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Convex sets, fixed points, variational and minimax inequalities

Published online by Cambridge University Press:  17 April 2009

Tzu-Chu Lin
Tzu-Chu Lin
Affiliation:
Department of Mathematical Sciences, The University of Winconsin-Milwaukee, P.O. Box 413, Milwaukee, Wisconsin 53201, U.S.A.
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Abstract

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Recently, Ky Fan extended his will known lemma (which is an extension of the classical theorem of Knaster, Kuratowski and Mazurkiewica) to the noncompact case. Using this result, another interesting lemma of Fan is generalized in this paper. As applications of our theorem, we obtain a generalizationof Browder's variational inequality and derive Fan's other recent results directly from our theorem. Also, in this paper, we give a slight extension recent results of K. K. Tan, which themselves are generalizations of many well-known results on minimax and variational inequalities.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 34 , Issue 1 , August 1986 , pp. 107 - 117
Copyright
Copyright © Australian Mathematical Society 1986

References

[1]Allen, G., “Variational inequalities, complementarity problems, and duality theorems”, J. Math. Anal. Appl. 58 (1977), 1–10CrossRefGoogle Scholar
[2]Browder, F.E., “The fixed point theory of multi-valued mappings in topological vestor spaces”, Math. Ann. 177 (1968), 283–301.CrossRefGoogle Scholar
[3]Dugundji, J. and Granas, A., Fixed point theory, Vol. 1 (PWN-Polish Scientific Publishers, Warszawa, 1982).Google Scholar
[4]Fan, K., “A generalization of Tychonoff's fixed point theorem”, Math. Ann. 142 (1961), 305–310.Google Scholar
[5]Fan, K., “Extensions of two fixed point theorems of F. E. Bowder”, Math. Z. 112 (1969), 234–240.CrossRefGoogle Scholar
[6]Fan, K., “A minimax inequality and applications”, Inequalities III (ed.). Shisha, , Academic Press, New York, 1972), 103–113.Google Scholar
[7]Fan, K., “Some properties of convex sets related to fixed point theorems”, Math. Ann. 266 (1984), 519–537.CrossRefGoogle Scholar
[8]Knaster, B., Kuratowsi, C. and Mazurkiewicz, S., Ein Beweis des Fixpunktsatzes fur n-dimensionale Simplexe”, Fund. Math. 14 (1929), 132–137.CrossRefGoogle Scholar
[9]Lin, T. C., “A note on a theorem of Ky Fan”, Canad. Math. Bull. 22 (1979), 513–515.CrossRefGoogle Scholar
[10]Lin, T. C. and Yen, C. L., “Applications of the proximity map to fixed point theorems in Hilbert space”, J. Approx. Theory (to appear).Google Scholar
[11]Tan, K. K., “Comparison theorems on minimax inequalities, variational inequalities, and fixed point theorems”, J. London Math. Soc. 28 (1983), 555–562.CrossRefGoogle Scholar
[12]Yen, C. L., “A minimax inequality and its applications to variational inequalities”, Pacific J. Math. 97 (1981), 477–481.Google Scholar