變異型態的最小最大定理

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2002

Abstract

The socalled minimax theorem means that if $X$ and $Y$ are two sets, and $f$ and $g$ are two real-valued functions defined on $X imes Y$, under some conditions the following inequality holds: $$inf_{yin Y}sup_{xin X}f(x,y)leq sup_{xin X}inf_{yin Y}g(x,y).$$ We will extend the two functions version of minimax theorems. Our purpose of this paper is three folds:\ (1)According to Lin and Yu's thesis: Two Functions Generalization of Horvath's Minimax Theorem, we will extend some theorems without convexity.\ (2)Without the condition of usual two functions version of minimax theorem: $f$ must be strictly lesser or equal to $g$, we replace it by a milder condition: $$sup_{xin X}f(x,y)leq sup_{xin X}g(x,y), orall yin Y.$$ However, we require some restrictions; such as, the functions $f$ and $g$ are {it jointly upward}, and their upper sets are connected.\ (3)Sometimes on some given region, the two functions version of minimax theorems is failure. By use of the properties of multifunctions, we define the {it X-quasiconcave} set, so that we can extend the two functions minimax theorem to the graph of the multifunction. In fact, we get the inequality: $$inf_{yin T(X)}sup_{xin T^{-1}(y)}f(x,y)leq sup_{xin X}inf_{yin T(x)}g(x,y),$$ where $T$ is a multifunction from $X$ to $Y$, and ${g}$ is a {it X- quasiconcave} set of $T$.