多值映射
正文
從集X到集Y的多值映射是一個對應規律F,按照這個規律,對於X的每個元素x,都能相應地得到Y的一個非空子集F(x),稱為x對於F的像。對於任何![多值映射](/img/e/b9e/ml2ZuM3XxQDN3EzNwMzNwgDM5ETMwADMwADMwADMwADMxAzL3AzLxQzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
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![多值映射](/img/e/b9e/ml2ZuM3XxQDN3EzNwMzNwgDM5ETMwADMwADMwADMwADMxAzL3AzLxQzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![多值映射](/img/0/b6d/ml2ZuM3X0YjN4EzNwMzNwgDM5ETMwADMwADMwADMwADMxAzL3AzL0YzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
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![多值映射](/img/4/02a/ml2ZuM3X3IjMyMTO5ITNxgDM5ETMwADMwADMwADMwADMxAzL1EzL3IzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
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多值映射的一般理論自然是單值映射相應理論的推廣,但前者顯然不如後者那么豐富多彩。多值映射理論的重要性在於它對其他數學分支的套用,特別值得一提的,是多值映射的不動點理論對博弈論的完美套用。x∈X稱為F:X→2X的不動點,如果x∈F(x)。角谷靜夫於1941年首先把關於單值映射的布勞威爾不動點定理推廣到多值映射,下面是一個等價形式:
角谷不動點定理 假設K嶅Rn是非空有界閉凸集,F:K→2K是上半連續多值映射,使得對每個p∈K,F(p)都是K的非空閉凸集,於是F有不動點。
命
![多值映射](/img/5/bdc/ml2ZuM3X1YzNwQTO5ITNxgDM5ETMwADMwADMwADMwADMxAzL1EzL1YzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![多值映射](/img/d/76a/ml2ZuM3X0MTOyQTO5ITNxgDM5ETMwADMwADMwADMwADMxAzL1EzL0MzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![多值映射](/img/3/2aa/ml2ZuM3XxkTOzEzNwMzNwgDM5ETMwADMwADMwADMwADMxAzL3AzLxkzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![多值映射](/img/3/2aa/ml2ZuM3XxkTOzEzNwMzNwgDM5ETMwADMwADMwADMwADMxAzL3AzLxkzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![多值映射](/img/3/2aa/ml2ZuM3XxkTOzEzNwMzNwgDM5ETMwADMwADMwADMwADMxAzL3AzLxkzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![多值映射](/img/3/2aa/ml2ZuM3XxkTOzEzNwMzNwgDM5ETMwADMwADMwADMwADMxAzL3AzLxkzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
參考書目
E.Michael,Topologies on Spaces of Subsets,Tran. Amer.Math. Soc., Vol.71, pp.152~182,1951.
E.Michael, A Survey of Continuous Selections,Lecture Notes in Math.,Vol.171, Springer-Verlag, Berlin, 1970.
C.Berge,Topological Spaces, Oliver and Boyd, Edinbergh and London, 1963.
C. Berge,Théorie Générale des Jeux ╜ n Personnes,Gauthier-Villars, Paris, 1957.