Journal of Optimization Theory and Applications

, Volume 163, Issue 3, pp 719–736 | Cite as

A Fixed-Point Theorem and Equilibria of Abstract Economies with Weakly Upper Semicontinuous Set-Valued Maps

  • Carlos Hervés-Beloso
  • Monica Patriche


The main purpose of this paper is to introduce the notion of weakly upper semicontinuous set-valued maps and to establish a new fixed-point theorem. The set-valued maps with an approximating upper semicontinuous selection property are also defined. Next, we use our fixed-point result to obtain equilibrium existence in abstract economies with two constraints, which provide a natural scenario for potential applications of our approach to general equilibrium theory. In this regard, we set models of economies with asymmetric informed agents, who are able to improve their initial information through market signals. These economies offer examples in which the informational feasibility requirement represents an additional constraint.


Fixed-point theorem W-upper semicontinuous set-valued maps Set-valued maps with e-USS property Abstract economy Equilibrium 

Mathematics Subject Classification (2010)

47H10 91A47 91A80 



We are grateful to the editor and two anonymous reviewers for their valuable suggestions that have improved the quality of the paper. The first author thanks the support by Research Grants ECO2012-38860-C02-02 (Ministerio de Economia y Competitividad), RGEA and 10PXIB300141PR (Xunta de Galicia and FEDER).


  1. 1.
    Nash, J.F.: Non-cooperative games. Ann. Math. 54, 286–295 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Debreu, G.: A social equilibrium existence theorem. Proc. Nat. Acad. Sci. USA 38, 886–893 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Arrow, K.J., Debreu, G.: Existence of an equilibrium for a competitive economy. Econometrica 22, 265–290 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Shafer, W., Sonnenschein, H.: Equilibrium in abstract economies without ordered preferences. J. Math. Econ. 2, 345–348 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Borglin, A., Keiding, H.: Existence of equilibrium action and of equilibrium: a note on the “new” existence theorem. J. Math. Econ. 3, 313–316 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Yannelis, N.C., Prabhakar, N.D.: Existence of maximal elements and equilibrium in linear topological spaces. J. Math. Econ. 12, 233–245 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Yuan, X.Z.: The study of minimax inequalities and applications to economies and variational inequalities. Mem. Am. Soc. 132, 625 (1998)Google Scholar
  8. 8.
    Agarwal, R.P., O’Regan, D.: A note on equilibria for abstract economies. Math. Comput. Model. 34, 331–343 (2001)CrossRefzbMATHGoogle Scholar
  9. 9.
    Chang, S.Y.: Inequalities and nash equilibria. Nonlinear Anal. Theor. Meth. Appl. 73(9), 2933–2940 (2010)CrossRefzbMATHGoogle Scholar
  10. 10.
    Tan, K.K., Wu, Z.: A note on abstract economies with upper semicontinuous set-valued map. Appl. Math. Lett. 5(11), 21–22 (1998)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Wu, X.: A new fixed point theorem and its applications. Proc. Am. Math. Soc. 125, 1779–1783 (1997)CrossRefzbMATHGoogle Scholar
  12. 12.
    Yuan, X.Z., Taradfar, E.: Maximal elements and equilibria of generalized games for U-majorized and condensing set-valued maps. Int. J. Math. Sci. 22(1), 179–189 (1999)CrossRefzbMATHGoogle Scholar
  13. 13.
    Barbagallo, A., Mauro, P.: Evolutionary variational formulation for oligopolistic market equilibrium problems with production excesses. J. Optim. Theory Appl. 155, 288–314 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Patriche, M.: Equilibrium in Games and Competitive Economies. The Publishing House of the Romanian Academy, Bucharest (2011)Google Scholar
  15. 15.
    Patriche, M.: Fixed point and equilibrium theorems in a generalized convexity framework. J. Optim. Theory Appl. 156(3), 701–715 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Stefanescu, A., Ferrara, M., Stefanescu, M.V.: Equilibria of the games in choice form. J. Optim. Theory Appl. 155(3), 1060–1072 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Himmelberg, C.J.: Fixed points of compact multifunctions. J. Math. Anal. Appl. 38, 205–207 (1972)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Donato, M.B., Milasi, M., Vitanza, C.: A new contribution to a dynamic competitive equilibrium problem. Appl. Math. Lett. 23(2), 148–151 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Donato, M.B., Milasi, M., Scrimali, L.: Walrasian equilibrium problem with memory term. J. Optim. Theory Appl. 151, 64–80 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Zheng, X.: Approximate selection theorems and their applications. J. Math. Anal. Appl. 212, 88–97 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Chateauneuf, A.: Continuous representation of a preference relation on a connected topological space. J. Math. Econ. 16, 139–146 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Fan, K.: A generalization of Tychonoff’s fixed point theorem. Math. Ann. 142, 305–310 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Radner, R.: Competitive equilibrium under uncertainty. Econometrica 36(1), 31–58 (1968)CrossRefzbMATHGoogle Scholar
  24. 24.
    Radner, R.: Rational expectations equilibrium: generic existence and the information revealed by prices. Econometrica 47(3), 655–678 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Correia-da-Silva, J., Hervés-Beloso, C.: General equilibrium in economies with uncertain delivery. Econ. Theor. 51(3), 729–755 (2012)CrossRefzbMATHGoogle Scholar
  26. 26.
    Correia-da-Silva, J., Hervés-Beloso, C.: Irrelevance of private information in two-period economies with more goods than states of nature. Econ. Theor. 55(2), 439–455 (2014)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.RGEA, Facultad de EconómicasUniversidad de VigoVigoSpain
  2. 2.Faculty of Mathematics and Computer ScienceUniversity of BucharestBucharestRomania

Personalised recommendations