On Some Generalized Ky Fan Minimax Inequalities
© Xianqiang Luo. 2009
Received: 31 October 2008
Accepted: 21 April 2009
Published: 2 June 2009
Some generalized Ky Fan minimax inequalities for vector-valued mappings are established by applying the classical Browder fixed point theorem and the Kakutani-Fan-Glicksberg fixed point theorem.
It is well known that Ky Fan minimax inequality  plays a very important role in various fields of mathematics, such as variational inequality, game theory, mathematical economics, fixed point theory, control theory. Many authors have got some interesting achievements in generalization of the inequality in various ways. For example, Ferro  obtained a minimax inequality by a separation theorem of convex sets. Tanaka  introduced some quasiconvex vector-valued mappings to discuss minimax inequality. Li and Wang  obtained a minimax inequality by using some scalarization functions. Tan  obtained a minimax inequality by the generalized G-KKM mapping. Verma  obtained a minimax inequality by an R-KKM mapping. Li and Chen  obtained a set-valued minimax inequality by a nonlinear separation function . Ding [8, 9] obtained a minimax inequality by a generalized R-KKM mapping. Some other results can be found in [10–16].
In this paper, we will establish some generalized Ky Fan minimax inequalities forvector-valued mappings by the classical Browder fixed point theorem and the Kakutani-Fan-Glicksberg fixed point theorem.
Now, we recall some definitions and preliminaries needed. Let and be two nonempty sets, and let be a nonempty set-valued mapping, if and only if , . Throughout this paper, assume that every space is Hausdorff.
Definition 2.1 (see ).
(iii)continuous, if it is both (usc) and (lsc);
Definition 2.2 (see ).
Lemma 2.3 (see ).
Lemma 2.4 (see ).
Lemma 2.5 (see [2, Theorem 3.1]).
Let be a topological vector space, let be a topological vector space with a closed pointed convex cone , , let and be two nonempty compact subsets of , and let be a continuous mapping. Then both defined by and defined by are upper semicontinuous and compact-valued.
We quote some of their properties as follows (see ):
Definition 2.7 (see ).
The following two propositions are very important in proving Proposition 2.10.
Proposition 2.8 (see ).
Let be a topological vector space, let be a topological vector space with a closed pointed convex cone , , and let be a nonempty compact convex subset of , be a vector mapping. Then the following two statements are equivalent:
(i) (ii) for all and for all , let . By Proposition 2.8, we have is quasiconcave, that is, for any , is convex, then by Proposition 2.9, we have for any , is convex. Thus, is convex. Therefore, we have is convex since by property (i) of .
3. Generalized Ky Fan Minimax Inequalities
In this section, we will establish some generalized Ky Fan minimax inequalities and a corollary by Propositions 1.1, 1.3 and Lemmas 3.1, 3.2.
Lemma 3.1 (see ).
Lemma 3.2 (see , Kakutani-Fan-Glicksberg fixed point theorem).
Let be a locally convex topological vector space and let be a nonempty compact and convex set. If is upper semicontinuous, and for any , is a nonempty, closed and convex subset, then has a fixed point.
In fact, firstly, by , we have , and for each , is open since is continuous. Secondly, by condition (i) and Proposition 2.8, we have is quasiconcave in , that is, for any , is convex. Thus, by Proposition 2.9, is convex.
Let be a locally convex topological vector space, let be a topological vector space with a closed convex pointed cone , , let be a nonempty compact and convex subset of , let be a continuous mapping, and let such that
Let be a locally convex topological vector space, let be a topological vector space with a closed convex pointed cone , , let be a nonempty compact and convex subset of , and let be a continuous mapping such that
(ii)From Figure 1, we can check that is properly -quasiconcave in for each .
(iii)From Figure 1, we can check that for each . Thus, for each .
Finally, from Figure 1, we can check that , that is, Corollary 3.8 holds.
The author gratefully acknowledges the referee for his/her ardent corrections and valuable suggestions, and is thankful to Professor Junyi Fu and Professor Xunhua Gong for their help. This work was supported by the Young Foundation of Wuyi University.
- Fan K: A minimax inequality and applications. In Inequalities, III (Proc. Third Sympos., Univ. California, Los Angeles, Calif., 1969; Dedicated to the Memory of Theodore S. Motzkin). Academic Press, New York, NY, USA; 1972:103–113.Google Scholar
- Ferro F: A minimax theorem for vector-valued functions. Journal of Optimization Theory and Applications 1989,60(1):19–31. 10.1007/BF00938796MathSciNetView ArticleMATHGoogle Scholar
- Tanaka T: Generalized quasiconvexities, cone saddle points, and minimax theorem for vector-valued functions. Journal of Optimization Theory and Applications 1994,81(2):355–377. 10.1007/BF02191669MathSciNetView ArticleMATHGoogle Scholar
- Li ZF, Wang SY: A type of minimax inequality for vector-valued mappings. Journal of Mathematical Analysis and Applications 1998,227(1):68–80. 10.1006/jmaa.1998.6076MathSciNetView ArticleMATHGoogle Scholar
- Tan K-K: G-KKM theorem, minimax inequalities and saddle points. Nonlinear Analysis: Theory, Methods & Applications 1997,30(7):4151–4160. 10.1016/S0362-546X(96)00129-0MathSciNetView ArticleMATHGoogle Scholar
- Verma RU: Some results on R-KKM mappings and R-KKM selections and their applications. Journal of Mathematical Analysis and Applications 1999,232(2):428–433. 10.1006/jmaa.1999.6302MathSciNetView ArticleMATHGoogle Scholar
- Li SJ, Chen GY, Teo KL, Yang XQ: Generalized minimax inequalities for set-valued mappings. Journal of Mathematical Analysis and Applications 2003,281(2):707–723. 10.1016/S0022-247X(03)00197-5MathSciNetView ArticleMATHGoogle Scholar
- Ding XP: New generalized R-KKM type theorems in general topological spaces and applications. Acta Mathematica Sinica 2007,23(10):1869–1880. 10.1007/s10114-005-0876-yMathSciNetView ArticleMATHGoogle Scholar
- Ding XP, Liou YC, Yao JC: Generalized R-KKM type theorems in topological spaces with applications. Applied Mathematics Letters 2005,18(12):1345–1350. 10.1016/j.aml.2005.02.022MathSciNetView ArticleMATHGoogle Scholar
- Chang SS: Variational Inequality and Complementary Problem Theory with Applications. Shanghai Science and Technology Press, Shanghai, China; 1991.Google Scholar
- Jahn J: Mathematical Vector Optimization in Partially Ordered Linear Spaces, Methoden und Verfahren der Mathematischen Physik. Volume 31. Peter Lang, Frankfurt, Germany; 1986:viii+310.Google Scholar
- Gerstewitz C: Nichtkonvexe trennungssatze und deren Anwendung in der theorie der Vektoroptimierung. Seminarberichte der Secktion Mathematik 1986, 80: 19–31.MathSciNetMATHGoogle Scholar
- Browder FE: The fixed point theory of multi-valued mappings in topological vector spaces. Mathematische Annalen 1968, 177: 283–301. 10.1007/BF01350721MathSciNetView ArticleMATHGoogle Scholar
- Ha CW: Minimax and fixed point theorems. Mathematische Annalen 1980,248(1):73–77. 10.1007/BF01349255MathSciNetView ArticleMATHGoogle Scholar
- Agarwal RP, O'Regan D: Variational inequalities, coincidence theory, and minimax inequalities. Applied Mathematics Letters 2001,14(8):989–996. 10.1016/S0893-9659(01)00077-5MathSciNetView ArticleMATHGoogle Scholar
- Zeng L-C, Wu S-Y, Yao J-C: Generalized KKM theorem with applications to generalized minimax inequalities and generalized equilibrium problems. Taiwanese Journal of Mathematics 2006,10(6):1497–1514.MathSciNetMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.