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A minimax inequality with applications to existence of equilibrium point and fixed point theorems. (English) Zbl 0833.49009

Ky Fan’s minimax inequality has become a versatile tool in nonlinear and convex analysis. In this paper, we obtain a minimax inequality which generalizes those generalizations of Ky Fan’s minimax inequality due to G. Allen, C. L. Yen, K. K. Tan, J. S. Bae, W. K. Kim and K. K. Tan and Fan himself. 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.

MSC:

49J35 Existence of solutions for minimax problems
47H10 Fixed-point theorems
47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics
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