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Minimax theorems without changeless proportion

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Chu L.-J., Tsai C.-N.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

2003

tom 23

nr 1

55-92

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: $inf_{y∈Y} sup_{x∈X} f(x,y) ≤ sup_{x∈X} inf_{y∈Y} g(x,y)$. We will extend the two functions version of minimax theorems without the usual condition: f ≤ g. We replace it by a milder condition: $sup_{x∈X} f(x,y) ≤sup_{x∈X}g(x,y)$, ∀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 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: $inf_{y∈T(X)} sup_{x∈T^{-1}(y)} f(x,y) ≤ sup_{x∈X} inf_{y∈T(x)} g(x,y)$, where T is a multifunction from X to Y.

Ograniczanie wyników

1 Discussiones Mathematicae, Differential Inclusions, Control and Optimization

1 Chu L.-J.

1 Tsai C.-N.

1 2003

Wyniki wyszukiwania

Minimax theorems without changeless proportion