Minimax theorems without changeless proportion

• Autorzy: Liang-Ju Chu , Chi-Nan Tsai
• 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

minimax theorems; t-convex functions; upward functions; jointly upward functions; X-quasiconcave sets

Kategorie tematyczne

• 47H04: Set-valued operators
• 49J35: Minimax problems
• 47H10: Fixed-point theorems
• 49J40: Variational methods including variational inequalities

• Wydawca: Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra
• Czasopismo: Discussiones Mathematicae, Differential Inclusions, Control and Optimization
• Rocznik: 2003
• Tom: 23
• Numer: 1
• Strony: 55-92

Daty

wydano

2003

otrzymano

2003-03-10

poprawiono

2003-10-10

Twórcy

autor

Liang-Ju Chu

• Department of Mathematics, National Taiwan Normal University, Taipei, Taiwan, Republic of China

autor

Chi-Nan Tsai

• 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.

Identyfikatory

DOI

10.7151/dmgt.1047