Kakutani fixed-point theorem
theorem that a function f: S→Pow(S) on a compact nonempty convex subset S⊂ℝⁿ, whose graph is closed and whose image f(x) is nonempty and convex for all x∈S, has a fixed point
In mathematical analysis, the Kakutani fixed-point theorem is a fixed-point theorem for set-valued functions. It provides sufficient conditions for a set-valued function defined on a convex, compact subset of a Euclidean space to have a fixed point, i.e. a point which is mapped to a set containing it. The Kakutani fixed point theorem is a generalization of the Brouwer fixed point theorem. The Brouwer fixed point theorem is a fundamental result in topology which proves the existence of fixed points for continuous functions defined on compact, convex subsets of Euclidean spaces. Kakutani's theorem extends this to set-valued functions.
The theorem was developed by Shizuo Kakutani in 1941, and was used by John Nash in his description of Nash equilibria. It has subsequently found widespread application in game theory and economics.
Statement
Kakutani's theorem states:
Let S be a non-empty, compact and convex subset of some Euclidean space Rn.
Let φ: S → 2S be a set-valued function on S with the following properties:
φ has a closed graph;
φ(x) is non-empty and convex for all x ∈ S.
Then φ has a fixed point.
The public source identifies “Kakutani fixed-point theorem” as theorem that a function f: S→Pow(S) on a compact nonempty convex subset S⊂ℝⁿ, whose graph is closed and whose image f(x) is nonempty and convex for all x∈S, has a fixed point. This brief keeps that definition visible, then builds a research path around Kakutani, fixed-point and theorem.
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