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Discussiones Mathematicae, Differential Inclusions, Control and Optimization

2003 | 23 | 1 | 55-92
Tytuł artykułu

Minimax theorems without changeless proportion

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Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
EN
Kategorie tematyczne
Rocznik
Tom
Numer
Strony
55-92
Opis fizyczny
Daty
wydano
2003
otrzymano
2003-03-10
poprawiono
2003-10-10
Twórcy
autor
• Department of Mathematics, National Taiwan Normal University, Taipei, Taiwan, Republic of China
autor
• Department of Mathematics, National Taiwan Normal University, Taipei, Taiwan, Republic of China
Bibliografia
• [1] F.E. Browder, Coincidence theorems, minimax theorems, and variational inequalities, Contemp. Math. 26 (1984), 67-80.
• [2] L.J. Chu, Unified approaches to nonlinear optimization, Optimization 46 (1999), 25-60.
• [3] K. Fan, Minimax theorems, Proc. Nat. Acad. Sci. U.S.A. 39 (1953), 42-47.
• [4] M.A. Geraghty and B.L. Lin, On a minimax theorem of Terkelsen, Bull. Inst. Math. Acad. Sinica. 11 (1983), 343-347.
• [5] M.A. Geraghty and B.L. Lin, Topological minimax theorems, Proc. AMS 91 (1984), 377-380.
• [6] B.L. Lin and F.S. Yu, A two functions minimax theorem, Acta Math. Hungar. 83 (1-2) (1999), 115-123.
• [7] S. Simons, On Terkelsen minimax theorems, Bull. Inst. Math. Acad. Sinica. 18 (1990), 35-39.
• [8] F. Terkelsen, Some minimax theorems, Math. Scand. 31 (1972), 405-413.
• [9] M. Sion, On general minimax theorem, Pacific J. Math. 8 (1958), 171-176.
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Bibliografia
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