Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-28T18:55:54.875Z Has data issue: false hasContentIssue false

A fixed point theorem and existence of equilibrium for abstract economies

Published online by Cambridge University Press:  17 April 2009

Dong Il Rim
Affiliation:
Department of Mathematics and Mathematics EducationChungbuk National UniversityKorea
Won Kyu Kim
Affiliation:
Department of Mathematics and Mathematics EducationChungbuk National UniversityKorea
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we shall prove a generalisation of Himmelberg's fixed point theorem and as applications, the existence of equilibrium points for abstract economies given by preference correspondences and utility functions have been established.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1992

References

[1]Arrow, K.J. and Debreu, G., ‘Existence of an equilibrium for a competitive economy’, Econometrica 22 (1954), 265290.CrossRefGoogle Scholar
[2]Aubin, J.P. and Ekeland, I., Applied nonlinear analysis (John Wiley and Sons, New York, 1984).Google Scholar
[3]Borglin, A. and Keiding, H., ‘Existence of equilibrium actions and of equilibrium: A note on the ‘new’ existence theorem’, J. Math. Econom. 3 (1976), 313316.CrossRefGoogle Scholar
[4]Debreu, G., ‘A social equilibrium existence theorem’, Proc. Nat. Acad. Sci. U.S.A. 38 (1952), 886893.CrossRefGoogle ScholarPubMed
[5]Ding, X.P., Kim, W.K. and Tan, K.-K., ‘Equilibria of non-compact generalized games with L*-majorized preferences’, J. Math. Anal. Appl. (to appear).Google Scholar
[6]Dugundji, J., Topology (Allyn and Bacon Inc., Boston, 1970).Google Scholar
[7]Fan, K., ‘Fixed-point and minimax theorems in locally convex topological linear spaces’, Proc. Nat. Acad. Sci. U.S.A. 38 (1952), 131136.CrossRefGoogle ScholarPubMed
[8]Fan, K., ‘Some properties of convex sets related to fixed point theorems’, Math. Ann. 266 (1984), 519537.CrossRefGoogle Scholar
[9]Gale, D. and Mas-Colell, A., ‘An equilibrium existence for a general model without ordered preferences’, J. Math. Econom. 2 (1975), 915.CrossRefGoogle Scholar
[10]Greenberg, J., ‘Quasi-equilibrium in abstract economy without ordered preferences’, J. Math. Econom. 4 (1977), 163165.CrossRefGoogle Scholar
[11]Himmelberg, C.J., ‘Fixed points of compact multifunctions’, J. Math. Anal. Appl. 38 (1972), 205207.CrossRefGoogle Scholar
[12]Im, S.M. and Kim, W.K., ‘An application of Himmelberg's fixed point theorem to non-compact optimization problems’, Bull. Inst. Math. Acad. Sinica 19 (1991), 147151.Google Scholar
[13]Mas-Colell, A., ‘An equilibrium existence theorem without complete or transitive preferences’, J. Math. Econom. 1 (1974), 237246.CrossRefGoogle Scholar
[14]Nash, J.F., ‘Equilibrium states in N-person games’, Proc. Nat. Acad. Sci. U.S.A. 36 (1950), 4849.CrossRefGoogle Scholar
[15]Shafer, W. and Sonnenschein, H., ‘Equilibrium in abstract economies without ordered preferences’, J. Math. Econom. 2 (1975), 345348.CrossRefGoogle Scholar
[16]Sonnenschein, H., ‘Demand theory without transitive preference with applications to the theory of competitive equilibrium’, in Preferences, utility and demand, Editors Chipman, J., Hurwich, L., Richter, M.K. and Sonnenschein, H.F. (Harcourt Brace Jovanovich, New York, 1971).Google Scholar
[17]Tarafdar, E., ‘An extension of Fan's fixed point theorem and equilibrium point of abstract economy’, Comment. Math. Univ. Carolin. 31 (1990), 723730.Google Scholar
[18]Toussaint, S., ‘On the existence of equilibria in economies with infinitely many commodities and without ordered preferences’, J. Econom. Theory 33 (1984), 98115.CrossRefGoogle Scholar
[19]Tulcea, C.I., ‘On the equilibriums of generalized games’, in The Center for Mathematical Studies in Economics and Management Science, Paper No. 696, 1986.Google Scholar