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Generalized KKM theorem, minimax inequalities and their applications

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  • Tian, Guoqiang

Abstract

This paper extends the well-known KKM theorem and variational inequalities by relaxing the closedness of values of a correspondence and lower semicontinuity of a function. The approach adopted is based on Michael's continuous selection theorem. As applications, we provide theorems for the existence of maximum elements of a binary relation, a price equilibrium, and the complementarity problem. Thus our theorems, which do not require the openness of lower sections of the preference correspondences and the lower semicontinuity of the excess demand functions, generalize many of the existence theorems such as those in Sonnenschein (Ref. 1), Yannelis and Prabhakar (Ref. 2), and Border (Ref. 3).

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Bibliographic Info

Paper provided by University Library of Munich, Germany in its series MPRA Paper with number 41217.

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Date of creation: 1994
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Publication status: Published in Journal of Optimization Theory and Applications 2.83(1994): pp. 375-389
Handle: RePEc:pra:mprapa:41217

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Keywords: KKM theorem; Variational inequalities; Complementarity problem; Price equilibrium; Maximal elements ; Binary relations;

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References

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  1. Wayne Shafer & Hugo Sonnenschein, 1974. "Equilibrium in Abstract Economies Without Ordered Preferences," Discussion Papers, Northwestern University, Center for Mathematical Studies in Economics and Management Science 94, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  2. Yannelis, Nicholas C. & Prabhakar, N. D., 1983. "Existence of maximal elements and equilibria in linear topological spaces," Journal of Mathematical Economics, Elsevier, vol. 12(3), pages 233-245, December.
  3. Tian, Guoqiang, 1993. "Necessary and Sufficient Conditions for Maximization of a Class of Preference Relations," Review of Economic Studies, Wiley Blackwell, Wiley Blackwell, vol. 60(4), pages 949-58, October.
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Cited by:
  1. Ruscitti, Francesco, 2012. "On the boundary behavior of the excess demand function," Research in Economics, Elsevier, Elsevier, vol. 66(4), pages 371-374.
  2. Tian, Guoqiang, 2010. "On the Existence of Price Equilibrium in Economies with Excess Demand Functions," MPRA Paper 57930, University Library of Munich, Germany, revised Jul 2014.
  3. Tian, Guoqiang, 2012. "A Full Characterization on Fixed-Point Theorem, Minimax Inequality, Saddle Point, and KKM Theorem," MPRA Paper 57929, University Library of Munich, Germany, revised Jul 2014.

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