Minimax theorems without changeless proportion
Liang-Ju Chu ; Chi-Nan Tsai
Discussiones Mathematicae, Differential Inclusions, Control and Optimization, Tome 23 (2003), p. 55-92 / Harvested from The Polish Digital Mathematics Library
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The so-called minimax theorem means that if X and Y are two sets, and f and g are two real-valued functions defined on X×Y, then under some conditions the following inequality holds: infy∈Ysupx∈Xf(x,y)≤supx∈Xinfy∈Yg(x,y). We will extend the two functions version of minimax theorems without the usual condition: f ≤ g. We replace it by a milder condition: supx∈Xf(x,y)≤supx∈Xg(x,y), ∀y ∈ Y. However, we require some restrictions; such as, the functions f and g are jointly upward, and their upper sets are connected. On the other hand, by using some properties of multifunctions, we define X-quasiconcave sets, so that we can extend the two functions minimax theorem to the graph of the multifunction. In fact, we get the inequality: infy∈T(X)supx∈T-1(y)f(x,y)≤supx∈Xinfy∈T(x)g(x,y), where T is a multifunction from X to Y.

Publié le : 2003-01-01
EUDML-ID : urn:eudml:doc:271509 
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Liang-Ju Chu; Chi-Nan Tsai. Minimax theorems without changeless proportion. Discussiones Mathematicae, Differential Inclusions, Control and Optimization, Tome 23 (2003) pp. 55-92. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1047/

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