在Nash的论文“Equilibrium Points in N-person Games”曾经提到了Kakutani Fixed Point theorem（角谷不动点定理），并将这个作为他理论证明的重要基础。而在他的另一篇非常重要的论文“Non-cooperative Games”里面提到了另一个定理Brouwer fixed point theorem。以下是从Wiki上找到的定义，中文Wiki没有-_-#。
角谷不动点定理：
Let S be a non-empty, compact convex subset of Euclidean space and let be an upper semicontinuous correspondence (that is, a function which maps points in S to subsets of S). Assume that for all the set is convex, closed and non-empty. Then j has at least one fixed point, i.e., there is an such that

Brouwer不动点定理：

The theorem states that every continuous function from the closed unit ball D ⁿ to itself has at least one fixed point. In this theorem, n is any positive integer, and the closed unit ball is the set of all points in Euclidean n-space Rⁿ which are at distance at most 1 from the origin. A fixed point of a function f : D ⁿ → D ⁿ is a point x in D ⁿ with f(x) = x.

The function f in this theorem is not required to be bijective or even surjective.

Because the properties involved (continuity, being a fixed point) are invariant under homeomorphisms, the theorem equally applies if the domain is not the closed unit ball itself but some set homeomorphic to it (and therefore also closed, bounded, connected, without holes, etc.).

The statement of the theorem is false if formulated for the open unit disk, the set of points with distance less than 1 from the orign. Consider for example

which maps every point of the open unit disk in R² to another point of the open unit disk slightly to the right of the given one.

关于不动点的定理很多。不动点是这样定义的，如果对于一个函数F，存在一点满足x＝F(X)，那么这个点就是F的不动点。简单的说，就是 y=F(x) 跟 y=x 的交点。在不动点定理里面都提到了Euclidean Space，即欧几里德空间，这是2维、3维欧式空间向n维的拓展。

在Brouwer不动点定理里面提到的Unit Ball，即欧式空间的单位圆，其满足：在二维空间，我们可以把它画作以原点为圆心1为半径的圆的内部区域。而这个定理要说的就是，从单位圆到其本身的连续映射（continuous function）都存在不动点。

而在角谷不动点定理中，他表述的是对于一个非空的紧凸集（compact convex subset）到自身的上半连续（upper semicontinuous）集值映射都存在不动点。 

这两者的不同在于Brouwer指的是单值映射，而Kakutani指的是集值映射。后者与Nash论文中所提到的N元策略组合（N-tuple of strategies），即N人博弈中每个人纯策略（pure strategy）的组合正好相符。