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研究生:鄭蕙琳
研究生(外文):Huilin-Cheng
論文名稱:固定點定理及其相關於緊緻凸集合定理之研究
論文名稱(外文):A Study of Fixed Point Theorem and Related Theorems for Compact Convex Sets
指導教授:李是男
指導教授(外文):Shihnan-Li
學位類別:碩士
校院名稱:中原大學
系所名稱:數學系
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2000
畢業學年度:88
語文別:英文
論文頁數:44
中文關鍵詞:固定點相關定理
外文關鍵詞:fixed point theorem
相關次數:
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  • 下載下載:10
  • 收藏至我的研究室書目清單書目收藏:1
本論文旨在研究Ky Fan之最大最小不等式[3]。該結果賦予Ky Fan所統合Nash平衡點定理[6]及Von Neumann最小最大原理[9]之相關定理[2]新的證明方法,同時亦有非常廣泛的應用[3]。
Von Neumann最小最大原理指出雙人零和對局有鞍點的條件[9],Nash則提供n人非合作對局具有Nash平衡點之條件[6]。雖然Nash平衡點在雙人零和對局時與鞍點具有完全相同的意義,但Nash平衡點定理中所要求的條件稍強,不能視為Von Neumann最小最大原理之推廣[6][9]。Ky Fan於1964年統合上述兩人的結果,提出一個蘊含Nash平衡點定理及Von Neumann最小最大原理的定理[2],在1972年更提出具廣泛應用的Ky Fan最小最大不等式[3],於是先前的結果成了該不等式的推論之一。
本論文的主體分為三個部分。第一部份為預備知識,介紹後兩部分會用到的定義、術語與結果,並引用Brouwer固定點定理證明Knaster-Kuratowski-Mazurkiewicz定理[5]及Ky Fan對其之推廣[1],該部分結束前介紹雙人零和對局中所見最大最小不等式。在第二部分中,我們詳細討論一些有關半連續性及擬凸性的擴充實數函數的性質,這些結果是為下一部份預做準備的,因此第二部分可說是本論文的中心。第三部分先證明將Ky Fan的兩個結果考慮成為擴充實數值函數也會成立,再證明Nash平衡點定理及Von Neumann最小最大原理可以由Ky Fan的結果得到。論文結束前,我們舉一例子來凸顯Ky Fan最小最大不等式還是在擴充實數系中討論較為恰當。我們相信這個自然推廣後的結果是學界所企盼或學者已有的認知,但目前我們尚未發現有文獻將其很仔細的做個探討,因此我們的工作或許對此一領域的研究有一些幫助。
The purpose of this master’s thesis is to study Ky Fan’s minimax inequality [3], which gives a new proof to the unification of Nash’s equilibrium point theorem [6] and Von Neumann’s minimax principle [9] that was raised by Ky Fan [2]. It also has broad usage [3] in many fields.
Von Neumann’s minimax principle [9] indicates the conditions that a two-person zero-sum game has a saddle point and Nash’s equilibrium point theorem [6] indicates the conditions that n-person non-cooperative game has a Nash’s equilibrium point. Although in two-person zero-sum game, Nash’s equilibrium point has the same meaning as the saddle point, Nash’s equilibrium point theorem is not an extension of Von Neumann’s minimax principle because its demandable conditions are stronger. In 1964, Ky Fan gave a common generalization [2] of Nash’s equilibrium point theorem and Von Neumann’s minimax principle. Furthermore, an extensive used theorem─Ky Fan’s minimax inequality [3] was published in 1972, which, of course, implies his previous result. Ky Fan’s minimax inequality is an inevitable result of real valued functions, which satisfies some particular conditions. In this thesis, we find it will be more natural to extend these functions to extended real valued functions.
The body of this thesis was divided to three parts. The first part is preliminaries, it introduces definitions, terminologies and results that will be used in the thesis, then we use Brouwer fixed point theorem to prove Knaster-Kuratowski-Mazurkiewicz theorem [5] and Ky Fan’s generalization [1]. In the end of the first part, we give a detailed discussion of semicontinuity and quasiconvexity of extended real valued functions for the preparation of the last part of this thesis. Then, in the last part, we first prove Ky Fan’s two results for extended real valued functions and then prove Nash’s equilibrium point theorem and Von Neumann’s minimax principle by applying Ky Fan’s results. In the end of this thesis, we give an example to show that it will be better to discuss the Ky Fan’s minimax inequality by considering extended real number system.
目錄:
1.預備知識-----------------------------------------------1
2.連續性與擬凸性-----------------------------------------6
3.Von Neumann最大最小定理與Nash平衡點定理---------------10
附錄:
1.Preliminaries-----------------------------------------15
2.Semicontinuity and Quasiconvexity---------------------23
3.Von Neumann''s Minimax Theorem and Nash''s Equilibrium Point
Theorem-------------------------------------------------32
Reference-----------------------------------------------44
[1].K. Fan:A Generalization of Tychonoff''s Fixed Point Theorem, Ann. of Math. 142(1961), pp.305-310
[2].K. Fan:Sur un Theorem Minimax, C.R.Acad. Sci Paris, 259(1964), pp. 3925-3928
[3].K. Fan: A Minimax Inequality and Appilcations, in O.Shisha, ed., Inequalities 3, Academic Press, New York and London, 1972, pp.103-113
[4].James R.Munkres: Topology: a first course, Prentice-Hall, Inc.Englewood Cliffs, New Jersey, 1975
[5].B.Knaster, C.Kuratowski and S.Mazurkiewicz: Ein Beweis des Fixpunktsatzes fur n-dimensional Simplexe, Fund. Math. 14(1929),pp. 132-137
[6].J.Nash: Non-Cooperative Games, Ann. of Math. 54(1951), pp.286-295
[7].G.Owen: Game Theory, Academic Press, Inc. 1995
[8].W.Rudin:Real and Complex Analysis, McGraw-Hall Book Company, 1987
[9].J. Von Neumann and O.Morgenstern: Theory of Games and Economic Behavior, Princeton University Press, 1944
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