983.Google Scholar
  12. 12.
    Clarke, F., Ledyaev, Y. S. and Stern, R. J.: Fixed points and equilibria in nonconvex sets, Nonlinear Analysis 25 (1995), 145-161.Google Scholar
  13. 13.
    Cornet, B.: Paris avec handicap et théorèmes de surjectivité de correspondances, Comptes Rendus de l'Académie des Sciences de Paris, Série I 281 (1975), 479-482.Google Scholar
  14. 14.
    Cornet, B.: Regularity properties of open tangent cones, Mathematical Programming Study 30 (1987), 17-33.Google Scholar
  15. 15.
    Cornet, B.: Topological properties of the attainable set in a non-convex economy, Journal of Mathematical Economics 17 (1988), 275-292.Google Scholar
  16. 16.
    Cornet, B. and Czarnecki, M.-O.: Représentation lisse de sous-ensembles épi-lipschitziens de R n , Comptes Rendus de l'Académie des Sciences de Paris, Série I 325 (1997), 475-480.Google Scholar
  17. 17.
    Cornet, B. and Czarnecki, M.-O.: Approximation normale de sous-ensembles épi-lipschitziens de R n , Comptes Rendus de l'Académie des Sciences de Paris, Série I 325 (1997), 538-588.Google Scholar
  18. 18.
    Cornet, B. and Czarnecki, M.-O.: Méthodes d'approximation en théorie du point fixe et de l'équilibre, Comptes Rendus de l'Académie des Sciences de Paris, Série I 327 (1998) 917-922.Google Scholar
  19. 19.
    Cornet, B. and Czarnecki, M.-0.: Smooth representations of epi-Lipschitzian subsets of R n , Nonlinear Analysis 37 (1999), 139-160.Google Scholar
  20. 20.
    Cornet, B. and Czarnecki, M.-0.: Smooth approximations of epi-Lipschizian sets, SIAM Journal of Control and Optimization 37 (1999), 710-730.Google Scholar
  21. 21.
    Cornet, B. and Czarnecki, M.-O.: Necessary and sufficient conditions for the existence of (generalized) equilibria on a compact epi-Lipschitzian domain, Communications on Applied Nonlinear Analysis 7 (2000), 21-53.Google Scholar
  22. 22.
    Cornet, B. and Czarnecki, M.-O.: Existence of generalized equilibria of correspondences, Nonlinear Analysis 44 (2001), 555-573.Google Scholar
  23. 23.
    Cornet, B. and Czarnecki, M.-O.: Approximation methods in equilibrium and fixed-point theory: proximally nondegenerate sets, Proceedings of the 9th Belgian-French-German Conference on Optimization, Namur, Lecture Notes in Economics and Mathematical Sciences, Springer-Verlag, Eds. V.H. Nguyen, J.J. Strodiot and P.Tossings, 2000, 91-110.Google Scholar
  24. 24.
    Fan, K.: Extensions of two fixed points theorem of F. E. Browder, Math. Z. 112 (1969), 234-240.Google Scholar
  25. 25.
    Fan, K.: A minimax inequality and applications, in Inequalities III, O. Shisha (ed.) New York and London: Academic Press, 1972, 103-113.Google Scholar
  26. 26.
    Granas, A.: Sur la notion du degré topologique pour une certaine classe de transformations multivalentes dans les espaces de Banach, Bull. Acad. Polon. Sci. 7 (1959), 271-275.Google Scholar
  27. 27.
    Halpern, B.: Fixed point theorems for set-valued maps in infinite dimensional spaces, Math. Ann. 189 (1989), 87-98.Google Scholar
  28. 28.
    Michael, E.: Continuous selections. I, Ann. of Math. 63 (1956), 361-382.Google Scholar
  29. 29.
    Milnor, J.: Topology from the Differentiable Viewpoint, University Press of Virginia, CharlottesvilleGoogle Scholar
  30. 30.
    Ortega, J. M., and Rheinboldt, W. C.: Iterative Solution of Nonlinear Equations in Several Variables, New York: Academic Press, 1970.Google Scholar
  31. 31.
    Reich, S.: Fixed-point theorems for set-valued mappings, Journal of Mathematical Analysis and Applications 69 (1979), 353-358.Google Scholar
  32. 32.
    Rockafellar, R.T.: Convex Analysis, Princeton: Princeton University Press, 1970.Google Scholar
  33. 33.
    Rockafellar, R. T.: Clarke's tangent cones and the boundaries of closed sets in R n , Nonlinear Analysis 3 (1979), 145-154.Google Scholar
  34. 34.
    Rockafellar, R. T.: The Theory of Subgradients and its Applications to Problems of Optimization: Convex and Nonconvex Functions, Helderman, Berlin, 1981.Google Scholar
  35. 35.
    Schwartz, J. T.: Nonlinear Functionnal Analysis, New York: Gordon and Breach Sciences Publication, 1969.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • B. Cornet
    • 1
  1. 1.CERMSEM, Maison des Sciences EconomiquesUniversité Paris 1Paris Cedex 13France

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