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1992 | 63 | 2 | 233-247
Tytuł artykułu

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

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
Rocznik
Tom
63
Numer
2
Strony
233-247
Opis fizyczny
Daty
wydano
1992
otrzymano
1990-02-07
poprawiono
1991-03-04
Twórcy
  • Department of Mathematics, Sichuan Normal University, Chengdu, Sichuan, China,
  • Department of Mathematics, Statistics and Computing Science, Dalhousie University, Halifax, Nova Scotia, Canada, B3h 3j5
Bibliografia
  • [1] G. Allen, Variational inequalities, complementarity problems, and duality theorems, J. Math. Anal. Appl. 58 (1977), 1-10.
  • [2] J. P. Aubin, Mathematical Methods of Games and Economic Theory, revised ed., Stud. Math. Appl. 7, North-Holland, 1982.
  • [3]J. S. Bae, W. K. Kim and K. K. Tan, Another generalization of Fan's minimax inequality and its applications, submitted.
  • [4] F. E. Browder, The fixed point theory of multi-valued mappings in topological vector spaces, Math. Ann. 177 (1968), 283-301.
  • [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] P. Deguire et A. Granas, Sur une certaine alternative non-linéaire en analyse convexe, Studia Math. 83 (1986), 127-138.
  • [7] K. Fan, A generalization of Tychonoff's fixed point theorem, Math. Ann. 142 (1961),305-310.
  • [8] K. Fan, A minimax inequality and applications, in: Inequalities III, O. Shisha (ed.), Acad. Press, 1972, 103-113.
  • [9] K. Fan, Some properties of convex sets related to fixed point theorems, Math. Ann. 266 (1984), 519-537.
  • [10] A. Granas and F. C. Liu, Coincidences for set-valued maps and minimax inequalities, J. Math. Pures Appl. 65 (1986), 119-148.
  • [11] J. Jiang, Fixed point theorems for multi-valued mappings in locally convex spaces, Acta Math. Sinica 25 (1982), 365-373.
  • [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] 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] 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] 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] K. K. Tan, Comparison theorems on minimax inequalities, variational inequalities, and fixed point theorems, J. London Math. Soc. 23 (1983), 555-562.
  • [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] C. L. Yen, A minimax inequality and its applications to variational inequalities, Pacific J. Math. 97 (1981), 477-481.
  • [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.
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.bwnjournal-article-cmv63i2p233bwm
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