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Minimax theorems without changeless proportion

Liang-Ju ChuChi-Nan Tsai — 2003

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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: i n f y Y s u p x X f ( x , y ) s u p x X i n f y Y g ( 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: s u p x X f ( x , y ) s u p x X g ( 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...

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