A minimax inequality with applications to existence of equilibrium point and fixed point theorems

Xie Ding; Kok-Keong Tan

Colloquium Mathematicae (1992)

  • Volume: 63, Issue: 2, page 233-247
  • ISSN: 0010-1354

Abstract

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Ky Fan’s minimax inequality [8, Theorem 1] has become a versatile tool in nonlinear and convex analysis. In this paper, we shall first obtain a minimax inequality which generalizes those generalizations of Ky Fan’s minimax inequality due to Allen [1], Yen [18], Tan [16], Bae Kim Tan [3] and Fan himself [9]. Several equivalent forms are then formulated and one of them, the maximal element version, is used to obtain a fixed point theorem which in turn is applied to obtain an existence theorem of an equilibrium point in a one-person game. Next, by applying the minimax inequality, we present some fixed point theorems for set-valued inward and outward mappings on a non-compact convex set in a topological vector space. These results generalize the corresponding results due to Browder [5], Jiang [11] and Shih Tan [15] in several aspects.

How to cite

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Ding, Xie, and Tan, Kok-Keong. "A minimax inequality with applications to existence of equilibrium point and fixed point theorems." Colloquium Mathematicae 63.2 (1992): 233-247. <http://eudml.org/doc/210149>.

@article{Ding1992,
abstract = {Ky Fan’s minimax inequality [8, Theorem 1] has become a versatile tool in nonlinear and convex analysis. In this paper, we shall first obtain a minimax inequality which generalizes those generalizations of Ky Fan’s minimax inequality due to Allen [1], Yen [18], Tan [16], Bae Kim Tan [3] and Fan himself [9]. Several equivalent forms are then formulated and one of them, the maximal element version, is used to obtain a fixed point theorem which in turn is applied to obtain an existence theorem of an equilibrium point in a one-person game. Next, by applying the minimax inequality, we present some fixed point theorems for set-valued inward and outward mappings on a non-compact convex set in a topological vector space. These results generalize the corresponding results due to Browder [5], Jiang [11] and Shih Tan [15] in several aspects.},
author = {Ding, Xie, Tan, Kok-Keong},
journal = {Colloquium Mathematicae},
keywords = {Ky Fan's minimax inequality; maximal element version; equilibrium point; one-person game; fixed point theorems; set-valued inward and outward mappings},
language = {eng},
number = {2},
pages = {233-247},
title = {A minimax inequality with applications to existence of equilibrium point and fixed point theorems},
url = {http://eudml.org/doc/210149},
volume = {63},
year = {1992},
}

TY - JOUR
AU - Ding, Xie
AU - Tan, Kok-Keong
TI - A minimax inequality with applications to existence of equilibrium point and fixed point theorems
JO - Colloquium Mathematicae
PY - 1992
VL - 63
IS - 2
SP - 233
EP - 247
AB - Ky Fan’s minimax inequality [8, Theorem 1] has become a versatile tool in nonlinear and convex analysis. In this paper, we shall first obtain a minimax inequality which generalizes those generalizations of Ky Fan’s minimax inequality due to Allen [1], Yen [18], Tan [16], Bae Kim Tan [3] and Fan himself [9]. Several equivalent forms are then formulated and one of them, the maximal element version, is used to obtain a fixed point theorem which in turn is applied to obtain an existence theorem of an equilibrium point in a one-person game. Next, by applying the minimax inequality, we present some fixed point theorems for set-valued inward and outward mappings on a non-compact convex set in a topological vector space. These results generalize the corresponding results due to Browder [5], Jiang [11] and Shih Tan [15] in several aspects.
LA - eng
KW - Ky Fan's minimax inequality; maximal element version; equilibrium point; one-person game; fixed point theorems; set-valued inward and outward mappings
UR - http://eudml.org/doc/210149
ER -

References

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  2. [2] J. P. Aubin, Mathematical Methods of Games and Economic Theory, revised ed., Stud. Math. Appl. 7, North-Holland, 1982. 
  3. [3]J. S. Bae, W. K. Kim and K. K. Tan, Another generalization of Fan's minimax inequality and its applications, submitted. Zbl0794.52003
  4. [4] F. E. Browder, The fixed point theory of multi-valued mappings in topological vector spaces, Math. Ann. 177 (1968), 283-301. Zbl0176.45204
  5. [5] F. E. Browder, On a sharpened form of the Schauder fixed point theorem, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), 4749-4751. Zbl0375.47028
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  10. [10] A. Granas and F. C. Liu, Coincidences for set-valued maps and minimax inequalities, J. Math. Pures Appl. 65 (1986), 119-148. Zbl0659.49007
  11. [11] J. Jiang, Fixed point theorems for multi-valued mappings in locally convex spaces, Acta Math. Sinica 25 (1982), 365-373. Zbl0569.47051
  12. [12] M. H. Shih and K. K. Tan, A further generalization of Ky Fan's minimax inequality and its applications, Studia Math. 77 (1984), 279-287. Zbl0554.49005
  13. [13] M. H. Shih and K. K. Tan, The Ky Fan minimax principle, sets with convex sections, and variational inequalities, in: Differential Geometry, Calculus of Variations and Their Applications, G. M. Rassias & T. M. Rassias (eds.), Lecture Notes in Pure Appl. Math. 100, Dekker, 1985, 471-481. 
  14. [14] M. H. Shih and K. K. Tan, Covering theorems of convex sets related to fixed point theorems, in: Nonlinear and Convex Analysis, B.L. Lin and S. Simons (eds.), Dekker, 1987, 235-244. 
  15. [15] 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. Zbl0664.52001
  16. [16] K. K. Tan, Comparison theorems on minimax inequalities, variational inequalities, and fixed point theorems, J. London Math. Soc. 23 (1983), 555-562. Zbl0497.49010
  17. [17] N. C. Yannelis and N. D. Prabhakar, Existence of maximal elements and equilibria in linear topological vector spaces, Math. Economics 12 (1983), 233-245. Zbl0536.90019
  18. [18] C. L. Yen, A minimax inequality and its applications to variational inequalities, Pacific J. Math. 97 (1981), 477-481. Zbl0493.49009
  19. [19] J. X. Zhou and G. Chen, Diagonal convexity conditions for problems in convex analysis and quasi-variational inequalities, J. Math. Anal. Appl. 132 (1988), 213-225. Zbl0649.49008

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