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ArticleOriginal scientific text

Title

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

Authors 1, 2

Affiliations

  1. Department of Mathematics, Sichuan Normal University, Chengdu, Sichuan, China,
  2. Department of Mathematics, Statistics and Computing Science, Dalhousie University, Halifax, Nova Scotia, Canada, B3h 3j5

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.

Bibliography

  1. [1] G. Allen, Variational inequalities, complementarity problems, and duality theorems, J. Math. Anal. Appl. 58 (1977), 1-10.
  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.
  4. [4] F. E. Browder, The fixed point theory of multi-valued mappings in topological vector spaces, Math. Ann. 177 (1968), 283-301.
  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.
  6. [6] P. Deguire et A. Granas, Sur une certaine alternative non-linéaire en analyse convexe, Studia Math. 83 (1986), 127-138.
  7. [7] K. Fan, A generalization of Tychonoff's fixed point theorem, Math. Ann. 142 (1961),305-310.
  8. [8] K. Fan, A minimax inequality and applications, in: Inequalities III, O. Shisha (ed.), Acad. Press, 1972, 103-113.
  9. [9] K. Fan, Some properties of convex sets related to fixed point theorems, Math. Ann. 266 (1984), 519-537.
  10. [10] A. Granas and F. C. Liu, Coincidences for set-valued maps and minimax inequalities, J. Math. Pures Appl. 65 (1986), 119-148.
  11. [11] J. Jiang, Fixed point theorems for multi-valued mappings in locally convex spaces, Acta Math. Sinica 25 (1982), 365-373.
  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.
  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.
  16. [16] K. K. Tan, Comparison theorems on minimax inequalities, variational inequalities, and fixed point theorems, J. London Math. Soc. 23 (1983), 555-562.
  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.
  18. [18] C. L. Yen, A minimax inequality and its applications to variational inequalities, Pacific J. Math. 97 (1981), 477-481.
  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.
Colloquium Mathematicum
1992/63/2
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Pages:
233-247
Main language of publication
English
Received
1990-02-07
Accepted
1991-03-04
Published
1992
Exact and natural sciences