Some minimax theorems with strictly monotone transformation

Abstract

Some two-function minimax theorems are proved. In these results, the staircase and quantitative-topological conditions of both functions involve strictly monotone transformation and mixing of functional values. Consequently, Lin Quan and Kindler's minimax theorems are generalized. §1 IntroductionThe minimax inequalities resulting from game theory play an important role in nonlinear and convex analysis. Let X and Y be two nonempty sets, and f, g :holds under certain conditions.A two-function minimax theorem is a generalized form of a classical minimax theorem. When f = g, a two-function minimax inequality becomes a one-function minimax equality. On the one hand, we can obtain new two-function theorems by generalizing one-function theorems. The first two-function minimax theorem was established by Fan [1] in 1964. However, the counterexamples in [2,3] show that some one-function results fail to have straightforward generalization of two functions in certain ways. On the other hand, we can also obtain some two-function minimax inequalities by weakening the concavity-convexity, linear structure of spaces and functions to some given two-function minimax inequalities. Thereafter many nonlinear minimax theorems were proved. Simons [4] gave a two-function minimax theorem under the conditions that f is 1 2 -convexlike on Y and g is 1 2 -concavelike on X; Lin Quan in [5] gave another result in which the concavity-convexity conditions of functions are described by mixed functional values of f and g; Forgó and Joó's two-function minimax theorems [6] is based on the conception of averaging functions proposed by Irle [7] and the mixed method used by Lin Quan; Forgó in [8]