Abstract
Some new weak Knaster-Kuratouski-Mazurkiewicz (KKM) theorems are proved under the noncompact situation in the generalized finitely continuous space (GFC-space) without any convexity. As applications, the minimax inequalities of the Ky Fan type are also given under some suitable conditions. The results unify and generalize some known results in recent literatures.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Knaster, B., Kuratowski, K., and Mazurkiewicz, S. Ein beweis des fixpunksates fur n-dimensional simplexe. Fund. Math., 14, 132–137 (1929)
Horvath, C. Some results on multivalued mappings and inequalities without convexity. Nonlinear and Convex Analysis, Lecture Notes in Pure and Applied Mathematics, Dekker, New York, 99–106 (1987)
Park, S. and Kim, H. Coincidence theorems for admissible multifunctions on generalized convex spaces. J. Math. Anal. Appl., 197, 173–187 (1996)
Ben-El-Mechaiekh, H., Chebhi, S., Florenzano, M., and Linewes, J. V. Abstract convexity and fixed points. J. Math. Anal. Appl., 222, 138–150 (1998)
Ding, X. P. Maximal element theorems in product FC-space and generalized games. J. Math. Anal. Appl., 305, 29–42 (2005)
Khanh, P. Q. and Quan, N. H. Intersection theorem, coincidence theorems and maximal-element theorems in GFC-space. Optimization, 59, 29–42 (2010)
Khanh, P. Q. and Quan, N. H. General extence theorems, alternative theorems and applications to minimax problems. Nonlinear Anal., 72, 2706–2715 (2010)
Khanh, P. Q. and Quan, N. H. Existence results for general inclusions using generalized KKM theorems with applications to minimax problem. J. Optim. Theory Appl., 146, 640–650 (2010)
Khanh, P. Q. and Quan, N. H. Generic stability and essential components of generalized KKM points and applications. J. Optim. Theory Appl., 148, 488–504 (2011)
Khanh, P. Q., Long, V. S. T., and Quan, N. H. Continuous selection, collectively fixed points and weak Knaster-Kuratowki-Mazurkiewicz mapping in optimization. J. Optim. Theory Appl., 151, 552–572 (2011)
Khanh, P. Q., Quan, N. H., and Yao, J. C. Generalized KKM type theorems in GFC-space and applications. Nonlinear Anal., 71, 1227–1234 (2009)
Liu, L. T., Ko, C. J., and Park, S. Coincidence theorems for set-valued mapping with G-KKM property on generalized convex space. Discussiones Mathematicae Differential Inclusions, Control and Optimization, 18, 69–85 (1998)
Balaj, M. Weakly G-KKM mapping, G-KKM property and minimax inequalities. J. Math. Anal. Appl., 294, 237–245 (2004)
Balaj, M. and Regan, D. Q. Weak-equilibrium problems in G-convex space. Rendiconti del Circolo Mathmatcio di Palermo, 57, 103–117 (2008)
Ding, X. P. Coincidenc theorems in topological spaces and their applications. Appl. Math. Lett., 12, 99–105 (1999)
Tang, G. S., Zhang, Q. B., and Cheng, C. Z. W-G-F-KKM mapping, intersection theorems and minimax inequalities in FC-space. J. Math. Anal. Appl., 334, 1481–1491 (2007)
Author information
Authors and Affiliations
Corresponding author
Additional information
Project supported by the National Natural Science Foundation of China (No. 11126346)
Rights and permissions
About this article
Cite this article
Fang, M., Wang, L. Weak KKM theorems in generalized finitely continuous space with applications. Appl. Math. Mech.-Engl. Ed. 34, 1291–1296 (2013). https://doi.org/10.1007/s10483-013-1745-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10483-013-1745-x
Key words
- weak Knaster-Kuratouski-Mazurkiewicz (KKM) mapping
- generalized finitely continuous space (GFC-space)
- minimax inequality
- GFC-quasicovex