# Minimax theorems without changeless proportion

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2003)

- Volume: 23, Issue: 1, page 55-92
- ISSN: 1509-9407

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topLiang-Ju Chu, and Chi-Nan Tsai. "Minimax theorems without changeless proportion." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 23.1 (2003): 55-92. <http://eudml.org/doc/271509>.

@article{Liang2003,

abstract = {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.},

author = {Liang-Ju Chu, Chi-Nan Tsai},

journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},

keywords = {minimax theorems; t-convex functions; upward functions; jointly upward functions; X-quasiconcave sets; -convex functions; quasiconvex sets; multifunctions},

language = {eng},

number = {1},

pages = {55-92},

title = {Minimax theorems without changeless proportion},

url = {http://eudml.org/doc/271509},

volume = {23},

year = {2003},

}

TY - JOUR

AU - Liang-Ju Chu

AU - Chi-Nan Tsai

TI - Minimax theorems without changeless proportion

JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization

PY - 2003

VL - 23

IS - 1

SP - 55

EP - 92

AB - 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.

LA - eng

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

UR - http://eudml.org/doc/271509

ER -

## References

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- [8] F. Terkelsen, Some minimax theorems, Math. Scand. 31 (1972), 405-413. Zbl0259.90042
- [9] M. Sion, On general minimax theorem, Pacific J. Math. 8 (1958), 171-176.

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