JOURNAL BROWSE
Search
Advanced SearchSearch Tips
GENERALIZED MINIMAX THEOREMS IN GENERALIZED CONVEX SPACES
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
  • Journal title : Honam Mathematical Journal
  • Volume 31, Issue 4,  2009, pp.559-578
  • Publisher : The Honam Mathematical Society
  • DOI : 10.5831/HMJ.2009.31.4.559
 Title & Authors
GENERALIZED MINIMAX THEOREMS IN GENERALIZED CONVEX SPACES
Kim, Hoon-Joo;
  PDF(new window)
 Abstract
In this work, we obtain intersection theorem, analytic alternative and von Neumann type minimax theorem in G-convex spaces. We also generalize Ky Fan minimax inequality to acyclic versions in G-convex spaces. The result is applied to formulate acyclic versions of other minimax results, a theorem of systems of inequalities and analytic alternative.
 Keywords
Minimax Theorem;Generalized Convex Space;Acyclic;-map;f-quasiconcave;Transfer l.s.c.;Intersection Theorem;Systems of Inequalities;Analytic Alternative;
 Language
English
 Cited by
 References
1.
M. Balaj, Acyclic versions for some minimax inequalities, Nonlinear Anal. Forum 10 (2005), 169-174.

2.
M. Balaj, Two minimax inequalities in G-convex space, Applied Math. Letters 19(2006), 235-239. crossref(new window)

3.
H. Ban-El-Mechaiekh, Note on a class of set-valued maps having continuous selections, Fixed point Theory and Applications (M.A. Th/'era and J.-B. Baillon, Eds.), Longman Sci. & Tech., Essex (1991), 33-43.

4.
H. Ben-El-Mechaiekh, P. Deguire, A. Oeanas, Points fixes et coincidences pour les fonctions multivoques II (Application de type ${\phi}$ et ${\phi}^{*}$), C.R. Acad. Sci. Paris 295 (1982), 381-384.

5.
R. Bielawski, Simplicial convexity and its applications, J. Math. Anal. Appl. 127(1987), 155-171. crossref(new window)

6.
H. Brezis, L. Nirenberg and G. Stampachia, A remark on Ky Fan's minimax principle, Boll. Un. Mat. Ital. 6 (1972), 293-300.

7.
X.-P. Ding, Continuous selection theorem, coincidence theorem and intersection theorems concerning sets with H-convex sections, J. Austral. Math. Soc. (Series A) 52 (1992), 11-25. crossref(new window)

8.
Ky Fan, Applications of a theorem concerning sets with convex sections, Math. Ann. 163 (1966), 189-203. crossref(new window)

9.
A. Granas, Sur Quelques methodes topologiques en analyse convexe, Methodes Topolo- giques en Analyse Convexe, Sem. Math. sup. 110 (1990), Press. Univ. Montreal, 11-77.

10.
A. Granas and F .C. Liu, Coincidences for set-valued maps and minimax inequlities, J. Math. Pures et Appl. 65 (1986), 119-148.

11.
J.Guillerme,, Les inegalites "Inf-Sup ${\leq}$ Sup-Inf", Fixed Point Theory and Applications (M.A. Thera and J.-B. Baillon, Eds.), Longman Sci. & Tech., Essex (1991), 193-214.

12.
C.-W. Ha, A non-compact minimax theorem, Pacific J. Math. 97 (1981), 115-117. crossref(new window)

13.
C.D. Horvath, Contractibility and generalized convexity, J. Math. Anal. Appl. 156(1991), 341-357. crossref(new window)

14.
H. Komiya, Convexity on a topological space, Fund. Math. 111 (1981), 107-113.

15.
M. Lassonde, On the use of KKM multifunctions in fixed point theory and related topics, J. Math. Anal. Appl. 97 (1983), 151-201. crossref(new window)

16.
L.J. Lin, Applications of a fixed point theorem in G-convex spaces, Nonlinear Anal. 46 (2001), 601-608. crossref(new window)

17.
F.C. Liu, A note on the von Neumann-Sion minimax principle, Bull. Inst. Math. Acad. Sinica 6 (1978), 517-522.

18.
J. von Neumann, Zur Theorie der Gesellschaftsspiele, Math. Ann. 100 (1928), 295-320. crossref(new window)

19.
H, Nikaido, On von Neumann's minimax theorem, Pacific J. Math. 4 (1954), 65-72. crossref(new window)

20.
S. Park, Continuous selection theorems in generalized convex spaces, Numer. Funct. Anal. Optimiz. 20 (1999), 567-583. crossref(new window)

21.
S. Park, Elements of th e KKM theory for generalized convex spaces, Korean J. Comp. Appl. Math. 7 (2000), 1-28.

22.
S. Park, Remarks on topologies of generalized convex spaces, Nonlinear Funct. Anal. Appl. 5 (2000), 67-79.

23.
S. Park, Fixed point theorems in locally G-convex spaces, Nonlinear Anal. 48(2002), 869-879. crossref(new window)

24.
S. Park, J. S. Bae and H. K. Kang, Geometric properties, minimax inequalities, and fixed point theorems on convex spaces, Proc. Amer. Math. Soc. 121 (1994), 429-440. crossref(new window)

25.
S. Park and H. Kim, Coincidence theorems of admissible maps on generalized convex spaces, J. Math. Anal. Appl. 191 (1996), 173-187.

26.
S. Park and H. Kim, Foundations of the KKM theory on generalized convex spaces, J. Math. Anal. Appl. 209 (1997), 551-571. crossref(new window)

27.
A. Pietsch, Operator Ideals, North-Holland, Amsterdam (1980).

28.
M.-H.Shih and K.-K.Tan, The Ky Fan minimax principle, sets with convex sections, and variational inequalities, Differential Geometry, Calculus of Variations, and Their Applications (G.M. Rassias and T.M. Rassias, Eds.), Marcel Dekker Inc, New York (1984).

29.
M.-H.Shih and K.-K.Tan, Non-compact set s with convex sections, II, J. Math. Anal. Appl. 120 (1986), 264-270. crossref(new window)

30.
M.-H.Shih and K.-K.Tan, 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. crossref(new window)

31.
S. Simons, Two-function minimax theorems and variational inequalities for functions on compact and noncompact sets, with some comments on fixed- point theorems, Proc. Symp. Pure Math. 45 (1986 Pt.2), 377-392.

32.
M. Sion, On general minimax theorems, Pacific J. Math. 8 (1958), 171-176. crossref(new window)

33.
G.Q. Tian, Generalizations of the FKKM Theorem and the Ky Fan minimax inequality, with applications to maximal elements, price equilibrium, and complementarity, J. Math. Anal. Appl. 170 (1992), 457-471. crossref(new window)

34.
H. Zhang and X. Wu, A new existence theorem of maximal elements in noncompact H-spaces with applications to minimax inequalities and variational inequalities, Acta. Math. Hungar. 80 (1998), 115-127. crossref(new window)