Kakutani Fixed-point Theorem - Statement

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 a closed graph and the property that φ(x) is non-empty and convex for all x ∈ S. Then φ has a fixed point.

When we say that the graph of is closed, we mean that for all sequences and such that, and for all, we have .

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