所謂兩個函數的最小最大定理(minimax theorem) , 是指在給定的兩個集合$X$ 和 $Y$
中 , 研究定義在 $X imes Y$ 上的兩個實值函數 $f$ 和 $g$ , 是否可以得到下列不等
式
$$inf_{yin Y}sup_{xin X}f(x,y)leq sup_{xin X}inf_{yin Y}g(x,y).$$
此篇論文將做進一步推廣 , 主要的推論有三層 :\
(1) 根據Lin和Yu的研究論文 : Two Functions Generalization of Horvath's
Minimax Theorem, 我們將推廣出一些不需要凸性的最小最大定理.
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(2) 打破一般關於兩個函數的最小最大定理中所規定 $f$ 必須嚴格小於或等於 $g$ 的條
件 ,
取而代之的是
$$sup_{xin X}f(x,y)leq sup_{xin X}g(x,y), orall yin Y.$$
當然 , 其中的兩個函數需稍作限制 , 包括 : 兩個函數形成聯合向上($jointly\nupward$)
函數關係 , 以及它們所形成的上集合($upper set$)必需為連通的$dots$ 等.
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(3) 有時候在某個定義域上兩個函數的最小最大定理不會成立 ,
但是若在此時稍微限制定義域的範圍後 , 最小最大定理便可以成立了 !
於是我們利用了多值函數的一些性質 , 定義$X$- 擬凹集合 ,
推廣出在多值函數上的最小最大定理 , 而得到下列變異型態的最小最大不等式
$$inf_{yin T(X)}sup_{xin T^{-1}(y)}f(x,y)leq
sup_{xin X}inf_{yin T(x)}g(x,y).$$
其中 , $T$ 為由 $X$ 對應到 $Y$ 的多值函數 , ${g}$ 則是相應於 $T$
的$X$- 擬凹集合.

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 $T$.

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