# Minimax theorems without changeless proportion

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

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## Abstract

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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: $in{f}_{y\in Y}su{p}_{x\in X}f\left(x,y\right)\le su{p}_{x\in X}in{f}_{y\in Y}g\left(x,y\right)$. We will extend the two functions version of minimax theorems without the usual condition: f ≤ g. We replace it by a milder condition: $su{p}_{x\in X}f\left(x,y\right)\le su{p}_{x\in X}g\left(x,y\right)$, ∀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: $in{f}_{y\in T\left(X\right)}su{p}_{x\in {T}^{-1}\left(y\right)}f\left(x,y\right)\le su{p}_{x\in X}in{f}_{y\in T\left(x\right)}g\left(x,y\right)$, where T is a multifunction from X to Y.

## How to cite

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Liang-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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1. [1] F.E. Browder, Coincidence theorems, minimax theorems, and variational inequalities, Contemp. Math. 26 (1984), 67-80. Zbl0542.47046
2. [2] L.J. Chu, Unified approaches to nonlinear optimization, Optimization 46 (1999), 25-60. Zbl0953.47044
3. [3] K. Fan, Minimax theorems, Proc. Nat. Acad. Sci. U.S.A. 39 (1953), 42-47.
4. [4] M.A. Geraghty and B.L. Lin, On a minimax theorem of Terkelsen, Bull. Inst. Math. Acad. Sinica. 11 (1983), 343-347. Zbl0521.49010
5. [5] M.A. Geraghty and B.L. Lin, Topological minimax theorems, Proc. AMS 91 (1984), 377-380. Zbl0512.90095
6. [6] B.L. Lin and F.S. Yu, A two functions minimax theorem, Acta Math. Hungar. 83 (1-2) (1999), 115-123.
7. [7] S. Simons, On Terkelsen minimax theorems, Bull. Inst. Math. Acad. Sinica. 18 (1990), 35-39. Zbl0714.49010
8. [8] F. Terkelsen, Some minimax theorems, Math. Scand. 31 (1972), 405-413. Zbl0259.90042
9. [9] M. Sion, On general minimax theorem, Pacific J. Math. 8 (1958), 171-176.

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