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  • Bulletin of the Australian Mathematical Society, Volume 47, Issue 3
  • June 1993, pp. 483-503

A minimax inequality with applications to existence of equilibrium points

  • Kok-Keong Tan (a1) and Zian-Zhi Yuan (a1)
  • DOI: http://dx.doi.org/10.1017/S0004972700015318
  • Published online: 01 April 2009
Abstract

A new minimax inequality is first proved. As a consequence, five equivalent fixed point theorems are formulated. Next a theorem concerning the existence of maximal elements for an Lc-majorised correspondence is obtained. By the maximal element theorem, existence theorems of equilibrium points for a non-compact one-person game and for a non-compact qualitative game with Lc-majorised correspondences are given. Using the latter result and employing an “approximation” technique used by Tulcea, we deduce equilibrium existence theorems for a non-compact generalised game with LC correspondences in topological vector spaces and in locally convex topological vector spaces. Our results generalise the corresponding results due to Border, Borglin-Keiding, Chang, Ding-Kim-Tan, Ding-Tan, Shafer-Sonnenschein, Shih-Tan, Toussaint, Tulcea and Yannelis-Prabhakar.

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[3]A. Borglin and H. Keiding , ‘Existence of equilibrium actions of equilibrium, A note the ‘new’ existence theorems’, J. Math. Econom. 3 (1976), 313316.

[4]F.E. Browder , ‘The fixed point theory of multi-valued mappings in topological vector space’, Math. Ann. 177 (1968), 283301.

[8]X.P. Ding , W.K. Kim and K.K. Tan , ‘Equilibria of non-compact generalized games with L*-majorized preference correspondences’, J. Math. Anal. Appl. (1992), 508517.

[10]K. Fan , ‘A generalization of Tychonoff's fixed point theorem’, Math. Ann. 142 (1961), 305310.

[15]G. Mehta and E. Tarafdar , ‘Infinite-dimensional Gale-Dubreau theorem and a fixed point theorem of Tarafdar’, J. Econom. Theory 41 (1987), 333339.

[16]W. Shafer and H. Sonnenschein , ‘Equilibrium in abstract economies without ordered preferences’, J. Math. Econom. 2 (1975), 345348.

[19]E. Tarafdar , ‘On nonlinear variational inequalities’, Proc. Amer. Math. Soc. 67 (1977), 9598.

[21]S. Toussaint , ‘On the existence of equilibria in economies with infinitely many commodities and without ordered preferences’, J Econom. Theory 33 (1984), 98115.

[23]C.I. Tulcea , ‘On the approximation of upper-semicontinuous correspondences and the equilibriums of generalized games’, J. Math. Anal. Appl. 136 (1988), 267289.

[25]N.C. Yannelis , ‘Maximal elements over non-compact subsets of linear topological spaces’, Econom. Lett. 17 (1985), 133136.

[26]N.C. Yannelis and N.D. Prabhakar , ‘Existence of maiximal elements and equilibria in linear topological spaces’, J. Math. Econom. 12 (1983), 233245. Erratum 13 (1984), 305.

[27]C. L. Yen , ‘A minimax inequality ans its applications to variational inequalities’, Pacific J. Math. 97 (1981), 477481.

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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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