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Matching theorems, fixed point theorems and minimax inequalities without convexity

Part of: Nonlinear operators and their properties Maps and general types of spaces defined by maps Connections with other structures, applications

Published online by Cambridge University Press:  09 April 2009

Xie-Ping Ding  and
Kok-Keong Tan
Xie-Ping Ding
Affiliation:
Department of MathematicsSichuan Normal UniversityChengdu, Sichuan People's Republic of China
Kok-Keong Tan
Affiliation:
Department of Mathematics, Statistics and Computing ScienceDalhousie UniversityHalifax, Nova ScotiaCanada, B3H 3J5
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Abstract

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Matching theorems, fixed point theorems and minimax inequalities are obtained in H-spaces which generalize the corresponding results of Bae-Kim-Tan, Browder, Fan, Horvath, Kim, Ko-Tan, Shih-Tan, Takahashi, Tan and Tarafdar to non-compact and/or non-convex settings.

Keywords

matching theoremfixed point theoremminimax inequalityH-spacecontractibleH-convexweakly H-convexcompactly open (closed)H-compactH-KKMtopological vector spaceconvex hulllower semi-continuous.

MSC classification

Secondary: 47H10: Fixed-point theorems 54H25: Fixed-point and coincidence theorems 54C60: Set-valued maps
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 49 , Issue 1 , August 1990 , pp. 111 - 128
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Allen, G., ‘Variational inequalities, complementarity problems, and duality theorems’, J. Math. Anal. Appl. 58 (1977), 1–10.CrossRefGoogle Scholar
[2]Bae, J. S., Kim, W. K. and Tan, K. K., ‘Another generalization of Ky Fan's minimax inequality and its applications’, submitted.Google Scholar
[3]Bardaro, C. and Ceppitelli, R., ‘Some further generalizations of Knaster-Kuratowski-Mazurkiewicz theorem and minimax inequalities’, J. Math. Anal. Appl. 132 (1988), 484–490.CrossRefGoogle Scholar
[4]Ben-El-Mechaiekh, H., Deguire, P. and Granas, A., ‘Points fixes et coincidences pour les fonctions multivoques II. (Applications de type ø et ø*)’, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), 381–384.Google Scholar
[5]Brézis, H., Nirenberg, L. and Stampacchia, G., ‘A remark on Ky Fan's minimax principle’, Bull. Un. Mat. Ital. (4) 6 (1972), 293–300.Google Scholar
[6]Browder, F. E., ‘Coincidence theorems, minimax theorems, and variational inequalities’, Contemp. Math. 26 (1984), 67–80.CrossRefGoogle Scholar
[7]Ding, X. P., Kim, W. K. and Tan, K. K., ‘A new minimax inequality on H-spaces with applications’ to appear in Bull. Austral. Math. Soc.Google Scholar
[8]Ding, X. P. and Tan, K. K., ‘Fixed point theorems and minimax inequalities without convexity’, submitted.Google Scholar
[9]Ding, X. P. and Tan, K. K., ‘Minimax inequality of von Neumann type and system of inequalities without convexity’, submitted.Google Scholar
[10]Ding, X. P. and Tan, K. K., ‘Non-convex generalizations of results on set with convex sections’, submitted.Google Scholar
[11]Fan, K., ‘A generalization of Tychonoff's fixed point theorem’, Math. Ann. 142 (1961), 305–310.CrossRefGoogle Scholar
[12]Fan, K., ‘A minimax inequality and applications’, Inequalities, Vol. III, (edited by Shisha, O.), pp. 103–113, (Academic Press, New York, 1972).Google Scholar
[13]Fan, K., ‘Some properties of convex sets related to fixed point theorems’, Math. Ann. 226 (1984), 519–537.CrossRefGoogle Scholar
[14]Horvath, C., ‘Point fixes et coincidences pur les applications multivoques sans convexité’, C. R. Acad. Sci. Paris 296 (1983), 403–406.Google Scholar
[15]Horvath, C., ‘Point fixes et coincidences dans les espaces topologiques compacts contractiles’, C. R. Acad. Sci. Paris 299 (1984), 519–521.Google Scholar
[16]Horvath, C., ‘Some results on multivalued mappings and inequalities without convexity’, Non-linear and convex analysis, edited by Lin, B. L. and Simons, S., pp. 99–106, (Marcel Dekker, 1987).Google Scholar
[17]Kim, W. K., ‘Some applications of the Kakutani fixed point theorem’, J. Math. Anal. Appl. 121 (1987), 119–122.CrossRefGoogle Scholar
[18]Knaster, B., Kuratowski, C. and Mazurkiewicz, S., ‘Ein Beweis des Fixpunktsatzes für n-dimensionale Simplexe’, Fund. Math. 14 91929), 132–137.CrossRefGoogle Scholar
[19]Ko, H. M. and Tan, K. K., ‘Coincidence theorems and matching theorems’, submitted.Google Scholar
[20]Shih, M. H. and Tan, K. K., ‘A further generalization of Ky Fan's minimax inequality and its applications’, Studia Math. 78 (1984), 279–287.CrossRefGoogle Scholar
[21]Shih, M. H. and Tan, K. K., ‘The Ky Fan minimax principle, sets with convex sections, and variational in equalities’, Differential geometry, calculus of variations, and their applications, edited by Rassias, G. M. and Rassias, T. M., Rassias, T. M., pp. 471–481, (Marcel Dekker, 1985).Google Scholar
[22]Shih, M. H. and Tan, K. K., ‘Shapley selections and covering theorems of simplexes’, Nonlinear and convex analysis edited by Lin, B. L., and Simons, S., pp. 245–251, (Marcel Dekker, 1987).Google Scholar
[23]Shih, M. H. and Tan, K. K., ‘A minimax inequality and Browder-Hartman-Stampacchia variational inequalitiés for multi-valued monotone operators’, Proc. Fourth Franco-SEAMS Conference, (Chiang Mai, Thailand, 1988).Google Scholar
[24]Shih, M. H. and Tan, K. K., ‘A geometric property of convex sets with applications to minimax type inequalities and fixed point theorems’, J. Austral. Math. Soc. Ser. A 45 (1988), 169–183.CrossRefGoogle Scholar
[25]Takahashi, W., ‘Fixed point, minimax, and Hahn-Banach theorems’, Nonlinear functional analysis and its applications, Part 2, pp. 419–427, (Proc. Sympos. Pure Math., vol. 45, 1986).CrossRefGoogle Scholar
[26]Tan, K. K., ‘Comparison theorems on minimax inequalities, variational inequalities, and fixed point theorems’, J. London Math. Soc. 28 (1983), 555–562.CrossRefGoogle Scholar
[27]Tarafdar, E., ‘On minimax principles and sets with convex sections’, Publ. Math. Debrecen 29 (1982), 219–226.CrossRefGoogle Scholar
[28]Yen, C. L., ‘A minimax inequality and its applications to variational inequalities’, Pacific J. Math. 97 (1981), 477–481.CrossRefGoogle Scholar
[29]Zhou, J. X. and Chen, G., ‘Diagonal convexity conditions for problems in convex analysis and quais-variational inequalities’, J. Math. Anal. Appl. 132 (1988), 213–225.CrossRefGoogle Scholar