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

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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.
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  • Department of Mathematics, Sichuan Normal University, Chengdu, Sichuan, China,
  • Department of Mathematics, Statistics and Computing Science, Dalhousie University, Halifax, Nova Scotia, Canada, B3h 3j5
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  • [3]J. S. Bae, W. K. Kim and K. K. Tan, Another generalization of Fan's minimax inequality and its applications, submitted.
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  • [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.
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  • [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.
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